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Advanced Quantum Mechanics Problems

1) An electron is confined by box boundary conditions in one direction and periodic boundary conditions in another, with a simple harmonic oscillator potential in the third direction. The problem asks for the energy levels and ground state wavefunction. 2) The commutation relation for position and momentum operators is demonstrated in both the position and momentum representations by applying it to the wavefunction for a particle in a potential that is linearly proportional to position. 3) For a particle in a simple harmonic oscillator potential, the expectation values of the position and momentum operators are calculated for the lowest energy state.

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Gabriel Montero Hernandez
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32 views1 page

Advanced Quantum Mechanics Problems

1) An electron is confined by box boundary conditions in one direction and periodic boundary conditions in another, with a simple harmonic oscillator potential in the third direction. The problem asks for the energy levels and ground state wavefunction. 2) The commutation relation for position and momentum operators is demonstrated in both the position and momentum representations by applying it to the wavefunction for a particle in a potential that is linearly proportional to position. 3) For a particle in a simple harmonic oscillator potential, the expectation values of the position and momentum operators are calculated for the lowest energy state.

Uploaded by

Gabriel Montero Hernandez
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Advanced Quantum Mechanics Problems

Set 1

1) An electron of mass m is confined in a container with box boundary conditions of length


L1 in the x direction, periodic boundary conditions of length L2 in the y direction and a
Simple Harmonic Oscillator potential in the z direction with spring constant γ . Write
down the expression for all the energy levels and the ground state wave function (lowest
energy eigenstate).
2 marks

2) Demonstrate the commutation relation [ xˆ , pˆ ] = ih in the position and in the momentum


representation by applying it to the appropriate wavefunction. Write down the one
dimensional Schrödinger equation in both representations when the potential is given by
V ( x ) = kx .
2 marks

3) Consider a particle of mass m in a simple harmonic oscillator potential with spring


constant γ . Find the expectation values of the position and the momentum operators if
the system is in its lowest energy state.
2 marks

4) How does quantum mechanics predict the measurement outcome of an experiment?


Demonstrate that by taking as an observable the momentum operator p̂ when the
particle is at an eigenstate of the momentum.
2 marks
p2 h2 d 2
5) Show that the kinetic energy operator =− is a Hermitian operator for a
2m 2m dx 2
particle confined in a box.
2 marks

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