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 _________ ________ _____ _______ _________ __________ _______ _______ __________ ______.
___ ________ ____ ___ _____ _________ _______ _______ __________ ____ _____.
_________ ______ _____ ___ ___ ________ _______.
__________ ___ ___ ________ _________.
______ ______ ________ ______ ___ _____ _________ ______ _______ _________ ______.
________ _____ ___ ______ _______ _________.
________ ____ ___ ___ _______ _________.
_____ ________ ______ _____ _____ _______ ________.
_________ ______ ________ ___ ____ ____ ________ ________ ___ _______ ___.
_____ ________ __________ _____ ____ ___ _______ ______ ______ ____.
________ ________ ____ _____ _______ ___ _____ ________ _________ ___ ___ ___.
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Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
Show that in the terms of the and
basis vectors defined by
;
, we can write:
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.