Need help with proof for expectation value relation.

In summary, the conversation discusses a proof involving the commutator term and the use of Ehrenfest's theorem in relation to angular momentum and torque. The individual solving the problem initially had difficulty with the commutation but later realized the angular momentum operator is time-independent, resulting in a simplification of the equation and ultimately finding the solution.
  • #1
Lovemaker
11
0

Homework Statement


I have to prove the following:
[tex]
\hbar \frac{d}{dt}\langle L\rangle = \langle N \rangle
[/tex]

Edit: L = Angular Momentum & N = Torque

Homework Equations


I used Ehrenfest's theorem, and I've got the equation in the following form:

[tex]
\frac{1}{i} \left(\left[L,H\right]\right) + \hbar \left\langle \frac{\partial L}{\partial t}\right\rangle
[/tex]

The Attempt at a Solution



I pretty much need to prove the commutator term vanishes, but I'm not sure if it does. I've done the following with the commutation:

[tex]
i\hbar\frac{d}{dt}(\mathbf{r}\times\mathbf{p}) - i\hbar(\mathbf{r}\times\mathbf{p})\frac{d}{dt}
[/tex]

[tex]
-i^2\hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - -(i^2)\hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}
[/tex]

[tex]
\hbar^2\frac{d}{dt}(\mathbf{r}\times\mathbf{\nabla}_r) - \hbar^2(\mathbf{r}\times\mathbf{\nabla}_r)\frac{d}{dt}
[/tex]

[tex]
\hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) - \hbar^2\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) \frac{d}{dt}
[/tex]

Can I switch the order of the derivatives/operators in such a way that I get the following:
[tex]
\hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) - \hbar^2\frac{d}{dt}\left(\epsilon_{ijk}\mathbf{r}^j\mathbf{\nabla}^k\right) = 0
[/tex]
 
Last edited:
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  • #2
Never mind, figured it out. I forgot that the angular momentum operator is time-independent, so it disappears. Then I took the commutation of [L,H] a little differently then the way I did it above, and I got the answer.
 

Related to Need help with proof for expectation value relation.

1. What is the expectation value relation?

The expectation value relation is a mathematical relationship that allows us to calculate the average value of a physical quantity in quantum mechanics. It relates the expectation value of an operator to the probability of obtaining a particular eigenvalue when the system is measured.

2. How is the expectation value relation derived?

The expectation value relation is derived from the postulates of quantum mechanics, specifically the postulate that states the state of a system can be described by a wave function, and the postulate that the measurement of an observable quantity is represented by an operator acting on the wave function.

3. Can the expectation value relation be applied to any physical quantity?

Yes, the expectation value relation can be applied to any physical quantity that can be described by an operator in quantum mechanics. This includes position, momentum, energy, and other observables.

4. What is the significance of the expectation value relation in quantum mechanics?

The expectation value relation is significant because it allows us to predict the average value of a physical quantity in a quantum system. This is important for understanding the behavior of quantum systems and making predictions about their behavior.

5. Are there any limitations to the expectation value relation?

Yes, there are limitations to the expectation value relation. It only gives us the average value of a physical quantity and does not provide information about the spread or distribution of values. Also, it assumes that the system is in a stationary state, which is not always the case in quantum mechanics.

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