Constants of motion in quantum mechanics

  • #1
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?
 
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  • #2
Dario SLC said:
On the other hand, these constants they are a set of observables that commute.

It is fine?
It should be fine.
Dario SLC said:
In addition, why isn't ^S2S^2\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 S2S2S^2 it is not a constant of motion?
##\hat S^2## is also a constant of motion. It's, however, usually omitted in the notation of the eigenfunction because one typically only deals with one type of particle in such type of problem.
 

1. What are constants of motion in quantum mechanics?

Constants of motion in quantum mechanics refer to quantities that are conserved throughout a system's evolution in time. These quantities are associated with physical symmetries, such as translation or rotation, and play a crucial role in understanding the behavior of quantum systems.

2. What is the significance of constants of motion in quantum mechanics?

Constants of motion provide important information about the dynamics of a quantum system. They allow for the prediction of a system's behavior over time, and can also reveal underlying symmetries and conservation laws. Furthermore, they help to classify and distinguish different quantum systems.

3. How are constants of motion related to Heisenberg's uncertainty principle?

Constants of motion and Heisenberg's uncertainty principle are both fundamental principles in quantum mechanics. While constants of motion refer to quantities that remain constant over time, the uncertainty principle states that the more precisely we know the value of one quantity (such as position), the less precisely we can know the value of another (such as momentum).

4. Can constants of motion be measured in experiments?

Yes, constants of motion can be measured in experiments. However, this may require specialized techniques and equipment, as well as a thorough understanding of the system and its symmetries. In some cases, constants of motion can also be inferred from other measured quantities.

5. Are there different types of constants of motion in quantum mechanics?

Yes, there are several types of constants of motion in quantum mechanics, including Hermitian operators, eigenvalues, and symmetries. These different types play different roles in understanding and describing the behavior of quantum systems, and are related to each other through mathematical relationships.

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