Question

Use Heisenberg’s uncertainty principle to estimate the ground state energy of the
harmonic oscillator.

Answer 1

The energy of the simple harmonic oscillator

E=p^2/2m+(mω^2 x^2)/2

Use Heisenberg’s uncertainty principle

ΔpΔx≥ℏ/2

Thus

p~ℏ/2x

We get

E=ℏ^2/(8mx^2 )+(mω^2 x^2)/2

In the ground state energy is minimal, so

dE/dx=-ℏ^2/(4mx^3 )+mω^2 x=0

x^2=ℏ/2mω

Finally

E_0=ℏ^2/(8m ℏ/2mω)+(mω^2 ℏ/2mω)/2=ℏω/2