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Working out harmonic oscillator operators at ##L \rightarrow \infty##

  • Thread starter JD_PM
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  • #1
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Homework Statement:

Given



$$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _-$$



Where



$$A^{\mu} _+ = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r (\vec k) e^{-i \vec k \cdot \vec x}$$



$$A^{\mu} _- = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r^{\dagger} (\vec k) e^{i \vec k \cdot \vec x}$$



$$V = L^3, \ \ \ \ \ \ \ \ \ \omega_{\vec k} = ck^0 = c|\vec k|$$



Where ##a_r (\vec k)## and ##a_r^{\dagger} (\vec k)## are the harmonic oscillator operators, which satisfy the following commutation relations

$$[a_r(\vec k), a_s(\vec k')] = [a_r^{\dagger}(\vec k), a_s^{\dagger}(\vec k')] = 0$$


$$[a_r(\vec k), a_s(\vec k')] = \rho_r \delta_{r,s} \delta_{\vec k, \vec k'}$$



a) Show that it is necessary to rescale the harmonic oscillator operators (as shown below) if we want to take the limit ##L \rightarrow \infty##



$$a_r(\vec k) \rightarrow \tilde{a_r}(\vec k) = \sqrt{\frac{V}{(2\pi)^3}} a_r (\vec k), \ \ \ \ \tilde {a^{\dagger}_r}(\vec k) = (\tilde{a_r}(\vec k))^{\dagger}$$



b) Give the commutation relations satisfied by ##\tilde{a_r}(\vec k)## and ##\tilde {a^{\dagger}_r}(\vec k)## at ##L \rightarrow \infty##.



c) Give the expressions for ##A^{\mu} _+## and ##A^{\mu} _-## at ##L \rightarrow \infty##.

Relevant Equations:

$$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _-$$
Let's go step by step

a)

We know that the harmonic oscillator operators are

$$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( -ip + m \omega q)$$

$$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$

But these do not depend on ##L##, so I guess these are not the expressions we want to work out...

My guess is that I should first find L-dependent expressions for the harmonic oscillator operators and then work out the limit.

I guess ##A^{\mu} _+## and ##A^{\mu} _-## are not the operators themselves.

But what are these expressions?

A hint would be much appreciated.

Thank you.
 

Answers and Replies

  • #2
nrqed
Science Advisor
Homework Helper
Gold Member
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Homework Statement:: Given



$$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _-$$



Where



$$A^{\mu} _+ = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r (\vec k) e^{-i \vec k \cdot \vec x}$$



$$A^{\mu} _- = \sum_{r = 0}^3 \sum_{\vec k \in \frac{2 \pi}{L} Z} \sqrt{\frac{\hbar c^2}{2V \omega_{\vec k}}} \epsilon_r^{\mu} (\vec k) a_r^{\dagger} (\vec k) e^{i \vec k \cdot \vec x}$$



$$V = L^3, \ \ \ \ \ \ \ \ \ \omega_{\vec k} = ck^0 = c|\vec k|$$



Where ##a_r (\vec k)## and ##a_r^{\dagger} (\vec k)## are the harmonic oscillator operators, which satisfy the following commutation relations

$$[a_r(\vec k), a_s(\vec k')] = [a_r^{\dagger}(\vec k), a_s^{\dagger}(\vec k')] = 0$$


$$[a_r(\vec k), a_s(\vec k')] = \rho_r \delta_{r,s} \delta_{\vec k, \vec k'}$$



a) Show that it is necessary to rescale the harmonic oscillator operators (as shown below) if we want to take the limit ##L \rightarrow \infty##



$$a_r(\vec k) \rightarrow \tilde{a_r}(\vec k) = \sqrt{\frac{V}{(2\pi)^3}} a_r (\vec k), \ \ \ \ \tilde {a^{\dagger}_r}(\vec k) = (\tilde{a_r}(\vec k))^{\dagger}$$



b) Give the commutation relations satisfied by ##\tilde{a_r}(\vec k)## and ##\tilde {a^{\dagger}_r}(\vec k)## at ##L \rightarrow \infty##.



c) Give the expressions for ##A^{\mu} _+## and ##A^{\mu} _-## at ##L \rightarrow \infty##.
Relevant Equations:: $$A^{\mu} (x) = A^{\mu} _+ + A^{\mu} _-$$

Let's go step by step

a)

We know that the harmonic oscillator operators are

$$a^{\dagger} = \frac{1}{\sqrt{2 \hbar m \omega}} ( -ip + m \omega q)$$

$$a= \frac{1}{\sqrt{2 \hbar m \omega}} (ip + m \omega q)$$

But these do not depend on ##L##, so I guess these are not the expressions we want to work out...

My guess is that I should first find L-dependent expressions for the harmonic oscillator operators and then work out the limit.

I guess ##A^{\mu} _+## and ##A^{\mu} _-## are not the operators themselves.

But what are these expressions?

A hint would be much appreciated.

Thank you.
What is the meaning of ##\rho_r## in the commutation relation of the ##a's##?
 
  • #3
280
116
What is the meaning of ##\rho_r## in the commutation relation of the ##a's##?
I assume that is defined in the usual way, as ##\rho_0=-1##, ##\rho_1=\rho_2=\rho_3=1##
 
  • #4
545
40
I assume that is defined in the usual way, as ##\rho_0=-1##, ##\rho_1=\rho_2=\rho_3=1##
That's right.
 
  • #5
545
40
Any hint for a) would be appreciated.

Thanks :)
 
  • #6
strangerep
Science Advisor
3,061
884
You're told to work with ##A^\mu(x)##, i.e., in position space. Moreover, the implication of ##V## is that it's a harmonic oscillator restricted to a finite cubic volume of sidelength ##L##. That restricts the allowable frequencies (modes) based on boundary conditions at the walls. A general solution is a linear combination of these modes, but each mode has an appropriate normalization constant.

Now, regarding part (a), what happens as ##L\to\infty##? Does such normalization still make sense?
 
  • #7
545
40
That restricts the allowable frequencies (modes) based on boundary conditions at the walls.
Actually I think that the periodic boundary conditions should look like ##x' = x + L## , ##y' = y + L## and ##z' = z + L##

Does such normalization still make sense?
I would say it does not. I'd say we have to rescale the oscillator operators so that ##A^{\mu} _+## and ##A^{\mu} _-## have physical sense (i.e. they do not blow up).

I came to such a conclusion due to some reading (QFT; Mandl and Shaw):

While using a finite normalization volume ##V##, we should be summing over a group of allowed wave vectors ##\vec k##, where ##\vec k = \frac{2 \pi}{L}(n_1, n_2, n_3), \ \ \ \ n_1, n_2, n_3 = 0, \pm 1,... ##

For ##V \rightarrow \infty## we have

$$\frac{1}{V} \sum_{k} \rightarrow\frac{1}{(2 \pi)^3} \int d^3 \vec k$$

The normalization volume ##V## must then drop out of all physically significant quantities, such as transition rates.
 
  • #8
545
40
Alright I think I understand what we're doing at a). Still stuck in how to show the Math though...
 
  • #9
strangerep
Science Advisor
3,061
884
Try googling for "harmonic oscillator normalization video".

As for the math, try reviewing how the standard Riemann integral is defined, and then generalize to the 3D case.
 

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