 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-3/30e/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

For fixed

n

, the right-hand side of the quantization equation tends to zero with

ε

. If

π ε (n + \frac{1}{2}) < f (1) = \frac{16}{3},

the unique energy satisfies

E_{n} < 1

\frac{16}{3} E_{n}^{3/2} = π ε (n + \frac{1}{2})

gives

E_{n} = [\frac{3 π ε}{16} (n + \frac{1}{2})]^{2/3} .

Thus this formula is valid whenever

ε < \frac{16}{3 π ( n + 1/2 )} .

 New to topics? Read the docs here!