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.
The de Broglie wavelength of an electron is given by [ \lambda = \frac{h}{p} ] where (h = 6.626 \times 10^{-34} , \text{Js}) and (p) is the momentum of the electron. The momentum can be related to the accelerating voltage (V) by [ p ______ _____ ____ ___ ______ ________ _____ _________ __________ _______ ______.
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If an operator is Hermitian and an operator
is unitary, show that the operator
is Hermitian.
Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.
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
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
Estimate the uncertainty in the energy of a photon localized within a distance of .
For the simple harmonic oscillator
i) Show that .
ii) Calculate the matrix element
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.