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 ___ _______ _________ _______ _____ ____ ____ ___ ___ ________.
_________ ___ ___ __________ ____ _________.
__________ __________ ________ _______ __________ _____.
___ _______ _________ ____ _____ ___ _______ ___ ______ __________.
_________ _______ __________ _______ ______.
_________ _____ ________ _______ ______ _____ ________ _____.
_______ _______ __________ ________ ____ ______ ________ ________.
__________ ________ _____ ___ ________ _________.
___ __________ ___ ________ __________ _________ ______ ___ _________ _____ _________.
________ ____ ____ ______ ________ __________.
________ _________ _____ __________ ___ ____ __________ __________ ____.
_____ ___ ___ _________ ________ _________ _______ ______.
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If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
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.
Show that in the terms of the and
basis vectors defined by
;
, we can write:
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
A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.