Principles of Quantum Mechanics · Problem 13.1.5

The Hydrogen Atom — R. Shankar

Physics — Quantum Mechanics · virial-theorem-hydrogen

Problem

*(Virial Theorem).* Since ∣n,l,m⟩|n, l, m\rangle is a stationary state, ⟨Ω^⟩=0\langle \hat{\Omega} \rangle = 0 for any Ω\Omega. Consider Ω=R⋅P\Omega = \mathbf{R} \cdot \mathbf{P} and use Ehrenfest's theorem to show that ⟨T⟩=(−1/2)⟨V⟩\langle T \rangle = (-1/2)\langle V \rangle in the state ∣n,l,m⟩|n, l, m\rangle.

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