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Brian Dorney | USLHC | USA

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Higgs Dependence Day: The Nobel Perspective

Friday, July 20th, 2012

I recently traveled to Lindau, Germany for the 62nd meeting of Nobel Laureates (http://www.lindau-nobel.org/), an annual meeting of Nobel Laureates and young researchers from around the world. This year’s meeting, by sheer coincidence, was dedicated to Physics (ironic right?).

One of the afternoon sessions for Wednesday July 4th was a panel discussion titled simply as “CERN.”  Which was, by sheer coincidence, so well timed.  After all, CERN had just finished giving their scientific and public press-releases regarding the discovery of a new boson, with mass of 125.3+/-0.6 GeV, earlier that morning.  I had the opportunity to sit in the front row of a room filled with approximately 250 other young researchers, listening to top names in astro- & particle-physics discuss the recent CERN discovery.  What follows below is a brief review of the Laureates’ discussion.


CERN Panel Discussion on Wednesday, July 4th, 2012 (Higgs Dependence Day) at the 62nd Lindau Nobel Laureate Meeting in Lindau, Germany. The Nobel Laureates shown here are, from left to right, David Gross, Martinus Veltman, Carlo Rubbia, and George Smoot.

The session featured Nobel Laureates David Gross, Martinus Veltman, Carlo Rubbia, and George Smoot.  It was chaired by Prof. Dr. Felicitas Pauss, of CERN.  Additionally, we were also joined by several CERN Scientists (John Ellis counted among their number!).  The air in the room was tense with excitement; and rightly so considering roughly 100 of the young researchers in the room, myself included, participated in high energy physics research in one way or other.  And all of us glowed with sheer joy.

However, it was the Nobel Laureates who out shined us all, for they had been waiting for a discovery like this for the majority of their lives!  David Gross remarked “[This was a] great day for me, for physics, for all humanity!”  David Gross went on to proclaim that this discovery was a “Triumph for CERN…a triumph of theory!”  Martinus Veltman followed by saying that this “closes the last gap amongst the Standard Model.”

I don’t think there was any doubt in either Gross’ or Veltman’s mind that a particle like the Higgs Boson existed.  However, George Smoot originally had his doubts, “I was critical of the theorists not looking for other solutions,” to which David Gross jokingly forgave Smoot on center stage.  Smoot followed up by “commending CERN for being cautious.”  He was referring to the fact that both ATLAS and CMS Collaborations have simply stated that we have found a Boson, and that this particle has similar properties to the predicted Standard Model Higgs Boson, but we have not claimed to have found the Higgs Boson.  Smoot cautioned us all to be careful, not to rush to judgement, and to continue our studies and cross-checks.  Very sound advice in my opinion!

Carlo Rubbia chimed in at this point to say that the value of the experimental cross-section (or the rate of how often this Boson was produced) is almost a factor of 2 larger then the theoretical predictions (measurement from the CMS Collaboration shown in the plot below).

Ratio of the measured production rate of our new boson to the theoretical predictions. Notice that for the case where this new boson decays to a pair of photons, the measured rate is almost two times the predicted rate (with errors).

Rubbia commented that this was a “very important new element that warrants consideration,” and asked the CERN scientists who had joined us “what about this factor 2?”

The ATLAS Representative responded by saying that for the H-> gamma gamma channel the ratio measured by ATLAS was 1.9 +/- 0.5, within two sigma of the theoretical prediction, however the overall ratio was 1.3 +/- 1.2, consistent with the Standard Model.  The CMS Representative responded by saying that this slight excess we observed is compatible with the Standard Model, and that the CMS Collaboration measures this to be 1.5 +/- 0.4 for the H->Gamma Gamma channel, one sigma above the theoretical prediction.

Gross asked John Ellis what he concludes about the possibility for beyond the standard model physics in light of this Boson’s discovery.  Ellis replied by stating that this depends very much on the mass of the Higgs Boson, at 127 GeV the vacuum becomes unstable; and that additional physics is needed to prevent the universe from collapsing.

I found this idea very interesting since the current mass measurements of this new Boson by CMS and ATLAS places its mass between 125 and 126 GeV.  However, these results are preliminary, with more data we will be able to narrow down the mass measurement (it might even shift!!).  If this Boson we discovered  truly is the Higgs Boson, and if a precision mass measurement reveals it’s mass to be above 127 GeV, then we definitely need some new physics to keep the universe in its present state, just as Ellis said!

Martinus Veltman was very curious how CMS and ATLAS were able to make this discovery so quickly.  After all, data collection started in 2010 and this month we announced to the world that we had discovered a new Boson.  CMS & ATLAS responded by saying increasing the center of mass energy of the LHC beams from 7 to 8 TeV was predicted to give a 30% increase in the rate of Higgs Boson production.  Additionally, CMS & ATLAS researchers were able to reduce experimental backgrounds by 15% from 2011 to 2012.  On top of these two facts the number of collisions per second taking place in the CMS and ATLAS Detectors was increased dramatically at the end of 2011 and at the start of 2012.  All of these were factors contributing to the rapid discovery of this new Boson.

At this point Carlo Rubbia brought up the topic of what’s after the LHC.  Rubbia’s idea was to build a muon-muon collider, with a center of mass energy slightly higher then this new Boson’s mass.  Rubbia referred to this as a “Higgs Factory,”  since he believes that such a machine would be able to produce these new Bosons with a much lower background then what occurs at the LHC, allowing for precision measurements of this Boson’s properties.  Gross immediately chimed in with “[a] muon Higgs factory would be fantastic,  ideal project for the US to get back into the game if anyone from FermiLab is listening!”  which caused several moments of laughter in the lecture hall.

However, Rubbia’s comment was a very serious one and good one in my opinion.  Physics needs something after the LHC, many questions are and will still be unanswered.  George Smoot was of a similar opinion, stating that “completing the Standard Model is a great triumph, but everyone wants to see us go beyond.”  On this note John Ellis stated that he would also like to see a Higgs factory on the agenda in the future.  However, Ellis was uncomfortable with the idea of making a machine that would be limited to just producing this new Boson.

I think the final comments from the panel discussion summed up the feeling of everyone in that room, and those of all high energy physicists.  David Gross closed by saying “Congratulations to all of you at CERN, are you having a big party tonight!?”


Until Next Time,



New State Discovered by the ATLAS Collaboration!

Tuesday, January 10th, 2012

Over the Christmas Holiday the ATLAS Collaboration submitted an article to Physical Review Letters, a peer-review journal.  The article titled, “Observation of a New χb State in Radiative Transitions to Υ(1S) and Υ(2S) at ATLAS,” can be found on arXiv.

The processes under study in this paper are the following:

χb(nP)→ Υ(1S) γ → μ+μ- γ

χb(nP)→ Υ(2S) γ → μ+μ- γ

Where n = 1,2 or 3.

The focus of this paper was on finding a meson known as the χb(nP). Mesons are a class of particles formed by a bound state of a quark and an anti-quark; the χb(nP) happens to be a bound state of a b-quark (termed b, for beauty) and an anti-b-quark (termed b).  The (nP) part means that quark/anti-quark are bound together in a P-orbital of energy level n. As a consequence of the relativistic energy-momentum relation, different energy levels correspond to bound states with different rest masses.  So basically for each value of n you have a unique particle! The n = 3 particle has only ever been theoretically predicted, so in this paper a new particle was discovered!

Now the χb(nP) particles are very short lived and usually can’t be observed directly.  So to find them the ATLAS Collaboration has to infer their presence by summing up the energy of their decay products.  In the above two equations, the χb(nP) is decaying into another meson known as the Upsilon, Υ(kS), and in the process a photon is also emitted (hence the “radiative transition” in the title). Now the Upsilon is also made up by a bb pair.  The (kS) part means that the quark/anti-quark pair are bound together in an S-Orbital of energy level k = 1 or 2.

The Υ(kS) is also a very short lived particle (mean lifetime of approximately 10-20 seconds).  To identify the Upsilons needed for this study the ATLAS Collaboration had to look for two oppositely charged muons, called a di-muon or μ+μ- pair, having a summed rest mass (termed “invariant mass”) near the published mass values for the Υ(kS).  A plot of the di-muon invariant mass can be see at right [1].  From left to right the peaks in the graph represents di-muons originating from decays of the Υ(1S),  Υ(2S), and the Υ(3S), respectively. The muons in the shaded regions from the Υ(1S) and Υ(2S) decays were used in the search for the χb(nP) particles.

Then to find the χb(nP) particles, ATLAS researchers looked for a point in the detector from which a di-muon and a photon originated from.  This point is known as a vertex.

