Dear all, I'd be very grateful for some help on this question:(adsbygoogle = window.adsbygoogle || []).push({});

"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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Klein-Gordon Momentum Question

**Physics Forums | Science Articles, Homework Help, Discussion**