Show that for any stationary state of a symmetric potential well: .
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Estimate the uncertainty in the energy of a photon localized within a distance of .
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
Show that in the terms of the and
basis vectors defined by
;
, we can write:
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
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 .