Charged particles, such as muons, leave tracks in the ATLAS Detector’s inner tracking detector (which consists of a silicon pixel detector, a silicon microstrip detector, and a transition radiation tracker).  The inner tracking detector is like a giant CCD camera, and is based on the same technology.

However, neutral particles, like photons, do not leave a track in the tracker.  Photons are detected by energy depositions in the ATLAS Detector’s electro-magnetic calorimeter.  To see if an energy deposition marked as a photon comes from this di-muon vertex, you take every di-muon vertex, and you try and match it with one of your photon energy depositions.  If the match is “good enough” you call this di-muon plus photon a χb(nP) candidate.

Before we show you these χb(nP) candidates I want to talk about the di-muon invariant mass plot one more time.  Notice how the peaks in this plot have some width to them.  This has to do with the resolution of the ATLAS Detector.  The narrower the peaks are the better the resolution.  However, there is a limit to how thin these peaks can be.  For example, the Υ(1S) has its own natural width of  about 54 keV or 0.000054 GeV.  So suppose you had the perfect particle detector and made the measurement shown in the di-muon invariant mass plot.  Even using your perfect detector your Υ(1S) peak would still have a width of exactly 0.000054 GeV.  As you can see the peaks are no where near this, and as I said this is due to the finite resolution of the ATLAS Detector.  To account for this resolution, researchers at ATLAS worked with a variable defined as:

Δm = m(μ+μ-γ) - m(μ+μ-)

This takes the invariant mass (e.g. rest mass) of the di-muon and the photon, the χb(nP) candidates, and subtracts the di-muon mass.  Then the ATLAS researchers add the world average values of the Upsilon masses back to Δm.

m k = Δm + mΥ(kS) = m(μ+μ-γ) - m(μ+μ-) + mΥ(kS)

Note, for your perfect detector measuring the Υ(1S) this quantity: m(μ+μ-) – mΥ(1S) is approximately zero, but has a maximum value of 0.000027 GeV, e.g. this would be half the width of the Υ(1S)!  This is how the use of Δm and the world average value of the Upsilon minimizes the affect of the ATLAS detector’s resolution.

A little side note about world average values in particle physics.  They are a single value for some experimental observation, produced by the Particle Data Group [2], and take into account every experimental result that has ever been published.

A plot of mk is shown at left [1] for the χb(nP) candidates in which the photon was measured directly (as opposed to an indirect measurement from the photon splitting into an e+e- pair).  The first two peaks are the previously observed χb(1P) and χb(2P) particles.  The third peak is the first ever observation of the χb(3P)!

In case you all remember the golden rule of particle physics, the ATLAS Collaboration reports that:

“the significance of the χb(3P) signal is found to be in excess of six standard deviations in each of the unconverted and converted photon selections independently” [1]

Or put plainly, the probability that this third peak could have happened by coincidence is about 2 in one billion.  You literally have a higher probability of winning the lottery at 1 in approximately 16 million [3] or being struck by lightning this year at 1 in 775,000 [4].


So how about that? Not everyday a new particle is discovered!



Until Next Time,




[1] The ATLAS Collaboration, “Observation of a New State in Radiative Transitions to and at ATLAS,” arXiv:1112.5154v1 [hep-ex].

[2] K. Nakamura et al. (Particle Data Group), J. Phys. G 37, 075021 (2010). Note this may be found online here: http://pdg.lbl.gov/2011/tables/contents_tables.html

[3] wikiHow, “How to Calculate Lotto Odds,” http://www.wikihow.com/Calculate-Lotto-Odds, Jan 10th 2012.

[4] NOAA, “Medical Aspects of Lightning,” http://www.lightningsafety.noaa.gov/medical.htm, Jan 10th 2012.


An Update from OPERA!

Friday, November 18th, 2011

CERN provided an update regarding the muon-neutrino time of flight measurement performed by the OPERA Collaboration this morning, which I shall now quote:

“Following the OPERA collaboration’s presentation at CERN on 23 September, inviting scrutiny of their neutrino time-of-flight measurement from the broader particle physics community, the collaboration has rechecked many aspects of its analysis and taken into account valuable suggestions from a wide range of sources. One key test was to repeat the measurement with very short beam pulses from CERN. This allowed the extraction time of the protons, that ultimately lead to the neutrino beam, to be measured more precisely.

The beam sent from CERN consisted of pulses three nanoseconds long separated by up to 524 nanoseconds. Some 20 clean neutrino events were measured at the Gran Sasso Laboratory, and precisely associated with the pulse leaving CERN. This test confirms the accuracy of OPERA’s timing measurement, ruling out one potential source of systematic error. The new measurements do not change the initial conclusion. Nevertheless, the observed anomaly in the neutrinos’ time of flight from CERN to Gran Sasso still needs further scrutiny and independent measurement before it can be refuted or confirmed.

On 17 November, the collaboration submitted a paper on this measurement to the peer reviewed Journal of High Energy Physics (JHEP). This paper is also available on the ArXiv preprint server.” [1]

Comparison with the Original OPERA Measurement

If you’ll recall, the original OPERA experiment measured the time of flight, or how long it takes to go from point A to B, for a muon-neutrino beam traveling from CERN to Gran Sasso (this distance is ~730km).  The muon-neutrino beam under study was created using a proton beam taken from CERN’s Super Proton Synchrotron (SPS).  In this original measurement, the proton beam taken from the SPS was 10.5 microseconds long in time.  The OPERA Collaboration originally reported that the muon-neutrinos traveled 60.7 ± 10.1 nanoseconds faster than the speed of light (FTL)!

But one obvious criticism of the original OPERA Measurement was that they were unable to determine exactly which proton gave rise to a muon-neutrino that struck the OPERA detector.  This is a problem since the proton beam time was several orders of magnitude larger than the originally quoted FTL observation.  e.g. If a muon neutrino strikes the OPERA Detector and was thought to come from the start of the proton beam, but actually came from the end of the proton beam there would be no observed FTL behavior.

Now in the OPERA Collaboration’s newest measurement, they are still unable to determine which proton created a muon-neutrino that struck the OPERA detector. However, the quote above shows that the OPERA Collaboration repeated their measurement using a proton beam that is only three nanoseconds long and this change in proton beam length has not affected their results! So the fact that you don’t know which proton creates the muon neutrino that strikes the your detector no longer matters! This is because the reported excess in the time of flight of the neutrinos is twenty times the proton beam’s time in this new measurement; as opposed to being three orders of magnitude smaller in the original measurement.


The Future

Whether or not the OPERA publication will be accepted by JHEP, a peer-review journal, remains to be seen.  The OPERA measurement also needs to be confirmed by another experiment, to ensure the phenomenon is actually real.  But, rumor has it that the MINOS and T2K Experiments are gearing up to repeat the OPERA Measurement.  So stay tuned on this rapidly developing phenomenon!

For those of you interested the updated OPERA manuscript can be found on arXiv.org.


Until Next Time,




[1] CERN, “OPERA experiment reports anomaly in flight time of neutrinos from CERN to Gran Sasso,” http://press.web.cern.ch/press/pressreleases/releases2011/pr19.11e.html, November 18th, 2011.


Research Experiences for Undergraduates

Monday, November 14th, 2011

So this post goes out to all undergraduate students interested in pursuing research opportunities or graduate studies in the fields of astro-particle physics, particle physics and/or high energy physics.  The reason for this is that it’s the time of the year to start thinking about what you’re doing this summer; because various Research Experiences for Undergraduates (aka REU’s) are now accepting applications.

For those of you who may not know, REU’s are summer programs held by national laboratories and universities across the country (sometimes even around the world!).  Selected students have the opportunity to go to these institutions for 9-15 weeks (depending on the program) and work with expert researchers in the field.  They allow an undergraduate to get first hand experience of what it’s like to be a scientist, and develop valuable networking opportunities.  Most REU’s also provide room & board for accepted students, and some occasionally provide a stipend as well.

If you are considering pursuing graduate study, a career in scientific research, or just want to know what it’s like to be a physicist then I would definitely suggest applying to an REU program that interests you; and in that spirit I took the time to compile a list of REU’s in High Energy Physics (and related fields) that are now accepting applications for the summer of 2012!


The University of Michigan REU at CERN

Why save the best for last?  Because this, as they say, is the real deal.

Through an agreement between University of Michigan and CERN, undergraduate juniors and seniors studying physics (and related disciplines) have the opportunity to spend the summer in CERN!  This is where the all the action in particle physics is going to be this summer; because next year the LHC plans to collide our proton beams at either 7 or maybe even 8 TeV!  The ATLAS and CMS experiments each plan on recording ~7-13 inverse femtobarns of data (or 700-1300 trillion proton-proton collisions)!

