Consider a 3-dimensional, spherically symmetric, isotropic harmonic oscillator with a potential energy of [see the attachment for full equation]. The Hamiltonian in this case is: [attached]

a. Use the trial function [attached] and ﬁnd the value of the parameter a that the energy, and ﬁnd that minimum energy.
b. Repeat the process for the trial function [attached] and ﬁnd the parameter b and the minimum energy.
c. The actual minimum energy turns out to be [attached] (i.e. each dimension contributes the usual [attached]). Compare the values you calculated in parts a and b to this value. Which is closer? Why do you think that is the case?

The Hamiltonian is:
(1.1)
Which can be written as:
(1.2)
The expectation value of the energy is given by:
(1.3)
Where the integration is over the entire space.
First case:
(1.4)
Applying the Hamiltonian:

(1.5)
Then:
(1.6)
(1.7)
In our case this gives:

...

Solution Summary

The expert uses the trial function and finds the value of the parameter that the energy and the minimum energy is found.

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