A Modern Approach to Quantum Mechanics · Problem 7.6

Path Integrals — John S. Townsend

Physics — Quantum Mechanics · harmonic-oscillator-uncertainty-ground-state

Problem

Use the Heisenberg uncertainty relation

\Delta x\,\Delta p_x \geq \hbar/2

to express the expectation value of the energy (7.53) as an inequality involving the uncertainty in position

\Delta x

. Show that the

\Delta x

that minimizes this expectation value corresponds to a lower bound on the energy that is equal to the ground-state energy

E_0 = \hbar\omega/2

of the harmonic oscillator. From this result we can infer that the ground state is the minimum uncertainty state.