513.001 Molecular and Solid State Physics

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Physical
Constants

Periodic System
of Elements

      

Approximate wavefunction

Often in molecular or solid state physics we know the Hamiltonian but we can't solve the Schrödinger equation associated with this Hamiltonian. In these cases we often guess a solution and then calculate the corresponding energy.

Consider the Hamiltonian for a hydrogen atom. In spherical coordinates it is,

Find the expectation value of the energy,

for the wavefunction

Note that this wavefunction is not an eigenfunction of the Hamiltonian. Here a0 = 5.3 × 10-11 m is the Bohr radius.

(You may use a computer algebra program to solve this problem.)