Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 7 / a / Solution

Separate the Klein-Gordon equation as $\varphi (t,x)=e^{\pm i\omega t}X(x)$. The spatial equation is $X^{\prime \prime }+k^{2}X=0$, with ${\omega }^2=k^2+m^2$. The first Dirichlet boundary condition removes the cosine solution, and the second requires $\sin (kL)=0$. ThusThere is no nonzero $n=0$ Dirichlet mode. The real field is built from both complex-conjugate time dependences, with mode functionsThe sine functions satisfy ${\int }_0^L\sin (k_{n}x)\sin (k_{m}x)dx=(L/2){\delta }_{nm}$. Consequently the factor $1/\sqrt{{\omega }_{n}L}$ gives unit Klein-Gordon inner product for each positive-frequency mode and the canonical equal-time field commutator after quantization. A general solution is a superposition of these modes, rather than a claim that a single mode exhausts all solutions.