Derive the quantised Hamiltonian corresponding to the uncharged Klein-Gordon equation


 * 1) Set the Klein-Gordon equation, \(m^2c^4\phi = \hbar^2c^2\nabla^2\phi - \hbar^2\partial_t^2\phi\), in natural units: \(m^2\phi = \nabla^2\phi - \partial_t^2\phi\)
 * 2) Show that \(\mathcal{L}=\frac{1}{2}\left(\partial^{\mu}\phi\right)\left(\partial_{\mu}\phi\right)-\frac{1}{2}m^2\phi^2\) is the Lagrangian representation of the Klein-Gordon equation.
 * 3) Separate the time and space derivatives: \(\mathcal{L}=\frac{1}{2}\left(\left(\partial_t\phi\right)^2 - \left(\nabla\phi\right)^2\right)-\frac{1}{2}m^2\phi^2\)
 * 4) Substitute this Lagrangian into the Euler-Lagrange equation, \(\frac{\partial \mathcal{L}}{\partial\phi} - d_t\frac{\partial \mathcal{L}}{\partial\dot{\phi}} - \nabla\frac{\partial \mathcal{L}}{\partial\nabla\phi} = 0\): \(\frac{\partial}{\partial\phi}\frac{-1}{2}m^2\phi^2 - d_t\frac{\partial}{\partial\dot{\phi}}\frac{1}{2}\dot{\phi}^2 - \nabla\frac{\partial}{\partial\nabla\phi}\frac{-1}{2}\left(\nabla\phi\right)^2 = 0\)
 * 5) Solve: \(- m^2\phi - \ddot{\phi} + \nabla^2\phi=0\)
 * 6) Find the equivalent Hamiltonian.
 * 7) Find the canonical momentum: \(\Pi = \frac{\partial \mathcal{L}}{\partial \dot{\phi}} = \dot{\phi}\)
 * 8) Solve for the Hamiltonian density, \(\mathcal{H}=\Pi\dot{\phi} - \mathcal{L}\): \(\mathcal{H}=\frac{1}{2}\left(\dot{\phi}^2 + \left(\nabla\phi\right)^2 + m^2\phi^2\right)\)
 * 9) Integrate over \(x\) to find the Hamiltonian: \(\hat{\mathbf{H}}=\frac{1}{2}\,\int dx\left(\dot{\phi}^2 + \left(\nabla\phi\right)^2 + m^2\phi^2\right)\)
 * 10) Take the Fourier transform of the Hamiltonian: \(\hat{\mathbf{H}}=\frac{1}{2}\,\int dx\int \int dk dq \frac{1}{(2\pi)^6}\left(e^{-i(k+q)\cdot x}\dot{\phi}(k)\dot{\phi}(q) + (iq)(ik)e^{-i(k+q)\cdot x}\phi(k)\phi(q) + m^2e^{-i(k+q)\cdot x}\phi(k)\phi(q)\right)\)
 * 11) Solve the integral over \(x\): \(\hat{\mathbf{H}}=\frac{1}{2}\,\int \int dk dq \frac{1}{(2\pi)^3}\left(\delta_{x+k}\dot{\phi}(k)\dot{\phi}(q) + (iq)(ik)\delta_{x+k}\phi(k)\phi(q) + m^2\delta_{x+k}\phi(k)\phi(q)\right)\)
 * 12) Solve the integral over \(q\): \(\hat{\mathbf{H}}=\frac{1}{2}\,\int dk \frac{1}{(2\pi)^3}\left(-k\dot{\phi}(k)\dot{\phi}(q) - k^3\phi(k)\phi(q) - k m^2\phi(k)\phi(q)\right)\)
 * 13) Simplify: \(\hat{\mathbf{H}}=\frac{1}{2}\,\int dk \frac{1}{(2\pi)^3}\left(-k\dot{\phi}(k)\dot{\phi}(q) - k \left(k^2 + m^2\right)\phi(k)\phi(q)\right)\)
 * 14) Set the Hamiltonian in operator form, with \([\hat{\phi}(k),\hat{\Pi}(q)] = -i\hbar\delta_{k,q}\): \(\hat{\mathbf{H}}=\frac{1}{2}\,\int dk \frac{1}{(2\pi)^3}\left(-k\hat{\Pi}(k)\dot{\hat{\phi}}(q) - k \left(k^2 + m^2\right)\hat{\phi(k)}\hat{\phi(q)}\right)\)
 * 15) Find the ladder operators for the Hamiltonian: \(\hat{a} = \sqrt{\frac{\omega_k}{2\hbar}} \left ( \hat{\phi}(k) + i\frac{\hat{\Pi}(k)}{\omega} \right ) \) \(\hat{a}^{\dagger} = \sqrt{\frac{\omega_k}{2\hbar}} \left ( \hat{\phi}(k) - i\frac{\hat{\Pi}(k)}{\omega} \right ) \)
 * 16) Substitute these into the Hamiltonian: \(\mathbf{\hat{H}} = \int dk \hbar\omega_k \left(\hat{a}^{\dagger}_k\hat{a}_k+1/2\right)\)