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 given Hermitian operator is [ \hat{A} = 3|1\rangle\langle 1| - 1|2\rangle\langle 2| ] and the normalized state is [ |\psi\rangle = \frac{1}{2}|1\rangle + \frac{\sqrt{3}}{2}|2\rangle. ] First, the expectation value of (\hat{A}) in state (|\psi\rangle) is [ \langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle. ] Expanding using the basis states: [ \langle \psi | \hat{A} | \psi \rangle = \left(\frac{1}{2}\langle 1| + \frac{\sqrt{3}}{2}\langle 2|\right) \left(3|1\rangle\langle 1| - 1|2\rangle\langle 2|\right) \left(\frac{1}{2}|1\rangle + \frac{\sqrt{3}}{2}|2\rangle\right). ] Compute _________ __________ ________ ________ __________ ____ __________ _________ _______ _______ _______ _____.
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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
.
A quantum mechanical particle in one dimension has the wave function:
Use the Schrödinger equation to determine the corresponding potential V(x).
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 .
Estimate the uncertainty in the energy of a photon localized within a distance of .
Show that for any stationary state of a symmetric potential well: .