If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
Let (\hat O) be Hermitian and (\hat U) be unitary. By definition of a Hermitian operator, [ \hat O^\dagger=\hat O, ] and by definition of a unitary operator, [ \hat U^\dagger=\hat ____ _________ _________ __________ ________ ______ ________.
_________ ____ ______ ____ _____ ____ ________ _________ _________ _____.
_________ ___ __________ _________ _________ ____ _____.
___ __________ __________ _________ _______ ____ __________ ____ ______.
______ _________ ____ __________ __________ _____ ___ ______ _______ _____ _________ ___.
_____ _______ _________ ___ ___ ______ ______.
__________ ________ _____ ____ __________ ____ ______ _____ ___ _______.
______ _______ _________ ___ ________.
__________ ___ ______ _______ ___ ___ _____.
_______ ____ ________ __________ _______ ___ ______ _________ ___.
_________ ______ ________ ______ ________ _____ ___.
______ ____ ____ ________.
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The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .
Estimate the uncertainty in the energy of a photon localized within a distance of .
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.
A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).
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.
and
are the orthonormal basis states of a two-dimensional Hilbert space. A Hermitian operator
in this basis is given by the spectral representation:
. For a normalized state given by