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Theoretical Physics course of the University of Cambridge

Theoretical Physics II course of the University of Cambridge

Past exam of the theoretical physics II course of the University of Cambridge

2018

Using $[x,p]=i\hbar$, the <Heisenberg picture> equations for the <quantum harmonic oscillator> are
$$
\boxed{\dot x=\frac{i}{\hbar}[H_0,x]=\frac pm,\qquad
\dot p=\frac{i}{\hbar}[H_0,p]=-m\omega^2x}.
$$
Solving these coupled linear equations gives
$$
\boxed{\begin{aligned}
x(t)&=x(0)\cos\omega t+\frac{p(0)}{m\omega}\sin\omega t,\\
p(t)&=p(0)\cos\omega t-m\omega x(0)\sin\omega t.
\end{aligned}}
$$
The oscillator <annihilation operator> and <creation operator> are
$$
a=\sqrt{\frac{m\omega}{2\hbar}}x+\frac{i}{\sqrt{2m\hbar\omega}}p,\qquad
a^\dagger=\sqrt{\frac{m\omega}{2\hbar}}x-\frac{i}{\sqrt{2m\hbar\omega}}p.
$$
They obey $[a,a^\dagger]=1$ and $H_0=\hbar\omega(a^\dagger a+1/2)$. Substitution of the position and momentum solutions, or the corresponding <commutators> with $H_0$, yields
$$
\boxed{a(t)=a(0)e^{-i\omega t},\qquad a^\dagger(t)=a^\dagger(0)e^{i\omega t}}.
$$

Solution

Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 4 / a / Solution

Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 4 / a / Solution

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