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.
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
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
A 1.5 mA beam of electrons enters a sharply defined boundary with a velocity , and then its velocity reduces to
, due to the difference in potential. Calculate the transmitted and reflected currents.
i) Normalize the wave function:
ii) Calculate the expectation value of x for a particle in this state.
Derive the Heisenberg equations of motion for and
for the Hamiltonian
.
The spectral representation of an operator in a two-dimensional orthonormal basis
is
Determine the matrix elements of .