Tady je, dejte si to do wolfram mathematica aby se ukazalo formatovani,

To solve the problem and derive the equations of motion for the vector potential \(\vec{A}\), we need to use the given Hamiltonian and the definitions of \(\vec{Q}_k\) and \(\vec{P}_k\). Let's go through the steps:

1. **Hamiltonian:**
\[
\mathcal{E} = \frac{1}{2} \left( \vec{P}_k^2 + \omega_k^2 \vec{Q}_k^2 \right)
\]

2. **Definitions:**
\[
\vec{Q}_k = \sqrt{\epsilon_0 V} \left( \vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right)
\]
\[
\vec{P}_k = i \omega_k \sqrt{\epsilon_0 V} \left( -\vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right)
\]

3. **Expression for the vector potential \(\vec{A}\):**
\[
\vec{A} = \sum_{\vec{k}} \left( \vec{a}_k e^{-i \omega_k t - i \vec{k} \cdot \vec{x}} + \vec{a}_k^* e^{i \omega_k t + i \vec{k} \cdot \vec{x}} \right)
\]

Let's derive the equations of motion. We use Hamilton's equations:
\[
\dot{\vec{Q}}_k = \frac{\partial \mathcal{E}}{\partial \vec{P}_k}
\]
\[
\dot{\vec{P}}_k = -\frac{\partial \mathcal{E}}{\partial \vec{Q}_k}
\]

First, calculate the partial derivatives:

\[
\frac{\partial \mathcal{E}}{\partial \vec{P}_k} = \vec{P}_k
\]

\[
\frac{\partial \mathcal{E}}{\partial \vec{Q}_k} = \omega_k^2 \vec{Q}_k
\]

Using these in Hamilton's equations:

1. Equation for \(\dot{\vec{Q}}_k\):
\[
\dot{\vec{Q}}_k = \vec{P}_k
\]

2. Equation for \(\dot{\vec{P}}_k\):
\[
\dot{\vec{P}}_k = -\omega_k^2 \vec{Q}_k
\]

Next, substitute \(\vec{Q}_k\) and \(\vec{P}_k\) into these equations:

For \(\dot{\vec{Q}}_k = \vec{P}_k\):
\[
\dot{\vec{Q}}_k = \frac{d}{dt} \left[ \sqrt{\epsilon_0 V} \left( \vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right) \right]
\]
\[
= \sqrt{\epsilon_0 V} \left( -i \omega_k \vec{a}_k e^{-i \omega_k t} + i \omega_k \vec{a}_k^* e^{i \omega_k t} \right)
\]
\[
= i \omega_k \sqrt{\epsilon_0 V} \left( -\vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right)
\]
\[
= \vec{P}_k
\]

This confirms \(\dot{\vec{Q}}_k = \vec{P}_k\).

For \(\dot{\vec{P}}_k = -\omega_k^2 \vec{Q}_k\):
\[
\dot{\vec{P}}_k = \frac{d}{dt} \left[ i \omega_k \sqrt{\epsilon_0 V} \left( -\vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right) \right]
\]
\[
= i \omega_k \sqrt{\epsilon_0 V} \left( \omega_k \vec{a}_k e^{-i \omega_k t} + \omega_k \vec{a}_k^* e^{i \omega_k t} \right)
\]
\[
= \omega_k^2 \sqrt{\epsilon_0 V} \left( \vec{a}_k e^{-i \omega_k t} + \vec{a}_k^* e^{i \omega_k t} \right)
\]
\[
= -\omega_k^2 \vec{Q}_k
\]

This confirms \(\dot{\vec{P}}_k = -\omega_k^2 \vec{Q}_k\).

Therefore, the equations of motion for the vector potential \(\vec{A}\) derived from the Hamiltonian are:
\[
\dot{\vec{Q}}_k = \vec{P}_k
\]
\[
\dot{\vec{P}}_k = -\omega_k^2 \vec{Q}_k
\]

These equations describe the dynamics of the vector potential in terms of the canonical coordinates \(\vec{Q}_k\) and \(\vec{P}_k\).