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This document discusses several topics in quantum mechanics: 1. It verifies an integral relationship for an antisymmetric spatial state under particle exchange. 2. It determines the ground and first-excited energy states and wavefunctions for two identical spin-1/2 particles in a harmonic oscillator potential, both for a total-spin state of 0 and 1. It considers how an interacting potential would affect these energies. 3. It asks to write down the normalized wavefunctions for three identical bosons in given one-particle states. 4. It shows that the orbital angular momentum quantum number λ must be even or zero if the wavefunction is an eigenfunction of the relative orbital angular momentum of

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95 views1 page

Tut 1

This document discusses several topics in quantum mechanics: 1. It verifies an integral relationship for an antisymmetric spatial state under particle exchange. 2. It determines the ground and first-excited energy states and wavefunctions for two identical spin-1/2 particles in a harmonic oscillator potential, both for a total-spin state of 0 and 1. It considers how an interacting potential would affect these energies. 3. It asks to write down the normalized wavefunctions for three identical bosons in given one-particle states. 4. It shows that the orbital angular momentum quantum number λ must be even or zero if the wavefunction is an eigenfunction of the relative orbital angular momentum of

Uploaded by

LisWei
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOC, PDF, TXT or read online on Scribd
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PC 4201 Quantum Mechanics

Tutorial 1

1. Verify for an antisymmetric spatial state under exchange of two particles that

1
A d
3
= x 1d 3 x 2
2
 1
2
x 1, x 2  1
2
x 2 , x1  1
2
x 1, x 2  A  1
2
x 2 , x1  A 
=  d x 1d r2 x 1, x 2 x 1, x 2  A
3 3

2. Two identical, noninteracting spin - ½ particles of mass m are in the

one- dimensional harmonic oscillator for which the Hamiltonian is

P̂12x 1 P̂ 2 1
Ĥ   m2 x̂12  2 x  m2 x̂ 22
2m 2 2m 2

(a) Determine the ground-state and first-excited-state kets and corresponding


energies when the two particles are in a total-spin-0-state. What are the lowest
energy states and corresponding kets for the particles if they are in a total-spin-1
state?

(b) Suppose the two particles interact with a potential energy of interaction
x1  x 2  a
V( x 1  x 2 )    V0
0
elsewhere

Argue what the effect will be on the energies that you determined in (a), that is,
whether the energy of each state moves up, moves down, or remains unchanged.

3. Write down the normalized wavefunctions of a system of three identical bosons,


which are in given one-particle states.

4 Show that if the wavefunction of a system of two identical spinless particles is


an eigenfunction of the orbital angular momentum of relative motion of the two
particles, then the quantum number  necessarily has an even (or zero) value.

5. Show that the symmetrization and the anti-symmetrization operators are the
orthogonal projection operators, i.e. that
S 2  S, A 2  A, SA  AS  0.

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