If a major discovery is made, would you rather hear about it on your Quantum Diaries RSS feed (all the cool kids have one) or be part of the discovery at CERN!?

You must be a US Citizen or permanent resident to be eligible, for more details see:


The deadline is December 20th!

And just in case you think this program is impossible to get into, I’m good friends with one undergraduate student who was accepted to the program in the summer of 2010!  So apply, you have nothing to lose and so much to gain!


Summer Internships at Fermilab

Fermi National Accelerator Laboratory (Fermilab) has three summer programs for high school students and teachers, and five summer programs for undergraduate students!  These programs allow selected applicants to work on projects ranging from QuarkNet to LHC experiments.

Aside from the physics, Fermilab is a beautiful place in Batavia, IL; also very close to Chicago.  I first visited Fermilab for the First Beam Pajama Party in 2008, and it was there that I decided to pursue a life in high energy physics.  Needless to say, Fermilab, is a truly awe inspiring place, and I definitely encourage you to apply.  I mean they even have live buffalo on site!

Information on the programs I mentioned above can be found here:


With deadlines ranging from January to April for each:


I know another student who was accepted to this program also, their summer work at Fermilab has turned into several poster presentations, a peer-review publication, and a senior research project (for which they received actual college credits for!).  So definitely do apply, it can be so beneficial for your future career (wherever that may take you)!


Summer Student Programs at SLAC

The SLAC National Accelerator Laboratory (known as SLAC for short) has two summer programs for undergraduate students.  The first of which is the SLAC Summer Student Program.  The second of which is the SLAC Youth Opportunity Program.

The SLAC Summer Student Program is slightly different from the other REU’s I’ve listed.  They accept applications from high schoolers, undergraduates and graduate students.  But you must apply to a specific summer student job opportunity (rather than applying to the program and then be assigned to a research group once  you arrive).  However, this gives you the advantage of knowing what you’re doing before you arrive.  Not to mention it allows you some degree of choice on what you become involved in if accepted.

The SLAC Youth Opportunity Program targets students of low income families.  Positions within this program are full-time paid positions designed to give real world work experience to students ages 18-22.

More information on these two programs, and were to apply can be found below:


While I’ve never been to SLAC, I do know it is located in sunny California and operated by Stanford University.  So if you ever had dreams for attending Stanford, being a summer student at SLAC would give you some really great networking opportunities you might be able to exploit on your application to Stanford’s Graduate Program.


Science Summer Undergraduate Internship at BNL

Brookhaven National Laboratory (BNL) in New York also has a fantastic summer program for undergraduate students.  However this program, unlike most of the above programs, allows students to become involved in numerous research fields.  The REU available at BNL for selected students are chemistry, high energy physics, biology, engineering, and related disciplines.

This program is for all undergraduate students, and is now accepting applications until January.  More details on this opportunity, and how to apply, can be found here:



Search for an REU Site

In case none of the above opportunities appeal to you, the National Science Foundation (NSF) has this great resource for finding REU programs by field.


So spend an afternoon this week or weekend and find an REU program that would best fit your interests!

I really encourage any undergraduate students reading this to apply to one, if not all, of the above opportunities.  They are truly fantastic opportunities, they give you an idea of what a scientist actually does on a day to day basis; and might help you pick out a future career path.  The networking connections you can gain from these opportunities can really help you out later in life (i.e. recommendation letters).  I cannot put into words just how fantastic these opportunities are.

But hopefully you’ll have found this post helpful.  Good luck to all, I hope you are accepted!


Until next time,



The OPERA Measurement & the Peer-Review Process

Tuesday, November 1st, 2011

An article posted on engadget.com [1] regarding the OPERA Neutrino measurement has been making its way through the internet and various social network sites recently.  I’ve had several discussions on Facebook regarding this article, the most recent was on the Society of Physics Students Facebook Page.  And it looks like I’m starting yet another discussion regarding this article.

The article references a recent paper published on arXiv, which can be found here [2].  This arXiv reference states that the OPERA Collaboration have not properly accounted for relativistic effects [2] when calculating the time of flight of the neutrino’s in their recent pre-print (also published on arXiv, [3]).  But, let me be very explicit in this, the OPERA Collaboration has correctly taken into account relativistic effects in their calculation [3].

So why discuss this engadget.com article and the source it references?  Well, my goal with this post is to illustrate reasons why one should always remain skeptical regarding scientific news, and the importance of doing your own search for sources when a new result is presented.

The Argument

But let’s get back to talking about this engadget article.  I was immediately skeptical of this article, the source engadget.com referenced was a paper published solely on arXiv (which has NOT been submitted to a peer-review journal).

Now arXiv is a fantastic resource, which is used by scientists all over the world, and publishes world class research.  However, arXiv is not a peer-review journal, and anyone is able to submit to it.  As a result, arXiv is a resource that needs to be used with care, much like Wikipedia; arXiv’s own disclaimer reads:

Papers will be entered in the listings in order of receipt on an impartial basis and appearance of a paper is not intended in any way to convey tacit approval of its assumptions, methods, or conclusions by any agent (electronic, mechanical, or other). We reserve the right to reject any inappropriate submissions [4].

Meaning anyone is free to submit to arXiv, however if a document is published on arXiv it does not mean that the material covered in the document has been accepted by the scientific community.  Most documents that hit arXiv are also published in peer-review journals.  Which is what makes arXiv great, it provides free access to scientific publications, but not all papers on arXiv have been “published” in a peer-review journal.

But what does it mean to be published in a peer-review journal?  Well, it means that your research has been accepted, or at least acknowledge, by the scientific community.  This gives your research merit since other scientists reviewed your work (i.e. the peer-reviewers).  The peer-review process is often intense, and many publications submitted  to a journal are rejected, or asked to provide additional material to support their claims before they are published.  However, after your results have been published other scientists are still free to dispute your results/conclusions; but they must do so by submitting  their own work to the same (or other) peer-review journals.

Now Ref. [2] uses what I would call overly simplified model of a satellite in orbit.  The author makes use of only special relativity, and assumes the satellite moves in a straight line parallel to the earth’s surface.  Using this model of satellite motion the author claims that the OPERA Collaboration has miscalculated the time of flight for the neutrinos by 64 nanoseconds [2].  This discrepancy, if it was true, would mean that the neutrino’s did not travel faster than light.

Example of a typical elliptical orbit

However, if you know anything about orbital mechanics, you should already be raising your eyebrow.  When an object with very little mass, like a satellite, orbits another object with a lot of mass, like the earth, the small object makes an elliptical path.  The ellipse created by the small object’s orbit has one of its foci centered on the large object; take a look at the figure I’ve added for a graphical interpretation.  Now other types of orbits exist, but this orbit is one of the simplest types that is easiest to achieve.  Suffice to say in an elliptical orbit, a satellite is not moving parallel to the ground.

Also, if you want to talk about relativistic motion for objects in orbit, you need to worry about general relativity, and not just special relativity.  The reason for this is you need to be concerned with gravitational fields that cause “reference” frames to undergo uniform acceleration.

But in case you were wondering the OPERA Collaboration has correctly accounted for general relativity in their calculations [3].

So the assumptions made in Ref. [2] regarding satellite motion are not a very accurate representation of reality.

Additionally, Some criticism of the analysis used Ref. [2] was published by Dr. Ted Bunn, an associate professor of physics at Richmond University.  Dr. Bunn points out an error in the analysis performed in Ref. [2].   With Dr. Bunn’s correction, the OPERA Collaboration’s time of flight measurement is off by zero nanoseconds, and not 64 nanosecond [5]. e.g. the OPERA Collaboration did not make a mistake.

I should point out that Dr. Bunn’s paper has also yet to be published in a peer-review journal, however I cite him as a credible source due to his position within Richmond University.  Now some you may disagree with my decision to use Dr. Bunn’s paper as a credible source since it wasn’t published in a peer-review journal…

…and that is exactly what I would like you to do!  Be skeptical of what you read online and look for credible information on your own! The OPERA Collaboration has yet to even submit their measurement to a peer-review journal, but they plan to do so in the future.

Some Scientific Concerns

However, everything above being said, their are plenty of scientific concerns regarding the OPERA measurement.  Additionally, the measurement needs to be repeated by another neutrino beamline before it will be accepted by the scientific community.

