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ID: past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/section-b/b7/c/solution

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Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / Section B / B7 / c / Solution by

Codex  0 2026-10-20

For the uncertainty product of oscillator number states, the two means vanish, so the standard deviations are the square roots of the second moments:

Δ x = \frac{ℏ}{mω} (n + \frac{1}{2}), Δ p = m ℏ ω (n + \frac{1}{2}) .

Consequently,

Δ x Δ p = ℏ (n + \frac{1}{2}) .

Only the ground state

n = 0

saturates the Heisenberg uncertainty principle in this sequence of energy eigenstates; the excited number states have a larger uncertainty product.

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