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## Main Question or Discussion Point

Dear all, I'd be very grateful for some help on this question:

"The momentum operator is defined by: [tex] \displaystyle P = - \int_{0}^{L} dz \left(\frac{\partial \phi}{\partial t}\right) \left( \frac{\partial \phi}{\partial z} \right) [/tex]

Show that P can be written in terms of the operators [tex] a_n [/tex] and [tex] a^{\dagger}_n [/tex] as:

[tex] \displaystyle P = \sum_{n} k_n a_n^{\dagger} a_n [/tex] "

The KG field is given by: [tex] \displaystyle \phi(t,z) = \sum_{n} \frac{1}{\sqrt{2E_n L}} \left[a_n e^{-i(E_n t - k_n z)} + a^{\dagger}_n e^{+i(E_n t - k_n z)} \right] [/tex]

The following relations are true:

[tex] \displaystyle k_n = \frac{2 \pi n}{L} \;;\; \left[a_n, a_m^{\dagger} \right] = \delta_{nm} \;;\; \left[a_n, a_m \right] = \left[a_n^{\dagger}, a_m^{\dagger} \right] = 0 [/tex]

and [tex] E_n^2 = k_n^2 + m^2 [/tex]

[tex]\displaystyle \int_{0}^{L} dz e^{iz(k_n-k_m)} = L \delta_{nm} [/tex]

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I've fed all this information into the definition of the momentum operator and have the result:

[tex] \displaystyle P = \sum_{n} \frac{k_n}{2} \left[1 + 2 a_n^{\dagger} a_n - a_n a_{-n} - a^{\dagger}_n a^{\dagger}_{-n} \right][/tex]

but I am unsure of how to reduce this down even further.

Any help would be greatly appreciated.

"The momentum operator is defined by: [tex] \displaystyle P = - \int_{0}^{L} dz \left(\frac{\partial \phi}{\partial t}\right) \left( \frac{\partial \phi}{\partial z} \right) [/tex]

Show that P can be written in terms of the operators [tex] a_n [/tex] and [tex] a^{\dagger}_n [/tex] as:

[tex] \displaystyle P = \sum_{n} k_n a_n^{\dagger} a_n [/tex] "

The KG field is given by: [tex] \displaystyle \phi(t,z) = \sum_{n} \frac{1}{\sqrt{2E_n L}} \left[a_n e^{-i(E_n t - k_n z)} + a^{\dagger}_n e^{+i(E_n t - k_n z)} \right] [/tex]

The following relations are true:

[tex] \displaystyle k_n = \frac{2 \pi n}{L} \;;\; \left[a_n, a_m^{\dagger} \right] = \delta_{nm} \;;\; \left[a_n, a_m \right] = \left[a_n^{\dagger}, a_m^{\dagger} \right] = 0 [/tex]

and [tex] E_n^2 = k_n^2 + m^2 [/tex]

[tex]\displaystyle \int_{0}^{L} dz e^{iz(k_n-k_m)} = L \delta_{nm} [/tex]

----------------------

I've fed all this information into the definition of the momentum operator and have the result:

[tex] \displaystyle P = \sum_{n} \frac{k_n}{2} \left[1 + 2 a_n^{\dagger} a_n - a_n a_{-n} - a^{\dagger}_n a^{\dagger}_{-n} \right][/tex]

but I am unsure of how to reduce this down even further.

Any help would be greatly appreciated.

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