Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.
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i) Write down the eigenfunctions for the and
states of the hydrogen atom.
ii) Write the eigenvalues of and
for the states
and
.
iii) Calculate the most probable value of r for the hydrogen atom in the state .
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
Show that for any stationary state of a symmetric potential well: .
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
Show that in the terms of the and
basis vectors defined by
;
, we can write:
Estimate the uncertainty in the energy of a photon localized within a distance of .