Past exam of the mathematics course of the University of Cambridge / 2007 / ib / Paper 3 / 7B (Quantum Mechanics)

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- Solution 7B

Solution

For the quantum harmonic oscillator, split the two expectation values into their variances and squared means:

E = \frac{( Δ p ) ^{2}}{2 m} + \frac{m ω ^{2} ( Δ x ) ^{2}}{2} + \frac{⟨ p ⟩ ^{2}}{2 m} + \frac{m ω ^{2} ⟨ x ⟩ ^{2}}{2} .

The last two terms are nonnegative, so

E \geq \frac{( Δ p ) ^{2}}{2 m} + \frac{m ω ^{2} ( Δ x ) ^{2}}{2} .

Apply the arithmetic-geometric mean inequality to the two remaining nonnegative terms, then the Heisenberg uncertainty principle:

E \geq 2 \frac{( Δ p ) ^{2}}{2 m} \frac{m ω ^{2} ( Δ x ) ^{2}}{2} = ω Δ p Δ x \geq \frac{ℏ ω}{2} .

The argument applies to any normalized state with finite oscillator energy; the stated stationary state is a particular case. Equality requires zero position and momentum means, minimum quantum uncertainty, and balanced kinetic and potential contributions, as in the oscillator ground state.

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