But one of the major concerns of the community surrounding the OPERA measurement is that the proton beam used to make the muon-neutrino beam was 10.5 micro-seconds long [3].  Meaning, if you started a stop-watch the moment the first proton reached you, and stopped it the moment the last proton went past, this time interval would be 10.5 micro-seconds (hope you have good reflexes!!!).

Now why is this fact important?  The OPERA Collaboration at this time is unable to determine from which proton a muon-neutrino which struck the OPERA detector originated from [3].  To counteract this the OPERA Collaboration employs a widely accepted statistical technique, outlined in Ref [3], to determine the time of flight of the neutrino’s.  But as a result of the length of the proton beam, the reported excess of 60 nanoseconds in favor the neutrino’s over the photons has been met with skepticism.

If you are interested in additional concerns of the peer-review community regarding the OPERA measurement I can point you to Aidan’s live coverage of the CERN Press-Release on the OPERA measurement:



The Future of FTL Neutrino’s

You may have read on Jonathan’s blog that CERN plans to decrease the proton waveform from 10.5 microseconds to 1-2 nanosecond pulses separated by 500 nanosecond delays.  If the neutrinos were then found to travel faster than light by 60 nanoseconds it really would be something!

But in the spirit of this post, Jonathan and I are referencing a recent article published on the BBC’s science page, which can be found here:


This BBC article discusses the plans of the OPERA Collaboration to reduce the proton beam time as I mentioned above.  Additionally, this BBC article states that two additional neutrino beamlines plan on repeating the OPERA measurement.  These two beamlines are  these are Japan’s T2K experiment, and FermiLab’s MINOS experiment.

Once the OPERA Collaboration conducts their experiment using a shorter timed proton beam they plan to publish their results in a peer-review journal.  I should point out that OPERA will publish regardless of  their findings.  So even if they find that the neutrino’s don’t travel faster than light they will still publish; as they should!

With these new plans of the OPERA, T2K and the MINOS Collaborations we will really be able to say if the preliminary measurement published by OPERA (on arXiv) was a statistical fluke, or a real phenomenon.

Until then I will remain skeptical, but it will be a very interesting day if we find out that the neutrino’s really did go faster than light….


Before closing I would like to take the time to acknowledge and congratulate all scientific journalists.  These individuals do a truly fantastic job of bringing attention to exciting new research.  I myself rely on many scientific journalists (and my colleagues here at Quantum Diaries!)  to keep me informed about what happens in the scientific community; and for this I must extend my sincere gratitude to all of them.


Until Next Time,




[1]  Sharif Sakr, “Remember those faster-than-light neutrinos? Great, now forget ‘em,” Engadget.com, Oct 17th 2011, http://www.engadget.com/2011/10/17/remember-those-faster-than-light-neutrinos-great-now-forget-e/.

[2] Ronald A. J. van Elburg, “Time-of-flight between a Source and a Detector observed from a Satellite,” arXiv:1110.2685v3 [physics.gen-ph], http://arxiv.org/abs/1110.2685.

[3] OPERA Collaboration, “Measurement of the neutrino velocity with the OPERA detector in the CNGS beam,” arXiv:1109:4897v1 [hep-ex], http://arxiv.org/abs/1109.4897.

[4] arXiv, “General Information About arXiv,” http://arxiv.org/help/general.

[5] Ted Bunn, “Critique of ‘Time of flight between a sourceand a detector observed from a satellite’ (arxiv:1110.2685v3),” https://facultystaff.richmond.edu/~ebunn/vanelburg.pdf.


Angular Momentum in Quantum Mechanics: Spin Indepth

Monday, September 26th, 2011

Last week we began a journey through quantum wonderland with our discussion on Angular Momentum in Quantum Mechanics. We learned that for quantum angular momentum you can only ever know the total and one of its components (i.e. x, y or z) at any time t. We learned that this was strange result was due to the what the Generalized Uncertainty Principle has to say about the observables for operators that do not commute with one and other. Additionally we saw that angular momentum in quantum mechanics was a discrete variable that could only take certain quantized values, unlike its continuous counterpart in Classical Mechanics (CM).

For this week, as promised, we shall follow Alice’s footsteps deeper into wonderland and try to catch a glimpse of the probabilistic nature of Quantum Mechanics (QM). And for this journey we will further explore the nature of Spin Angular Momentum in QM. But before we begin, let’s arm ourselves with the notion of what physicists like to call an ensemble of identically prepared systems.


An Ensemble Romance

Let’s imagine we have some brave young female physicist, who happens to be single, let’s call her Juliet (we always need more women in science anyway, even fictitious Shakespearian women). Now Juliet has some dark-haired, “handsome,” physicist come to call upon her, his name is Romeo.

Juliet, being a scientist, wants to see if she and Romeo will make a good long-term couple. However, Juliet is rather impatient and doesn’t want to spend the months/years that it would take to learn this knowledge (she doesn’t have long to live after all, only three Acts!). She hatches a plan to assess whether or not the two of them will be a good couple. She’s discovered how to make a perfect clone of a person (not just genetically, she can also clone their consciousness, personality, memories, etc…).

So she asks our dear Romeo for a lock of his hair, a swab of the inside of his cheek, and an MRI of his brain. Romeo finding this all rather odd, but eager to please Juliet, agrees to all of the above. Juliet then takes these back to her laboratory, deep underground, and makes a countless number of identical Romeo Clones.

She places each Romeo Clone in an identically prepared, but separate room. In each room she walks in and performs a single action and records the Romeo Clone’s response. The actions she performs are, what she would consider, half the time pleasant and half the time unpleasant (see examples below). During this process Juliet ensures that each Romeo Clone has no knowledge of the other clones, rooms, or actions. All the Clones are blank slates with respect to Juliet’s actions (though all the clones, like the original Romeo, are romantically interested in Juliet at the start). When Juliet repeats this process on enough Romeo Clones she will learn if she and the original Romeo are compatible.

After her experiment, she decides to not to date him; thinking he will probably be the death of her anyway.


The Statistical Interpretation

While this story in the preceding section is absurd in numerous ways, it highlights several facts key ideas.

As I’ve said, Quantum Mechanics is a probabilistic theory. Physicists work within this theory much in the same way Juliet does for her love life. We prepare an ensemble of identically prepared systems (i.e. each identical Romeo Clone in an identical, but separate, room). With each system we make a single measurement (i.e. Juliet’s single action toward each Romeo Clone). And then from the results of the experiment on each single system we build a distribution which has an expectation value.

The expectation value is the average of all the independent measurements performed on each independent identically prepared system (i.e. Juliet’s decision not to date Romeo after she finished her experiment). You should not confuse the expectation value with the most probable value. For almost all but some very special cases, they are two different numbers.

Additionally, in Quantum Mechanics you could never say exactly what the outcome of a single experiment will be (just like Juliet did not know if she was compatible with a single Romeo Clone). However, as I outlined above, Quantum Mechanics is able to say what the average outcome for a series of measurements on a series of identically prepared systems will be.

This idea has no analog in Classical Mechanics (for those of you who know what a partition sum function is, you know more than what’s good for you; let’s just leave Statistical Mechanics out of this discussion [1]).

But what in Feynman’s name does all this have to do with Spin Angular Momentum!? Stay with me and you shall find out, I’ll bring this all together at the very end.


Spin Angular Momentum Revisited

Last week I mentioned that spin angular momentum exists in the abstract world of linear algebra (specifically something known as a 2×2 Hilbert Space).  Let’s learn a little more about that here.  We know from last week that the total spin angular momentum for a particle can have the value:

For particles known as fermions, s is a half-integer, with the lowest possible value being ½. We also know from last week that the component of the spin angular momentum along a given direction (let’s say, the z-direction) can be written as:

It should not shock you to learn that there is a relation between a particle’s spin s, and the component of spin in a given direction, ms (keep in mind we are measuring this component in units of ).  This relation can be described as:

so that there are 2s+1 values of ms for every value of s (hence the reason there are  two values for ms for spin ½ particles). This can be written very tidily if we use Dirac Notation:

Spin State = |s ms>

Where this term above is known as a “ket,” and shows the spin, s, and z-component of the spin, ms , for the state.  Then we have what is termed as “spin up” and “spin down:”

{Spin Up}z = |½ ½>z and {Spin Down}z = |½ -½>z

These two states form what is known as a “basis set,” any arbitrary spin state, |ψ> can be describe by a sum of these two states (called a linear combination):

|ψ> = α |½ ½>z + β |½ -½>z

For two constants α and β.

