Electronic Properties of Materials/Quantum Mechanics for Engineers/The Stern-Gerlach Experiment

Electronic Properties of Materials/Quantum Mechanics for Engineers
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We discussed in the first chapter a list of historical experiments that highlight the origins of quantum mechanics. In this lecture, I want to present one final experiment. The experiment itself just showed the origin of spin and orbital quantum numbers, but we're going to have to take it a step further and discuss a thought experiment that will demonstrate the fundamental working of quantum mechanics.

The Experiment edit

Stern–Gerlach experiment: Silver atoms travelling through an inhomogeneous magnetic field, and being deflected up or down depending on their spin; (1) furnace, (2) beam of silver atoms, (3) inhomogeneous magnetic field, (4) classically expected result, (5) observed result

As it happens, for reasons we will discuss during the second half of this class, the Silver (Ag) atom has a very simple magnetic nature. Each atom can be treated as a little dipole with magnetic moment  .


The force on a magnetic moment is:


In the z-direction:


The deflection of the Ag atom is proportional to the z-component of  .

Expected Results edit

Based on this, we expect to see atoms of all different orientations of  , and random magnetic moments, spread out in a single distribution.

<FIGURE> "Classic Theoretical Results of the Stern-Gerlach Experiment" (Atoms are of all different orientations of u, and there is a single distribution across the screen, centered on the main axis.

But this is not what we see...

Actual Results edit

Rather, we see two separate distributions on either side of the main beam.

<FIGURE> "Actual Results of the Stern-Gerlach Experiment" (Two separate distributions, not on the main axis, are seen instead of the single, classically predicted, distribution.)

As it happens, in quantum mechanics, magnetization is tied to angular momentum. (This of electrons zipping about in a circular orbit.) In Gold we are only looking at the spin of an electron. The directional component of  , say  , can only take two values, "up"  , or "down"  . What we just did was measure   of the Silver atoms (electrons?), and separated them into two beams, one with spin-up and the other with spin-down. Is this shocking? Yes. We just took a randomly oriented vector,  , and measured it's projection,  , and found it could only take two values.

Explaining Quantum Mechanics edit

Let's keep going. Now that (in principle) we can make a simple measurement we can make a series of thought experiments. Let's pass a beam through a filter, and see what happens...

<FIGURE> "Explaining Quantum Mechanics: The   Box" (Some beam,  , enters the box,  , and is separated based on up and down spin.)

Let's take some beam,  , have it enter the   box which separates the beam based on up and down spin. If we take the output from   measurement, discard the up elements, and remeasure down beam, the resulting beam will still be "down". This is good, no surprise here as this follows with classical logic.


Hypothesis - Polarized sunglasses all y-components are discarded.

  1. Not 50/50 in polarized light.
  2. Try rotating the box...

Now let's try rotating the   box into an   box. The   beam is still being split into up and down spin by the first  box, but now that down group is being filtered based on an   box, which is an   box that has been rotated 90°.

<FIGURE> "Explaining Quantum Mechanics: The   Component" (Note that the   box is the same as the   box, just rotated 90° to measure the y-component of the vector  .)

It looks like both boxes have a base probability of 50/50 for up or down spin. Does this make sense? Maybe?

<FIGURE> "Title" (Description)

Now we filter   to be either up or down 50/50 probability?

Something seems wrong with this picture...

Let's run one more experiment. This is the same as <FIGURE>, but now the up group coming out of the   box is again filtered through an   box. Looking at the problem, this should result in 100% down spin as the elements were tested to be 100% down spin before they entered the   box, but this is not what we see here. Instead the elements coming out of the second   box are 50/50 up and down spin.

<FIGURE> "Explaining Quantum Mechanics: The second   box." (Now the   up beam is filtered through a second   box.)

This is definitely weird.   is just some vector. If you measure the sign of  , and you can measure it again and again and again, it doesn't change. BUT after you go and measure  , if you look back at   it has once again randomized. Classically, this is like taking a bunch of marbles and splitting it into red and blue marbles. You then split the blue marbles in to large and small, but when you look back at the pile, half of the blue marbles have changed into red!

Why does this happen? edit

The components of   are "incompatible", as we can only know one component at a time. Before we measure   we can say that the atom's wave function is in a "superposition" of being up and down. By using Born's probabilistic interpretation, or psi wave, we know that the odds of measuring up or down is 50/50. We measure   and the psi wave "collapses" to  or  , depending on the measurement. Subsequent measurements have 100% chance to repeat the initial measurement according to the probabilistic interpretation of  . In  , the system is in a superposition of being  . If we measure   and find  , then we cause the wave function to collapse to  . In this state we have no information about  . We lost the information we had measured earlier when psi collapsed into  .

In the next section we will go over the formalism of quantum mechanics, and will readdress the Stern-Gerlach experiment mathematically.