Commutation Relations, 2D Harmonic Oscillator

In summary, the conversation discusses a two-dimensional harmonic oscillator described by the Hamiltonian ##\hat H_0 = \hbar \omega (\hat a_x \hat a_x ^{\dagger} + \hat a_y \hat a_y^{\dagger} + 1)## and the operators ##\hat H_0 \hat L | n_1, n_2 \rangle## and ##\hat L \hat H_0 |n_1, n_2 \rangle##. The question is whether the operator ##\hat L = \hat x \hat P_y - \hat y \hat P_x## changes the total ##n_1 + n_2## of a number state ##
  • #1
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Homework Statement


Consider a two-dimensional harmonic oscillator, described by the Hamiltonian

##\hat H_0 = \hbar \omega (\hat a_x \hat a_x ^{\dagger} + \hat a_y \hat a_y^{\dagger} + 1)##

Calculate ##\hat H_0 \hat L | n_1, n_2 \rangle## and ##\hat L \hat H_0 |n_1, n_2 \rangle##. What does this imply for ##[\hat H_0 , \hat L]##

Homework Equations

The Attempt at a Solution


[/B]
I can solve this problem by "brute force", having shown that ##\hat L = \hat x \hat P_y - \hat y \hat P_x = i \hbar (\hat a_y^{\dagger} \hat a_x - \hat a_x ^{\dagger} \hat a_y)##. I can apply one operator and then the other, but the results are messy making me think this can be done in a smarter manner.

Trying to combine these operators into a single simplified operator gives me

##H_0 \hat L = i \hbar^2 \omega (\hat a_x ^{\dagger} \hat a_x (\hat a_y ^{\dagger} \hat a_x - \hat a_x ^{\dagger} \hat a_y) + \hat a_y^{\dagger} \hat a_y (\hat a_y^{\dagger} \hat a_x - \hat a_x^{\dagger} \hat a_y) + (\hat a_y^{\dagger} \hat a_x - \hat a_x^{\dagger} \hat a_y))##

I need to deal with this ##(\hat a_y ^{\dagger} \hat a_x - \hat a_x ^{\dagger} \hat a_y)## term but I am unsure of how to do it. I know that ##[\hat a_i, \hat a_i^{\dagger}] = 1## and that the x and y operators are mutually commuting.

If I set ##\hat N = \hat a_y^{\dagger} \hat a_x##, then ##(\hat a_y ^{\dagger} \hat a_x - \hat a_x ^{\dagger} \hat a_y) = [\hat N, \hat N^{\dagger}]## but the numbered states or not eigenstates of my invented operator...

So, is there anything I can realize to make this problem less of an accounting exercise?

I hope it's clear what I'm asking; I can solve this problem, but it does not feel efficient.

Thank you for your help!
 
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  • #2
Do you realize that the Hamiltonian on a number state ##\mid n_1,n_2\rangle ## has eigenvalue related to the total ##n_1+n_2##?
\begin{align*} H& = \hbar \omega (a_x ^{\dagger}a_x + a_y^{\dagger}a_y \ + 1) \\
H &= \hbar\omega(N_x + N_y + 1)\\
H|n_1, n_2> &= \hbar\omega(n_1 + n_2+1)|n_1, n_2>\end{align*}
So a question you can ask is "does ##L## change the total ##n_1 + n_2## ?"
You can examine the operator ##L## or try it out.
 

1. What are commutation relations?

Commutation relations are mathematical expressions that describe the relationship between two operators in quantum mechanics. They dictate how the order in which two operators are applied affects the outcome of a measurement.

2. How do commutation relations relate to the 2D harmonic oscillator?

In the 2D harmonic oscillator, the position and momentum operators have specific commutation relations. These relations determine the allowed energy levels and the corresponding wave functions of the system.

3. What is the significance of commutation relations in quantum mechanics?

Commutation relations are essential in quantum mechanics as they allow us to calculate the uncertainty in measurements. They also help us understand the fundamental properties of quantum systems and their corresponding operators.

4. How do commutation relations differ from classical mechanics?

In classical mechanics, the order in which two operators are applied does not affect the outcome of a measurement. However, in quantum mechanics, the commutation relations show that the order does matter and can lead to different results.

5. Can commutation relations be applied to other systems besides the 2D harmonic oscillator?

Yes, commutation relations are a fundamental concept in quantum mechanics and are applicable to all quantum systems. They are particularly useful in analyzing the behavior of particles in potential wells and other complex systems.

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