Let’s expand a little bit on the what this idea of a basis set entails. In the above expression we have a set of objects (spin states), that are unique; meaning we can’t use one to make the other (i.e. you can’t mathematically make spin up from spin down). Mathematicians and physicists call such objects/states linearly independent. Furthermore, using these two unique spin states, I was able to form any arbitrary spin state. Mathematicians and physicists would then say these objects span the space (here the space in question is the space of all possible spin states).

So then a basis set is any set of objects that are all linearly independent of one and other and span the space those objects exist in.  Just to drive this idea of a basis home let’s take an example. If we look at the two points in the xy plane, (1,0) and (0,1), they are obviously linearly independent. There is no way to make (1,0) from a constant multiple of (0,1). Also, any arbitrary point, (x,y), in the plane can be made by adding the correct multiples of these two points, (1,0) and (0,1), together. Then these two points span the space and are linearly independent! Hence they form a basis set, and each of the points are known as basis elements. An important point which I must stress is that the set {(1,0), (0,1)} isn’t the only basis set that exists for the xy-plane! The points (1,1) and (1,-1) are also linearly independent and span the space, so they too form a basis set!

Returning to quantum mechanics, recall how last week we learned that any physical observable has a corresponding operator. Then if the total and one component of spin angular momentum take values according to the two equations I started this section with, there must be some operator that is responsible for these observed values! To see these operators in action we have:

S2 |s ms> = s (s + 1) ℏ2 |s ms>

Sj |s ms> = ms ℏ |s ms>      for j = x, y, or z

Then for a spin up electron (s = ½) it’s total spin angular momentum would be √(3/4) and its component in the z-direction is then +½ .

Now, this begs the question, what is the component of spin for this state (spin up along the z-direction) in the x-direction!?

For this we must express our spin up z-state in terms of the basis elements for spin in the x-direction. So we must make a change of basis!

Visualization of a fermion's spin angular momentum in the "spin-up" and "spin-down" orientations along the z-axis. Notice how the vector sweeps out a circle in the xy-plane. This causes the x & y components of the spin-angular momentum to be smeared all along this circle. Ref 2.

Our spin up z-state can be expressed as:

|½ ½>z = √(2)/2 |½ ½>x + √(2)/2 |½ -½>x

Where the states on the right hand side are now with respect to spin up and down along the x-axis  (so the subscripts are denoting which basis I’m using). Notice how a purely spin up z-state breaks into a combination of spin up and spin down x-states!! This is precisely what I spoke of last week, for a spin up z-state, the spin is exactly defined in the z-direction. But now, when we switch to expressing the state with respect to x-state basis elements we get a state that is smeared, i.e. it is made of both spin up and spin down x-components (as it must be according to the Generalized Uncertainty Principle!).

So for our spin up z-state, which has an amount of it’s spin, ½ , along the z-direction we get spin components along the x-direction that are + ½ and – ½ ! This result is seen from using the operator equation above, involving Sj, on our state expressed in terms of the x-spin basis states.

This is all well and good, but does this happen in nature? And how does this relate to an ensemble of identically prepared systems?

Bringing It All Together:  The Stern-Gerlach Experiment

In 1922, Germany was the center of the new dazzling theory of Quantum Mechanics. Otto Stern and Walther Gerlach decided to join the club with a brand new experiment. They decided to investigate the radical new theory of Erwin Schrödinger, by experimenting with a beam of silver atoms in a non-uniform magnetic field.  A sketch of their experimental apparatus can be seen here:


Experimental setup used by Walther Gerlach & Otto Stern. A furnace vaporized silver atoms and created a beam which was passed through a non-uniform magnetic field (oriented along the z-direction) toward a screen. Ref 3.


Classical Physics, states that this beam should be turned into a smeared line in the presence of the magnetic field due to the magnetic moment of the silver atom interacting with the field (as we can see in the above image).  Schrödinger’s wave theory (Quantum Physics) predicted that the beam would be split into 2l+1 pieces for a given orbital angular momentum l. Now for l=0, this gives one piece, l=1 gives three, l=2 gives five, etc… So for any orbital angular momentum the beam is predicted to split into an odd number of pieces.

Now silver is a “hydrogen like” atom, it has 47 electrons, but the first 46 are all paired up in their respective orbitals. If the silver atom is in its ground state, this lone 47th electron is in the 5s orbital (l=0), and has no partner (the fact that silver has one electron all by its lonesome in the outer shell makes it hydrogen like).  Now if you were to place a silver atom in a magnetic field, it’s magnetic moment is solely due to the 47th electron (because to a very good approximation, the magnetic moment of the other 46 electrons cancel each other out).

So Stern & Gerlach prepared an ensemble of identical systems.  Where one individual system is a single silver atom (and thankfully due to nature, all silver atoms are identical!).  Then the beam of silver atoms is an ensemble of systems! Stern & Gerlach, as I mentioned, sent this beam of silver through a non-uniform magnetic field that was aligned along the, you guessed it, z-direction.

What they observed however was utterly baffling, the beam split into exactly two pieces! As you can see in the figure from their original publication almost a century ago:


Stern & Gerlach's beam of silver atoms impacting a screen with no magnetic field (left) and with magnetic field (right), Ref 4.


This didn’t match either of the predictions of Classical Physics or Schrödinger’s wave theory (but keep in mind Schrödinger’s wave theory is correct, the silver atoms are just in their ground state.  If spin didn’t exist, the beam wouldn’t have split at all!).

So here is experimental proof for spin-angular momentum if you ever saw it (don’t let your physical chemistry professor tell you spin is not a valid quantum number, I certainly didn’t)!

What would later become the theory of spin in quantum mechanics gave rise to the prediction that the beam should split into 2s+1 pieces. The spin of the first 46 electrons in the silver atom cancel with each other; the lone 47th electron has spin s = ½, hence the theoretical prediction is that the beam will split into exactly two pieces. Which is confirmed by the experiment!

Let’s get philisophical for a moment to tie more of our discussion together.  The act of passing the silver beam through the field causes a single measurement to be performed on each of these atoms.  So the non-uniform magnetic field is applying the spin-angular momentum operator for the z-direction.  And from the application of this operator, we got a measurement, i.e. the deflected beams.


Probability At Its Finest

The Stern-Gerlach experiment is then capable of creating “spin-polarized” beams of atoms.  By putting a screen in front of part of the split beam you can select a beam of atoms that are all either spin up in the z-direction or spin down in the z-direction.

Here’s a question…what happens if we then pass a spin up z beam through a non-uniform magnetic field aligned along the x-direction?  Well we’d be applying the spin angular momentum operator for the x-direction.  But these operators do not commute!  So our single beam spin up z-beam, will be smeared into two beams, one spin up in x, the other spin down in x.  Nothing major right?  We knew that a spin up z-beam should have uncertainty in the spin along the x-direction.

So let’s just pass one of these spin up x and spin down x beams back through a non-uniform magnetic field aligned in the z-direction.  We’ll take the spin up x piece for simplicity, and then the non-uniform magnetic field aligned in the z-direction will apply the spin angular momentum operator for that direction.  Since this beam was originally pure spin up z, applying this operator should then return this beam back to how it was before the beam encountered the x-magnet, namely, pure spin up-z…..

But this cannot be done!

You will never recover your pure spin up z beam from the above procedure.  You will only ever get a smeared beam that is spin up z and spin down z.

By placing the non-uniform magnetic field in the x-direction.  You made a measurement, you learned some information about the spin along the x-direction.  In doing so you forever modified the silver atom’s wave function.  As a result you placed an amount of uncertainty into the spin along the z-direction.

But you were really really really careful right? Wrong!

The Generalized Uncertainty Principle forbids you from predicting a determinate outcome for such an experiment.  These two operators, Sx and Sz , do not commute; as such you will always have an irreducible uncertainty in your theoretical prediction/experimental measurement.  You can certainly measure this final spin in the z-direction, and you could certainly say, I predict it to be spin up z.  However, you would be wrong half the time.

What you can say, is that the expectation value for the final spin along the z-direction is half the time spin up, and half the time spin down.

To help you visualize this very confusing (and complicated arrangement) feel free to take a look at this image below:

Three Stern-Gerlach magnets in a row. The first & third magnets are aligned along the z-axis, the second magnet is aligned along the x-axis. Notice how the pure spin up-z beam was forever altered by the second magnet. We are left with two beams, a spin down z and spin up z beam. Ref 5



Finally I will leave you with this Java applet [6] so that you can get a “hands-on” feel for the experiment, and help yourself understand the consequences of the Generalized Uncertainty Principle:







Until Next Time,



(Special thanks to fellow physics graduate students Samaneh Sadighi and her husband Shahab “Sean” Arabshahi for playing Juliet & Romeo for this week)!



