Quantum Mechanics Solution Manual
© Leon van Dommelen
To:
Contents
SOLUTION MANUAL
Fundamental Quantum Mechanics for Engineers
Leon van Dommelen
Contents
1
. Mathematical Prerequisites
1
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1
Complex Numbers
1
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1
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1
Solution
1
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1
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2
Solution
1
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1
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3
Solution
1
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1
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4
Solution
1
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1
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5
Solution
1
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1
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6
Solution
1
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1
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7
Solution
1
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1
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8
Solution
1
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2
Functions as Vectors
1
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2
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1
Solution
1
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2
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2
Solution
1
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3
The Dot, oops, INNER Product
1
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3
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1
Solution
1
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3
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2
Solution
1
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3
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3
Solution
1
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3
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4
Solution
1
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3
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5
Solution
1
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3
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6
Solution
1
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3
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7
Solution
1
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4
Operators
1
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4
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1
Solution
1
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4
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2
Solution
1
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4
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3
Solution
1
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4
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4
Solution
1
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5
Eigenvalue Problems
1
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5
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1
Solution
1
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5
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2
Solution
1
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5
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3
Solution
1
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6
Hermitian Operators
1
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6
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1
Solution
1
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6
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2
Solution
1
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6
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3
Solution
1
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6
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4
Solution
1
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6
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5
Solution
1
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6
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6
Solution
1
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6
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7
Solution
1
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6
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8
Solution
1
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6
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9
Solution
1
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7
Additional Points
1
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7
.
1
Dirac notation
1
.
7
.
2
Additional independent variables
2
. Basic Ideas of Quantum Mechanics
2
.
1
The Revised Picture of Nature
2
.
2
The Heisenberg Uncertainty Principle
2
.
3
The Operators of Quantum Mechanics
2
.
4
The Orthodox Statistical Interpretation
2
.
4
.
1
Only eigenvalues
2
.
4
.
2
Statistical selection
2
.
5
A Particle Confined Inside a Pipe
2
.
5
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1
The physical system
2
.
5
.
2
Mathematical notations
2
.
5
.
3
The Hamiltonian
2
.
5
.
4
The Hamiltonian eigenvalue problem
2
.
5
.
5
All solutions of the eigenvalue problem
2
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5
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5
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1
Solution
2
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5
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5
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2
Solution
2
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5
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6
Discussion of the energy values
2
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5
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6
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1
Solution
2
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5
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6
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2
Solution
2
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5
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6
.
3
Solution
2
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5
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7
Discussion of the eigenfunctions
2
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5
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7
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1
Solution
2
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5
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7
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2
Solution
2
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5
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7
.
3
Solution
2
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5
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8
Three-dimensional solution
2
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5
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8
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1
Solution
2
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5
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8
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2
Solution
2
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5
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8
.
3
Solution
2
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5
.
9
Quantum confinement
2
.
6
The Harmonic Oscillator
2
.
6
.
1
The Hamiltonian
2
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6
.
2
Solution using separation of variables
2
.
6
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2
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1
Solution
2
.
6
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2
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2
Solution
2
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6
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2
.
3
Solution
2
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6
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2
.
4
Solution
2
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6
.
3
Discussion of the eigenvalues
2
.
6
.
3
.
1
Solution
2
.
6
.
3
.
2
Solution
2
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6
.
4
Discussion of the eigenfunctions
2
.
6
.
4
.
1
Solution
2
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6
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4
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2
Solution
2
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6
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4
.
3
Solution
2
.
6
.
5
Degeneracy
2
.
6
.
5
.
1
Solution
2
.
6
.
6
Non-eigenstates
3
. Single-Particle Systems
3
.
1
Angular Momentum
3
.
1
.
1
Definition of angular momentum
3
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1
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2
Angular momentum in an arbitrary direction
3
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1
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2
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1
Solution
3
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1
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2
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2
Solution
3
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1
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2
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3
Solution
3
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1
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3
Square angular momentum
3
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1
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3
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1
Solution
3
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1
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3
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2
Solution
3
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1
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3
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3
Solution
3
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1
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4
Angular momentum uncertainty
3
.
2
The Hydrogen Atom
3
.
2
.
1
The Hamiltonian
3
.
2
.
2
Solution using separation of variables
3
.
2
.
2
.
1
Solution
3
.
2
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2
.
2
Solution
3
.
2
.
2
.
3
Solution
3
.
2
.
3
Discussion of the eigenvalues
3
.
2
.
3
.
1
Solution
3
.
2
.
3
.
2
Solution
3
.
2
.
3
.
3
Solution
3
.
2
.
3
.
4
Solution
3
.
2
.
4
Discussion of the eigenfunctions
3
.
2
.
4
.
1
Solution
3
.
2
.
4
.
2
Solution
3
.
2
.
4
.
3
Solution
3
.
3
Expectation Value and Standard Deviation
3
.
