In: Physics
Consider a three-dimensional isotropic harmonic oscillator for which the Hamiltonian is given by H = p2 2m+ 1/2mω2r2. Use the variational method with the trial function u(r) = 1πa2 3/4 exp(−r2/2a2) and obtain E. Minimizing E with respect to a2, show that the upper bound for the ground-state energy reproduces the exact result for the energy given by a =(mω and Ea = 32ω. Substitute the above value of a in the trial function and show that it also reproduces the exact ground-state wave function.
The trial wave function:
The expectation of energy:
For each of the components:
The total expectation energy from all components:
Therefore,
This is exactly the ground state energy of a 3D Harmonic oscilator:
And the wave function:
This is also exactly the ground state wave function of the harmonic oscillator.