Stern Gerlach Experiment: Understanding the Unexpected Results

[MiCasilla]

In the Stern Gerlach experiment the Ag beam is split in two beams of half intensity of the original beam due to the equal distribution of spin up and spin down electrons.

Considering that each of the split beams is composed of only one type of electrons (let's say spin up), if we pass this beam through a second Stern Gerlach apparatus at some relative angle from the first one I would expect to see only one beam. Either up or down depending on the relative angle between the second apparatus to the first one. Nevertheless, I read that an experiment like this will produce actually two other beam of different intensity depending on the angle.

Why is this so?

 

[selfAdjoint]

The spins around lines in different directions do not commute. In terms of measurement this means that the second apparatus will treat the output of the first as unselected. Another thing we can infer is that the electrons don't have spin between measurements. This is analogous to the stement that they don't have trajectories or momenta, only values measured from time to time. At least as far as human experiments and quantum mechanics are concerned.

 

[Sonty]

You expect finding only one beam because you think of spin as just another vector which you project onto geometric axes. The names "up" and "down" can be misleading. Thei don't mean there is a vector (like angular momentum) pointed upwards or downwards. They mean there can be only two states and they probably named them that way because one beam went upwards and the other one downwards.
In Dirac notation this may seem clearer. If you fix an SG apparatus along some axis n you get two beams corresponding to the states: |S_n;+> and |S_n;->, the + and - signs coming from the eigenvalues.

 

[pmb]

Originally posted by MiCasilla
In the Stern Gerlach experiment the Ag beam is split in two beams of half intensity of the original beam due to the equal distribution of spin up and spin down electrons.

Considering that each of the split beams is composed of only one type of electrons (let's say spin up), if we pass this beam through a second Stern Gerlach apparatus at some relative angle from the first one I would expect to see only one beam. Either up or down depending on the relative angle between the second apparatus to the first one. Nevertheless, I read that an experiment like this will produce actually two other beam of different intensity depending on the angle.

Why is this so?

After the electrons pass through the first Stern Gerlach (SG) device then are in a particular quantum state, an "eigenstate." One beam will have all electrons in the spin up state "+", the other will all be in the spin down state "+." Note that "up" and "down" is with respect to the direction of the magnetic field of the SG device that it passed through. "+" means parallel to the magnetic field, i.e. in the +z direction, and "-" means anti-parallel, i.e. in the -z direction. All that refers to is the component of the electrons spin along the direction of the magnetic field. If the magnetic field points in the +z-direction the device is said to "measure the z-component" of the spin. If you pass the "+" beam from the SG device into another Stern Gerlach device which also measures the z-component, then all the electrons will come out in one beam, not two - all in the "+" sate. Once a system (e.g. the electron) is in an eigenstate it will remain there until a certain kind of measurement is done to take it out of that state. That means that you've measured an observable which is "incompatible" with the spin-z measurement.


If the beam with all electrons in the "+" is now passed through the SG device which does not have its magnetic field aligned in the z-direction then when the electron passes through the device its acting like there is another measurement taking place which is incompatible with the first one. Suppose the SG device has its field in the x-direction (the beam is traveling in the y-direction). The state of the electron, the "+" state, can be represented as a sum of two states. The two states are the x+ state and the x- states. That means that there are two possibilities upon measuring the x-component of the spin: the electron could be measured to be in the x+ state or the x- state. That means that the electron is measured to have it's spin in the "+" x direction or the "-" direction.

So basically what you're doing by passing the beam through another SG device is taking a measurement of the other components of the spin of the electron. If the electron is in the +z eigenstate then it can be represented as a sum of the +x eigenstate and the -x eigenstate. If the SG device has it's magnetic field in the x-direction then there is a 50/50 chance of measuring +x or -x spin. The +z state can be represented as

|+z> = A|+x> + B|-x>

where the magnitudes of A and B are the same.

Hope that helps

Pete