1. Adapted from footnote on page 81 of David J. Griffiths, “Introduction to Elementary Particles,” 2nd ed., John Wiley & Sons, Inc., 1987.

2. Theresa Knott, “Quantum projection of S onto z for spin half particles.PNG,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/File:Quantum_projection_of_S_onto_z_for_spin_half_particles.PNG, Sept. 27th 2011.

3. Theresa Knott, “Stern-Gerlach experiment.PNG,” Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/File:Stern-Gerlach_experiment.PNG, Sept 27th 2011.

4. Walther Gerlach, Otto Stern, “Der experimentelle Nachweis der Richtungsquantelung im Magnetfeld,” Zeitschrift fur Physik A Hadrons and Nuclei, Vol 9, No. 1, 349-352, 1922.

5. Techne, “Quantum Physics vs The Principle of Casuality,” Telic Thoughts, http://telicthoughts.com/quantum-physics-vs-the-principle-of-causality/, Sept. 27th 2011.

6. Doug Mounce, Chris Mounce, Michael Dubson, Sam McKagan, and Carl Wieman, “Stern-Gerlach Experiment,” http://phet.colorado.edu/en/simulation/stern-gerlach, Sept. 27th 2011.


Angular Momentum in Quantum Mechanics

Saturday, September 17th, 2011

If you wanted to nail down the fundamental difference between Classical and Quantum Mechanics it would be this: Classical Mechanics is an utterly deterministic theory whereas Quantum Mechanics is a probabilistic theory.

Meaning if you knew all the forces acting on all particles in the universe, their positions, and their momenta at some arbitrary time t, Classical Mechanics says you could determine the positions & momenta of all particles in the universe at any time t (from the birth to the death of the universe). Now whether a human or a computer could ever perform this calculation is another story; however, there is nothing in the theory of Classical Mechanics that prevents you from obtaining this knowledge.  Not so for Quantum Mechanics.

So I thought with this post that I would try to guide you on brief tour of this quantum wonder land (thankfully for you I’m not mad as a hatter…yet).  And I think the best trip to take down the rabbit hole is to investigate how angular momentum behaves in these two very different theories.

What is Angular Momentum?

In Classical Mechanics (CM), angular momentum is associated with rotational motion.  As an example, let’s look at the spinning tea cup ride available at most amusement parks/carnivals (i.e. something similar to the one seen in this YouTube Video).  Here the tea cups have what’s called an orbital angular momentum associated with their motion around the center of the ride (similar to the Earth revolving around the Sun).  Now the tea cups also have spin angular momentum due to the of the cup spinning on its own axis (similar to the masses that make up the Earth rotating about the planet’s axis).

Angular Momentum in classical mechanics. The left portion of the diagram shows a particle with both orbital angular momentum Lorbital (rotating about the dotted line in the center of the dotted circle), Ref 1.

These two types of momentum can be assigned a vector (having three components).  Thus, the total angular momentum for the attendees of the tea cup ride is then the (vector) sum of their orbital and spin angular momenta.  The diagram on the right should give you a nice graphical description of this.

Now in Quantum Mechanics (QM) it should not shock you to learn particles also have orbital and spin angular momentum (the sum of these two is the again total angular momentum for the particle).  In QM, orbital angular momentum is associated with a particle that is interacting with another particle (these interacting particles form what’s referred to as a bound state).  The electron and the proton are able to form a bound state, known as the hydrogen atom; here the electron in this state would have some orbital and spin angular momentum (the proton also, but we usually ignore the proton; in the hydrogen atom it just doesn’t do much).

Now another key difference is that elementary particles are true point particles, and thus have no internal substructure.  This causes a profound difference in how angular momentum is handled in QM versus how it is handled in CM.

Take for example spin angular momentum.  The notion of a piece of an electron rotating about an electron’s axis is nonsense, there simply isn’t “a piece of an electron!”  Thus, spin angular momentum in QM is an intrinsic property of a particle and is not associated with some spatial variables.  You cannot describe spin angular momentum in QM via a function of position variables or a vector in 3D space as we know it.  Spin angular momentum in QM exists in the abstract world of linear algebra (aka matrix algebra), for which I will try not to delve to far into here.


Angular Momentum and the Uncertainty Principle

The Generalized Uncertainty Principle (for which the Heisenberg Uncertainty Principle is a special case of!) says that you cannot simultaneously observe two quantities, if the operators for those two quantities do not commute. Well that’s a bit of mouthful, what does it mean?

Let’s start by describing what an operator is; mathematically, an operator is what you apply to a particle’s wavefunction when you want to know something about that particle.  The wavefunction for a particle in QM contains all possible information about a particle at some time t (however the wavefunction is not necessarily constant for all times t, it will generally change with time for all but special cases).

So suppose I wanted to know the position of a particle. I would then apply the position operator to the particle’s wavefunction, and the resulting calculation would give me the particle’s position!  Now in practice, when I am in the laboratory and I make a measurement, I am automatically “applying an operator” on a particle (this should tell you that all physically observable quantities have a corresponding operator).

Now returning to the statement I started this section with, what does it mean for operators to “commute?”  We have something called a “communtator” between two operators in QM.  If this commutator is zero, then the operators are said to commute.  The commutator for two operators, A & B is defined as:

Now operators are very slippery fellows, and the order in which operators are written always matters; i.e. AB does not necessarily equal BA, this is only true for two operators that commute!

So unless the commutator between two operators is zero, you can never observe both quantities at the same time.  Taking this back to the famous Heisenberg Uncertainty principle, the position operator (in the x direction)  does not commute with the momentum operator (in the x direction).  This is why in QM you cannot know a particle’s exact position (in the x direction) and it’s exact momentum (in the x direction) at the same time t.  There is no such analogous situation in Classical Mechanics!

So what does this have to do with angular momentum, and the differences between Quantum and Classical Mechanics?  As I mentioned above, in CM angular momentum is described by a vector that has three components.  The theory of CM allows me to know these three components exactly.  However, in QM it is impossible to know to know the three components of the angular momentum vector (which exists in the abstract space of linear algebra).  This is because of the Generalized Uncertainty Principle, evidently the operators for the angular momentum in the x, y, and z directions do not commute with each other!

To quote the famous Dr. David J. Griffiths of Reed College:

“It’s not merely that you don’t know all three components of [the angular momentum] L; there simply aren’t three components – a particle just cannot have a determinate angular momentum vector, any more than it can simultaneously have a determinate position and momentum.” [2]

This is a subtle statement with a profound affect, so let me elaborate.  Not even God (if such a being exists) knows all three components of a particle’s angular momentum.  To help us understand this, take a look at the figure is below.

Quantization of Angular Momentum in Quantum Mechanics, Ref 3.

Here we see some angular momentum vector (the blue arrow), but this vector isn’t really a vector at all.  The arrow acts only to show us the magnitude of the angular momentum of a particle.  It actually carves out a cone, with a specific radius.   In this diagram I know precisely the value of the z-component of angular momentum; it has a value of m, where m is some integer, in units of ℏ (Planck’s constant over ).  However, as a result I have no idea what the x & y components of angular momentum are!  These other two components are smeared all over the radius of that circle carved out by the blue arrow.

However, I can know both the total angular momentum of a particle and one of it’s components in a given direction.  From this we see that the total angular momentum operator (in actuality it is this operator squared) commutes with each component angular momentum operator!

This right here is one of the great differences between CM and QM!  How very strange it is that I can know a particles total angular momentum and one and only one component of that angular momentum at the same time!  If this disturbs you then do not worry.  For Nobel Laureate Neils Bohr said that “If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.”

Quantization of Angular Momentum

Additionally, in QM angular momentum is what’s called quantized. Meaning it comes in discrete amounts, as opposed to the classical case where angular momentum is a continuous variable.

Let’s take a moment to understand the differences between discrete and continuous variables.  Starting with the set of all integers.  Each integer has neighbors that are exactly ±1 away from it.  If I take two integers, say 7 and 9, there is one and only one integer between these two (i.e. 8).  Thus the set of all integers is quantized, and can be viewed as a discrete variable.  Now, let’s take the set of all rational numbers, this is a continuous set.  For example, the numbers 7.06 and 7.07 have the number 7.065 between the two of; they also have the numbers 7.06511, or 7.06512, or 7.06513, etc… between them.  There is in fact an infinite number of numbers between 7.06  and 7.07.  Hence the set is of all rational numbers can be viewed as a continuous variable.

Coming back to angular momentum in QM.  The mathematics for all types of angular momentum in QM is a carbon copy, once you understand how it works for one type (i.e. orbital, spin or total) you understand how it works for all types.  Quantization requires, that for some type of angular momentum a, the total angular momentum will have values of:

And that the component of angular momentum a in some given direction is:

Here a is an integer or half integer, and ma ranges from -a to +a.  Usually the factor of ℏ is dropped, and we say angular momentum is in units of ℏ.  Some of you might find this more recognizable if I had written a as l, s or j (for orbital, spin and total angular momentum, respectively).  But since the mathematics for each is literally identical, I prefer just one letter, for the sake of generality.

But as I said, this is another major difference between QM and CM.  In CM I am free to have any value of angular momentum vector.  However in QM, I can only have values of the total angular momentum (of type a) and the angugular momentum in one given direction (again of type a) that satisfy the above equations.  i.e. Angular momentum in QM is discrete, whereas in CM it is continuous!


Summarizing Wonderland

So from our discussion we can highlight several key differences between Quantum Mechanics and Classical Mechanics.

  • I can only ever know one component and the total angular momentum for a particle in QM, whereas in CM no such restriction exists
  • Angular momentum is a discrete, quantized variable for QM; whereas in CM it is a continuous variable free to take any value

For my next post we shall travel further into the quantum wonderland and try to understand the probabilistic nature of QM that I hinted at in the beginning of this post

Until next time,




[1] Maschen, “Angular momentum conservation,” Wikimedia Commons, http://commons.wikimedia.org/wiki/File:Angular_momentum_conservation.svg, Sept. 17th 2011.

[2] David J. Griffiths, “Introduction to Quantum Mechanics,” 2nd ed., Pearson Education, Inc.  Upper Saddle River, NJ, 2005.

[3] P. Wormer, “Quantum angular momentum,” Wikimedia Commons, http://commons.wikimedia.org/wiki/File:Quantum_angular_momentum.png, Sept. 17th 2011.


Summer Conference Coverage: Lepton Photon 2011

Monday, August 22nd, 2011

The summer conference season may be winding down, but that doesn’t mean we are quite done yet.  Today was the first day of the Lepton Photon 2011 (LP2011) Conference; which is taking place in Mumbai, India all this week.  The proceedings of LP2011 are available via webcast from CERN (although Mumbai is ~10 hours ahead if you are in the Eastern Standard Timezone).  But if you’re a bit of a night owl and wish to participate in the excitement, then this is the link for the webcast.

The complete schedule for the conference can be found here.

But what was shown today?  Today was a day of Higgs & QCD Physics.  I’ll try to point out some of the highlights of the day in this post.  So let’s get to it.

The Hunt for the Higgs

Today’s update on the CMS Collaboration’s search for the ever elusive Higgs boson made use of ~110-170 trillion proton-proton collisions (1.1-1.7 fb -1); covering eight separate decay channels and a Higgs mass range of 110-600 GeV.   The specific channels studied and the corresponding amount of data used for each are shown in the table at left.  Here l represents a charged lepton and v represents a neutrino.

The CMS Collaboration has not reported a significant excess of events in the 110-600 GeV range at LP2011.  However, the exclusion limits for the Higgs boson mass range were updated from our previously reported values at EPS2011.  By combining the results of the eight analyses mentioned above the CMS Collaboration produced the following plot summarizing the current state of Higgs exclusion (which I have taken from the Official CMS Press Release, Ref. 1; and CMS PAS HIG-11-022, Ref. 2.  Please see the PAS for full analysis details):


Standard Model Higgs boson combined confidence levels showing current exclusion regions, image courtesy of the CMS Collaboration (Ref 1 & 2).


But how do you interpret this plot?  Rather than re-inventing the wheel, I suggest you take a quick look at Aidan‘s nice set of instructions in this post here.

Now then, from the above plot we can see that the Standard Model Higgs boson has been excluded at 95% confidence level (C.L.) in the ranges of 145-216, 226-288 and 310-400 GeV [1,2].  At a lower CL of 90%, the Collaboration has excluded the SM Higgs boson for a mass window of 144-440 GeV [1,2].

These limits shown at LP2011 improve the previous limits shown at EPS2011 (using 1.1 fb-1).  The previous exclusion limits were 149-206 and 300-440 GeV at 95% C.L., or 145-480 GeV at 90% C.L.

While the LP2011 results did not show a Higgs discovery, the CMS Collaboration is removing places for this elusive boson to hide.

QCD Physics

Today’s other talks focused on quantum chromodynamics (QCD).  With the CMS Collaboration’s results shown for a variety of QCD related measurements.

One of the highlights of these results is the measurement of the inclusive jet production cross section.  The measurement was made for a jet transverse momentum over a range of ~20-1100 GeV.  The range in cross-section covers roughly ten orders of magnitude!

Measurement of the inclusive jet cross-section made with the CMS Collaboration, here data are the black points, the theoretical prediction is given by the red line. Image courtesy of the CMS Collaboration (Ref. 3).

In this plot above each of the data series are “binned” by what is known as a jet’s rapidity (denoted by the letter y). Or in this case the absolute value of the jets rapidity.  Rapidity is a measure of where a jet is located in space.

The CMS detector is a giant cylinder, with the collisions taking place in the center of the cylinder.  If I bisect the detector at the center with a plane (perpendicular to the cylinder’s axis), objects with lower rapidities make a small angle with this plane.  Whereas objects with higher rapidities make a large angle with this plane.

As we can see from the above plot, the theoretical prediction of QCD matches the experimental data rather well.

Another highlight of CMS Collaboration’s results shown at LP2011 is the measurement of di-jet production cross-section

Measurement of the dijet production cross-section made with the CMS Collaboration.  Again, data are the black points, the theoretical prediction is given by the red line.  Image courtesy of the CMS Collaboration (Ref. 3).

Here the CMS results shown cover an invariant dijet mass of up to ~4 TeV, that’s over half the CoM collision energy!  Again, the theory is in good agreement with the experimental data!

And the last highlight I’d like to show is the production cross section of isolated photons as recorded by the CMS Detector (this is a conference about leptons and photons after all!).

Measurement of the isolated photon production cross-section made with the CMS Collaboration. Again, data are the black points, the theoretical prediction is given by the red line.  Image courtesy of the CMS Collaboration (Ref. 3).

What happens in isolated photon production is a quark in one proton interacts with a gluon in the other proton.  This interaction is mediated by a quark propogrator (which is a virtual quark).  The outgoing particles are a quark and photon.  Essentially this process is a joining of QCD and QED, an example of the Feynman Diagram for isolated photon production is shown below (with time running vertically):

From the above plot, the theoretical predictions for isolated photon production are, again, in good agreement with the experimental data!

These and other experimental tests of QCD shown at LP2011 (and other conferences) are illustrating that the theory is in good agreement with the data, even at the LHC’s unprecedented energy level.  Some tweaks are still needed, but the theorists really deserve a round of applause.



But I encourage anyone with the time or interest to tune into the live webcast all this week!  Perhaps I’ll be able to provide an update on the other talks/poster sessions in the coming days (If not check out the above links!).

Until Next Time,




[1] CMS Collaboration, “New CMS Higgs Search Results for the Lepton Photon 2011 Conference,” http://cms.web.cern.ch/cms/News/2011/LP11/, August 22nd 2011.

[2] CMS Collaboration, “Combination of Higgs Searches,” CMS Physics Analysis Summary, CMS-PAS-HIG-11-022, http://cdsweb.cern.ch/record/1376643/, August 22nd 2011.

[3] James Pilcher, “QCD Results from Hadron Colliders,” Proceedings of the Lepton Photon 2011 Conference, http://www.ino.tifr.res.in/MaKaC/contributionDisplay.py?contribId=122&sessionId=7&confId=79, August 22nd 2011.


Look Mom No Nabla’s!

Monday, August 8th, 2011

From time to time I find myself looking back at my class notes from my undergraduate studies, just to brush up on a topic or two (usually when I am taking the graduate class on the subject matter). And I’ve begun to notice a trend while comparing my undergraduate and graduate notes. I’ve gotten lazier.

That is, the notation I use to describe mathematics has gotten simpler. I think the reason for this is because there has been simply more material to write down, and less time for me to do it. I’ve seen professors at least double (sometimes triple) my age move faster with a white board marker then I can move on a treadmill. I have a tough time keeping up. So to keep up with them (aside from nagging them to slow down) I’ve started adopting a shorthand notation.

But unlike the late Dr. Feynman, I’ve not come up with my own short hand notation [1]. Instead I’ve just tried to incorporate what’s known as four-vector notation.

