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:
Which can be written as:
The expectation value of the energy is given by:
Where the integration is over the entire space.
Applying the Hamiltonian:
In our case this gives:
The expert uses the trial function and finds the value of the parameter that the energy and the minimum energy is found.