Measuring Spin in the Stern Gerlach Experiment

[klen]

When we are measuring the spin of the electron in the experiment, we choose the spin property as its eigen state for the measurement. The eigen vectors corresponding to these states could be time dependent. Can we still break the problem into solving time independent Schrödinger Equation and then multiplying by a time dependent function, like we do for the case of measurement of energy? How do we calculate the spin eigen vectors using Schrödinger Equation?

 

[Simon Bridge]

In the representation you are talking about, it is the state vector that is time dependent... not the eigenstates.
$\psi = c_1(t)\psi_1+c_2(t)\psi_2: c_1^\star c_1+c_2^\star c_2 = 1$ ... so $\psi_1$ and $\psi_2$, the eigenstates, are solutions to the time independent Schrödinger equation.
... so what was your question there?

 

[klen]

In the representation you are talking about, it is the state vector that is time dependent... not the eigenstates.
$\psi = c_1(t)\psi_1+c_2(t)\psi_2: c_1^\star c_1+c_2^\star c_2 = 1$ ... so $\psi_1$ and $\psi_2$, the eigenstates, are solutions to the time independent Schrödinger equation.
... so what was your question there?

I believe only the eigen states of energy are the stationary states and do not depend on time, so if we are measuring spin eigen states, they could be time dependent. Isn't the time independent Schrödinger equation a energy eigen value equation?

 

[Nugatory]

I believe only the eigen states of energy are the stationary states and do not depend on time, so if we are measuring spin eigenstates, they could be time dependent. Isn't the time independent Schrödinger equation a energy eigenvalue equation?

It is. However, if two operators commute they will have shared eigenstates, and $S_z$ commutes with the Hamiltonian. Thus there are states that are eigenstates of both energy and spin, and both remain constant over time.