 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2012/paper-1/3/a/solution

[x^{'}, p^{'}] = i ℏ I

H^{'} = p^{'2} / (2 m) + m ω^{2} x^{'2} /2

give

[H^{'}, x^{'}] = - \frac{i ℏ}{m} p^{'}, [H^{'}, p^{'}] = i ℏ m ω^{2} x^{'} .

\overset{x}{˙}^{'} = p^{'} / m

and

\overset{p}{˙}^{'} = - m ω^{2} x^{'}

. Differentiating once more gives the harmonic-oscillator equation for each operator. Its initial-value solution is

x^{'} (t) p^{'} (t) = x^{'} (0) cos ω t + \frac{p ^{'} ( 0 )}{mω} sin ω t, = p^{'} (0) cos ω t - mω x^{'} (0) sin ω t .

Differentiating these expressions at zero verifies both prescribed initial derivatives. They also preserve

[x^{'} (t), p^{'} (t)] = i ℏ I

 New to topics? Read the docs here!