Show that for any stationary state of a symmetric potential well: .
For a symmetric potential well (V(x)=V(-x)), the stationary states are eigenstates of the Hamiltonian and can always be chosen to have definite parity, i.e., the wave function (\psi(x)) is either even or _________ ___ ___ ______ ________ __________.
_______ ______ _____ ____ ___ _______ _________ ___ ___ _____.
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__________ _________ _____ _______ ______ __________ ___ ________ _______ __________.
__________ ______ _________ ______ __________ _______ ________.
_____ ____ _________ ___ ______ ___ _______.
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____ _______ ______ ___ _______ _________ ________ ________.
_____ ______ _______ _____ ____ ___ _________.
________ ______ _______ ______ _________ ____ _____ ________ ____ ______ ____ ______.
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Show that in the terms of the and
basis vectors defined by
;
, we can write:
Calculate the expectation value of the potential energy for the first excited state of a simple harmonic oscillator.
i) Write down the angular momentum states and calculate the matrix elements of
and
for
.
ii) For the angular momentum state show that
and
.
Consider the following state vectors: .
Calculate (i) the norm of and (ii) the inner product
.
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
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.