Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
In the Heisenberg picture the time evolution of an operator (\hat A) with no explicit time dependence is given by [ \frac{d\hat A}{dt}=\frac{i}{\hbar}[\hat H,\hat A]. ] The Hamiltonian is (\hat H=\omega \hat S_z). ________ ___ ___ _____ ______ _______.
___ ____ ______ _____ ______.
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_____ ___ __________ _________ _______ ________ _________ __________.
__________ ______ ____ ___ __________ ___ ___ __________ _________ ________ ___.
_________ _____ __________ _________ ______ __________ _____ _____ _____ ________.
________ ______ _______ _____ _________ ____ _____ _______ ___ __________ ____ ________.
______ ____ ______ ______ ____ ___ __________ ____ ____ ______ _____.
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Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
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
The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .