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Brian Dorney | USLHC | USA

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Angular Momentum in Quantum Mechanics: Spin Indepth

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,

-Brian

 

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

 

References

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.

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6 Responses to “Angular Momentum in Quantum Mechanics: Spin Indepth”

  1. Ethaniel says:

    Hi Brian,

    In “S2 |s ms> = s (s + 1) ℏ |s ms>”, should not ℏ be squared?
    Thanks for this post!

    Regards.

  2. Gavin Flower says:

    Expected value = sum of probability(i) * value(i)

    Eg consider a system that can be in either of teo value states with a known probability distribution.

    i probability value
    1 0.4 5
    2 0.6 3

    expected value = probability(1) * value(1) + probability(2) * value(2)
    = 0.4 * 5 + 0.6 * 3
    = 2.0 + 1.8
    = 3.8

    Note this does not match either of the values, but is the asymptotic value of the arithmetic mean of an increasing numbers of independent observations of the system.

  3. salman says:

    Dear Brain,
    I have read your previous post on btagging.I donot understand the following tags which is given below,so Please send me the definitions as well as little explanations of these tags.

    TRACK COUNTING HIGH PUR Bjet
    TRACK COUNTING HIGH EFF Bjet
    JET PROBABILITY BJET
    JET BPROBABILITY BJET
    SECONDARY VERTEX HIGH EFF BJET
    SECONDARY VERTEX PUR BJET
    Best Regards
    salman

  4. Dominic says:

    Dear Brian,

    Thanks for this! Your breakdown of QM is a lot easier to read than most textbooks, and you present what’s going on in a much more accessible way. Now I can dive back into the textbooks and crank the handle on the math!

    Dominic

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