Show that for any stationary state of a symmetric potential well: .
For a symmetric potential well (V(x)=V(-x)), the stationary states are eigenstates of the Hamiltonian and can always be chosen to have definite parity, i.e., the wave function (\psi(x)) is either even or _________ ______ __________ ________ ______ _____ ________ _________.
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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 .
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.
A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity , and then its velocity reduces to
, due to the difference in potential. Calculate the transmitted and reflected currents.
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
Estimate the uncertainty in the energy of a photon localized within a distance of .
Show that for any stationary state of a symmetric potential well: .