Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.
For a one-dimensional simple harmonic oscillator with angular frequency ω, the energy eigenvalues are given by (E_n = \left(n+\tfrac{1}{2}\right)\hbar\omega). The potential energy operator is (V = \tfrac{1}{2}m\omega^2 x^2). For stationary ________ ________ ________ _________ ______ ________ ____ __________ ____ _____ _________ ___.
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Electrons are accelerated through a potential 150 V and incident on a crystal with interatomic spacing . Calculate the de Broglie wavelength and the first-order Bragg diffraction angle.
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
Show that for any stationary state of a symmetric potential well: .
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.
If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.