Question

Show that the minimum energy of a simple harmonic oscillator is hω/2 if ΔxΔp = h/2, where (ΔP)² = \(\langle(p - \langle p\rangle)^2\rangle\)..

Answer 1

For a harmonic oscillator, (x) = \(\langle p \rangle\) = 0, and so

\((\Delta x)^2 = \langle x^2 \rangle, (\Delta p)^2 = \langle p^2 \rangle, \)

Then the Hamiltonian of a harmonic oscillator, H = p²/2m + mω²x²/2, gives the average energy as