Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 2 / 17B / e

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- Solution e

Solution

The Hamiltonian of the quantum harmonic oscillator is the sum of two positive quadratic forms. For a nonzero eigenfunction in its domain,

E ∥ ϕ ∥^{2} = ⟨ ϕ, \hat{H} ϕ ⟩ = \frac{∥ p ^ ϕ ∥ ^{2}}{2 m} + \frac{m ω ^{2}}{2} ∥ \overset{x}{^} ϕ ∥^{2} \geq 0.

Thus

E \geq 0

. Part (d) supplies the stronger bound

E \geq ℏ ω /2

.

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