Four Vectors

Four vector notation is the notation of choice for quantum field theory. It allows a great simplification in how much you have to write (once you know the rules).

Let’s start with a simple example. Four vector notation allows me to describe a point in space-time (with respect to some reference frame), take the point:

(ct, x, y, z)

I can write this as:

Well that’s not astonishing in the least bit, I could have just as well labeled the point P.

Let’s take a second example. I can combine a scalar and a vector together in four-vector notation. For instance, if I wanted to describe a particle’s energy and it’s momentum (again, with respect to some reference frame) I could use a four-vector:

We can even go a bit more abstract and use four-vectors as mathematical operators:

Here we have a partial derivative with respect to time and the “del” operator (sometimes referred to as a nabla).

Now suppose I wanted to multiply two four-vectors, how would I do this? The product of two arbitrary four-vectors goes like this:

Notice how A and B have either a super-script or a sub-script in the above equations. In one case we have a contra-variant four-vector (super-script); and in the other we have a co-variant four-vector (sub-script).  However, their components are always labeled with super-scripts.  Notice how the product of four-vectors A & B is described by a “dot-product like” operation in which their respective components are multiplied together; but the last three are assigned a minus sign.

In fact I can only ever take the product of a contra-variant with a co-variant (nothing else); but the order in which one comes first doesn’t matter, their product is left invariant. I should also point out the name of the game is “summation over repeated index.” This means if I toss a third four-vector into the mix, if it has a different index (sub- or super-script) it’s ignored:

Notice how A & B have index μ and C has index ν. The μ is the “repeated” index, and the four-vector product acts between A & B. I realize this isn’t a true summation because there is a minus sign involved, but that’s just what the process is referred as.

Maxwell’s Equations – The Lazy Way

Now let’s dive into a serious example to really show the power of four-vector notation. And let’s go outside the realm of quantum field theory, instead let’s take Maxwell’s Equations:

With these four equations-and appropriate boundary conditions-I can describe all phenomenon in classical electrodynamics (I chosen to work in Heaviside-Lorentz units as opposed to the standard SI system, this causes the pesky μ’s & ε‘s to drop out. Remember I’m lazy!!).

These are four coupled first order differential equations that relate two vector fields (electric & magnetic). But from the theory of classical electrodynamics I can write these two vector fields as originating from a scalar and a vector potential (note, I did not say potential energy, which is very different from potential):

With this I can actually express Maxwell’s four first order equations as two second order equations:

Of course this is an awful mess when you look at it. Why on earth would anyone want to do this!? There are so many more terms and derivatives all over the place.

But, in physics there is something known as the “Lorentz Condition,” sometimes also called the Lorentz Gauge [2], which says:

(When I put this into Heaviside-Lorentz units the με again drop away).

Which simplifies the above two equations rather nicely:

Now this is truly enticing, these equations are almost identical! Suppose I made a set of four-vectors:

Notice how the last two are mathematical operators, one is a co-variant and the other is a contra-variant. They are just begging to be multiplied, so let’s do just that:

This is actually a new mathematical operator known as the d’Alembertian Operator, its usually represented by a square, but I don’t know the LaTeX command to make that. =(

But, with this set of four-vectors and the two equations above I can write mankind’s sum knowledge of all electromagnetic theory in one line:

Let’s pause on this for a moment.  I think this is really an astonishing miracle that physicists over the years have figured out how to write so much information about the natural world in such a small space (one line)!  Some of you might remember the Standard Model Lagrangian, which is conveniently written on a coffee mug should you forget.  That coffee mug contains A LOT of information, but it definitely cannot  fit on one line, at least with my handwriting (maybe someone someday will come up with some ingenious notation of their own?!).

But, just like that four vector notation has allowed physicists to simplify Maxwell’s Equations (all four of them) in a single concise statement. Talk about saving space on your final exam’s equation sheet! So hopefully you’ve come to appreciate the power of four-vector notation.

Until Next Time,




[1] Richard Feynman routinely used his own notation for trigonometric functions, logarithms and other common functions in mathematics, he did this because it was simpler & faster for him to write in such a fashion. For more details and other great stories, see Feynman’s own “Surely You’re Joking Mr. Feynman,” W. W. Norton & Company, Inc. 1985.

[2] See for instance D. J. Griffiths, “Introduction to Electrodynamics,” 2nd ed., Prentice-Hall, Inc., 1989.


Gell-Mann’s Eight Fold Way

Sunday, July 24th, 2011

I thought I might touch on the topic of a recent post by one of my fellow bloggers here at Quantum Diaries.  As was mentioned, researchers at Fermilab have recently discovered the Ξ0b (read Xi-0-b) a “heavy relative of the neutron.”  The significance of this discovery was found to 6.8σ [1]!  The CDF Collaboration has prepared a brief article regarding this discovery, which is is being submitted to Physical Review Letters (a peer-review journal).  A pre-print has been made available on arXiv.

But rather than talk about what’s already been written, let’s discuss something new.  Namely, how on earth did physicists know to look for such a particle?

The answer to this question takes us back to the year 1961.  An American physicist (and now Nobel Laureate, 1969)  by the name of Murray Gell-Mann proposed a way to arrange baryons and mesons based on their properties.  In a sense, Gell-Mann did for particle physics what Dmitri Mendeleev did for chemistry.

Gell-Mann decided to use geometric shapes for arranging particles.  He placed a baryon/meson onto these geometric shapes.  The location a particle was given went according to that particle’s properties.  All his diagrams however were incomplete.  Meaning, there were spaces on the shapes that a particle should have went, but the location was empty.  This was because Gell-Mann had a much smaller number of particles to work with, today more have been discovered; but we still have holes in the diagrams.

But to illustrate how Gell-Mann originally made these diagrams, I’ve shown an example using a triangle, which is part of a larger diagram that appeared in the previous post on this  subject.  I’ve also added three sets of colored lines to this diagram.

Let’s talk about the black set of lines first.  If you go along the direction indicated by each of these lines you’ll notice something interesting.  On the far right line (labeled Qe=+1, Up =2), there is only one particle along this direction, the Σ+b.  This baryon is composed of two up quarks, a beauty quark, and has an electric charge of +1.

Let’s go to the second black line (labeled Qe = 0, Up =1).  Here there are four particles (the Σ0b has yet to be discovered).  But all of these four particles have one up quark, and zero electric charge.

See the pattern?

But just to drive the point home, look at the orange lines.  Each line represents the number of strange quarks found in the particles along the line’s direction (0, 1 or 2 strange quarks!).  The blue lines do the same thing, only for the number of down quarks present in each particle. Also, for all the particles shown on this red triangle, each particle has one beauty quark present!

In fact, if you go back to the original post on the Ξ0b discovery, you’ll notice the diagram has three “levels.”  All the particles on the top level have two beauty quarks present.  Then the red triangle appears (that I’ve shown in detail above).  Then finally in the bottom level, all the particles have zero beauty quarks.

Also, if you spend some time, you can see the black, orange and blue lines I’ve drawn at right actually form planes in this 3D diagram.  And all the particles on one of these planes will have the properties of the plane (electric charge, quark content)!

So what’s the big deal about this anyway?

Well, when Gell-Mann first created the Eight-Fold-Way in the early 1960s, none of the shapes were “filled.”  But just like Dmitri Mendeleev, Gell-Mann  took this to mean that there were undiscovered particles that would go into the empty spots!!!!!

So this seemingly abstract ordering of particles onto geometric shapes (called the Eight-Fold-Way) gave Gell-Mann a way to theoretically predict the existence of new particles.  And just like Mendeleev’s periodic table, the Eight-Fold-Way went one step further, by immediately giving us knowledge on the properties these undiscovered particles would have!

If you’re not convinced, let’s come back to the experimental discovery of the Ξ0b, which is conveniently encompassed by the yellow star in the diagram above.  This particle was experimentally discovered just a few weeks ago.  But Murray Gell-Mann himself could have made the prediction that the Ξ0b existed decades earlier.  Gell-Mann would have even been able to tell us that it would have zero electric charge and be made of a u,s and b quark!!!

In fact, Gell-Mann’s Eight-Fold-Way tells high energy physicists that there is still one particle left to be discovered before this red triangle may be completed.  So, to all my colleagues in HEP, happy Σ0b hunting!



But in summary, it was the Eight-Fold-Way that gave physicists the clue that the Ξ0b was lurking out there in the void, just waiting to be discovered.

Until Next Time,




[1] T. Aaltonen (CDF Collaboration), “Observation of the Xi_b^0 Baryon,” arXiv:1107.4015v1[hep-ex], http://arxiv.org/abs/1107.4015