Introduction to Quantum Mechanics · Problem 4.48

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · virial_theorem

Problem

(a) Prove the three-dimensional virial theorem: 2⟨T⟩=⟨r⋅∇V⟩2\langle T\rangle = \langle \mathbf{r} \cdot \nabla V\rangle (4.217) (for stationary states). Hint: refer to Problem 3.37. (b) Apply the virial theorem to the case of hydrogen, and show that ⟨T⟩=−En;⟨V⟩=2En\langle T\rangle = -E_n; \quad \langle V\rangle = 2E_n. (4.218) (c) Apply the virial theorem to the three-dimensional harmonic oscillator (Problem 4.46), and show that in this case ⟨T⟩=⟨V⟩=En/2\langle T\rangle = \langle V\rangle = E_n/2. (4.219)

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