Solve Ehrenfest's Theorem: Find Position Expectation Value

In summary, the problem is finding an equation that describes the position of a particle confined to a circle. I looked at classical lagrangian mechanics and found the classical value of the force and energy. Then I compared this to the quantum ananlog.
  • #1
majeka
3
0
I have been asked to "find a solution to Ehrenfest's Theorem" (in this case for the expectation value of position, of a particle confined to a circle). What does this mean - what kind of answer should i find?
 
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  • #2
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  • #3
Thankyou, I have searched many times on google and other search engines, and have found (as on the websites you have shown me) that

m*(d^2/dt^2)<x> = <F>

which is a result of the form that I am looking for. But instead of <x> I have <exp(I*theta)>, so a function of theta, as I am interested in a particle confined to a circle. I have not been able to find out whether or not I can still use this same equation (despite my <exp(I*theta)> not being a linear variable like <x>). Please understand my problems with this; I am not trying to get someone else to do my homework for me - if you have never found it hard to do your 'homework', then you are very lucky.
 
  • #4
I don't remember all the details, but I thought for a potential proportional to r^2, you should expect out the classical value for such quantities as force and energy. You might want to try finding the classical value of the force for a particle confined to a circle using classical lagrangian mechanics, I don't think its that hard, and compare this answer to the quantum ananlog. Just an idea.
 
  • #5
thanks!

Thanks spdf13,
I've tried what you said and I think it has worked...
Cheers!
:smile:
 

1. What is Ehrenfest's Theorem?

Ehrenfest's Theorem is a mathematical theorem that relates the time evolution of a quantum mechanical expectation value to the classical equations of motion. It states that the rate of change of the expectation value of a quantum mechanical operator is equal to the expectation value of the commutator of the operator with the Hamiltonian.

2. What is the position expectation value?

The position expectation value is the average position of a particle in a quantum system, calculated by taking the expectation value of the position operator. It represents the most likely position of the particle when measured.

3. How do you solve Ehrenfest's Theorem?

To solve Ehrenfest's Theorem, you first need to determine the Hamiltonian of the system, which represents the total energy of the system. Then, you calculate the commutator of the position operator with the Hamiltonian. Finally, you take the expectation value of this commutator to find the rate of change of the position expectation value.

4. What is the significance of solving Ehrenfest's Theorem?

Solving Ehrenfest's Theorem allows us to understand the relationship between classical mechanics and quantum mechanics. It shows that, on average, a quantum system will behave similarly to a classical system, but with some key differences due to the inherent randomness of quantum mechanics.

5. Are there any limitations to Ehrenfest's Theorem?

Yes, there are limitations to Ehrenfest's Theorem. It only applies to systems that are in a stable state, and it does not take into account any quantum effects such as superposition or entanglement. Additionally, it is an approximation and may not accurately predict the behavior of highly complex quantum systems.

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