 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2013/paper-1/2/c/solution

p^{'} = p + α T

, and write

X = x - x^{'}

. The remaining momentum-dependent quantum phase is

pX - α T x^{'} - \frac{T}{2 m} (p + \frac{α T}{2})^{2} - \frac{α ^{2} T ^{3}}{24 m} .

Shifting the integration variable by

α T /2

reduces the integral to the free Gaussian integral. Therefore

K_{α} = θ (T) \frac{m}{2 πi ℏ T} exp {\frac{i}{ℏ} [\frac{m X ^{2}}{2 T} - \frac{α T ( x + x ^{'} )}{2} - \frac{α ^{2} T ^{3}}{24 m}]} .

The force is

- α

, not

+ α

. Both the linear endpoint term and the cubic-time term disappear in the zero-force limit. The prefactor remains the free one.

 New to topics? Read the docs here!