 Solution

ID: past-exam-of-the-theoretical-physics-ii-course-of-the-university-of-cambridge/2015/paper-1/2/e/solution

\int_{- L}^{L} cos \frac{( 2 j + 1 ) π x}{2 L} d x = \frac{4 L ( - 1 ) ^{j}}{( 2 j + 1 ) π},

t > 0

S (t) = \frac{4}{π} \sum_{j = 0}^{\infty} \frac{( - 1 ) ^{j}}{2 j + 1} e^{- (2 j + 1)^{2} u}, u = \frac{π ^{2} D t}{4 L ^{2}} .

The slowest Dirichlet mode dominates at large time:

S (t) \sim \frac{4}{π} exp (- \frac{π ^{2} D t}{4 L ^{2}}) .

The first relative correction is

- \frac{1}{3} e^{- 8 u}

. The approximation is a late-time result; extrapolating its prefactor to

t = 0

would incorrectly give a probability larger than one. The full alternating series tends to one there.

 New to topics? Read the docs here!