Basic question about how spin changes in time

[Turtle492]

We've been covering quantum spin lately in lectures and I'm a little confused about the time-dependence of it.

Basically what I want to know is - if you know that a particle (let's say an electron) is in a certain spin state (say it's spin-up) at one point, if you come back and look at it again some later, will it still be in the same spin state?

We've done about Stern-Gerlach filters and at the time we were taught that if you select only particles that have Sz = +h/2 and then put it through another identical Stern-Gerlach filter, all of the particles will come through, because they all still have Sz = +h/2, since you haven't measured any other components of spin.

But then last week we were learning about the time-dependence of spin, and they said that if we start in a definite value of Sx, the probability of finding it in Sx some time later varies with time, so that the probabilities of the particle being in spin-up or spin-down states oscillate with time.

So now I'm a bit confused as to whether starting in a certain spin state means that the particle will stay in that state until you measure another component, or if the probability of finding it in that state changes with time. Any clarification would be greatly appreciated.

 

[matonski]

So now I'm a bit confused as to whether starting in a certain spin state means that the particle will stay in that state until you measure another component, or if the probability of finding it in that state changes with time. Any clarification would be greatly appreciated.

It depends if the starting state is an eigenstate of the Hamiltonian (aka stationary state). For example, if the particle is all by itself in the absence of a magnetic field, then the probabilities for finding the particle spinning in different directions will not change. However, if there is an external magnetic field, then only those states that correspond to particles spinning parallel to the field will be stationary states.

 

[Turtle492]

It depends if the starting state is an eigenstate of the Hamiltonian (aka stationary state). For example, if the particle is all by itself in the absence of a magnetic field, then the probabilities for finding the particle spinning in different directions will not change. However, if there is an external magnetic field, then only those states that correspond to particles spinning parallel to the field will be stationary states.

So if we had a magnetic field in the z direction, the spin in the x and y directions can change in time but the z component can't?

Also, does saying 'we know the initial spin state of the particle' mean it must be in an eigenstate, or does it just mean we know how the eigenstates combine with the probability amplitudes and everything?

 

[matonski]

So if we had a magnetic field in the z direction, the spin in the x and y directions can change in time but the z component can't?

Yes. Well, of course you can't measure the spin in different directions at the same time. The probability that the spin will be up in the z-direction won't change but the probability that it will be up in the x and y directions will change.

Also, does saying 'we know the initial spin state of the particle' mean it must be in an eigenstate, or does it just mean we know how the eigenstates combine with the probability amplitudes and everything?

It's actually the same thing. Any state of a spin 1/2 particle is an eigenstate of spin in some direction. If you turn on a magnetic field, that direction will precess around the field, keeping it's angle with the field constant.