Consider a three-dimensional isotropic harmonic oscillator for which the Hamiltonian is given by H = p2...

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.

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Expert Solution

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.

genius_generous answered 1 year ago

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