Estimate the uncertainty in the energy of a photon localized within a distance of .
The uncertainty principle states that [ \Delta x , \Delta p \geq \frac{\hbar}{2}, ] where (\Delta x) is the uncertainty in position, (\Delta p) is the uncertainty in momentum, and (\hbar = \frac{h}{2\pi} \approx ___ _________ __________ ___ _______.
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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.
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
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
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
i) Write down the eigenfunctions for the and
states of the hydrogen atom.
ii) Write the eigenvalues of and
for the states
and
.
iii) Calculate the most probable value of r for the hydrogen atom in the state .