Dario SLC
Homework Statement
A particle of mass m and spin s, it's subject at next central potential:
##
\begin{equation*}
V(\mathbf{r})=
\begin{cases}
0\text{ r<a}\\
V_0\text{ a<r<b}\\
0\text{ r>b}
\end{cases}
\end{equation*}
##
Find the constants of motion of the system and the set of observables that commute.
Homework Equations
Ehrenfest's theorem
##\frac{\partial \left<\hat A\right>}{\partial t}=\frac{i}{\hbar}\left<[\hat H,\hat A]\right>+\left<\frac{\partial\hat A}{\partial t}\right>## when ##\hat A## it is an any observable.
The Attempt at a Solution
Because of that it's a problem of central potential, like hydrogen atom, ##\hat{L}_z## and ##\hat{L}^2## commute with hamiltonian ##\hat{H}##, and not depend explicity of the time, them are constant of motion (for the Ehrenfest's theorem), similar the hamiltonian because only depend of r coordinate and ##[\hat{H},\hat{H}]=0##. But the system has spin, therefore it add other degree of freedom, ie the system has four degree of freedom (if have'nt spin, only three degree of freedom), then the wave function will be:
$$
\psi(r,\theta,\phi,m_s)=R(r)Y_{lm}(\theta,\phi)g(m_s)
$$
when ##g(m_s)## will be ##\alpha## or ##\beta## corresponding at spin up or spin down respectly.
Like the operator ##S_z## corresponding to the observable relative to proyection about z-axis for the momentum of spin, it commute with hamiltonian, because it do not depends of ##m_s##.
Therefore, the constants of motion will be:
$$
\hat{L}_z\text{, }\hat{L}^2\text{, }\hat{H}\text{ and } \hat{S_z}
$$
On the other hand, these constants they are a set of observables that commute.
It is fine?
In addition, why isn't ##\hat{S}^2## a constant of motion?, this will have to do with the fact of that s=1/2, and no option to choose, therefore ##S^2## it is not a constant of motion?