3
.
1
Statistics of a die
3
.
3
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1
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1
Solution
3
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3
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1
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2
Solution
3
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3
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1
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3
Solution
3
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3
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1
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4
Solution
3
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3
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2
Statistics of quantum operators
3
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3
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2
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1
Solution
3
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3
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2
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2
Solution
3
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3
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3
Simplified expressions
3
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3
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3
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1
Solution
3
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3
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3
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2
Solution
3
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3
.
4
Some examples
3
.
4
The Commutator
3
.
4
.
1
Commuting operators
3
.
4
.
1
.
1
Solution
3
.
4
.
2
Noncommuting operators and their commutator
3
.
4
.
3
The Heisenberg uncertainty relationship
3
.
4
.
3
.
1
Solution
3
.
4
.
4
Commutator reference [Reference]
3
.
5
The Hydrogen Molecular Ion
3
.
5
.
1
The Hamiltonian
3
.
5
.
2
Energy when fully dissociated
3
.
5
.
3
Energy when closer together
3
.
5
.
4
States that share the electron
3
.
5
.
5
Comparative energies of the states
3
.
5
.
6
Variational approximation of the ground state
3
.
5
.
6
.
1
Solution
3
.
5
.
7
Comparison with the exact ground state
4
. Multiple-Particle Systems
4
.
1
Wave Function for Multiple Particles
4
.
1
.
1
Solution
4
.
1
.
2
Solution
4
.
2
The Hydrogen Molecule
4
.
2
.
1
The Hamiltonian
4
.
2
.
1
.
1
Solution
4
.
2
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1
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2
Solution
4
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2
.
2
Initial approximation to the lowest energy state
4
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2
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2
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1
Solution
4
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2
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2
.
2
Solution
4
.
2
.
3
The probability density
4
.
2
.
3
.
1
Solution
4
.
2
.
4
States that share the electrons
4
.
2
.
4
.
1
Solution
4
.
2
.
4
.
2
Solution
4
.
2
.
5
Variational approximation of the ground state
4
.
2
.
6
Comparison with the exact ground state
4
.
3
Two-State Systems
4
.
3
.
1
Solution
4
.
3
.
2
Solution
4
.
4
Spin
4
.
4
.
1
Solution
4
.
4
.
2
Solution
4
.
5
Multiple-Particle Systems Including Spin
4
.
5
.
1
Wave function for a single particle with spin
4
.
5
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1
.
1
Solution
4
.
5
.
2
Inner products including spin
4
.
5
.
2
.
1
Solution
4
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5
.
2
.
2
Solution
4
.
5
.
3
Commutators including spin
4
.
5
.
3
.
1
Solution
4
.
5
.
4
Wave function for multiple particles with spin
4
.
5
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4
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1
Solution
4
.
5
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4
.
2
Solution
4
.
5
.
5
Example: the hydrogen molecule
4
.
5
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5
.
1
Solution
4
.
5
.
6
Triplet and singlet states
4
.
5
.
6
.
1
Solution
4
.
6
Identical Particles
4
.
6
.
1
Solution
4
.
6
.
2
Solution
4
.
7
Ways to Symmetrize the Wave Function
4
.
7
.
1
Solution
4
.
7
.
2
Solution
4
.
8
Matrix Formulation
4
.
8
.
1
Solution
4
.
8
.
2
Solution
4
.
9
Heavier Atoms [Descriptive]
4
.
9
.
1
The Hamiltonian eigenvalue problem
4
.
9
.
2
Approximate solution using separation of variables
4
.
9
.
3
Hydrogen and helium
4
.
9
.
4
Lithium to neon
4
.
9
.
5
Sodium to argon
4
.
9
.
6
Potassium to krypton
4
.
10
Pauli Repulsion [Descriptive]
4
.
11
Chemical Bonds [Descriptive]
4
.
11
.
1
Covalent sigma bonds
4
.
11
.
2
Covalent pi bonds
4
.
11
.
3
Polar covalent bonds and hydrogen bonds
4
.
11
.
4
Promotion and hybridization
4
.
11
.
5
Ionic bonds
4
.
11
.
6
Limitations of valence bond theory
5
. Time Evolution
5
.
1
The Schrödinger Equation
5
.
1
.
1
Introduction to the equation
5
.
1
.
2
Some examples
5
.
1
.
2
.
1
Solution
5
.
1
.
2
.
2
Solution
5
.
1
.
2
.
3
Solution
5
.
1
.
3
Energy conservation [Descriptive]
5
.
1
.
4
Stationary states [Descriptive]
5
.
1
.
5
Particle exchange [Descriptive]
5
.
1
.
6
Energy-time uncertainty relation [Descriptive]
5
.
1
.
7
Time variation of expectation values [Descriptive]
5
.
1
.
8
Newtonian motion [Descriptive]
5
.
1
.
9
The adiabatic approximation [Descriptive]
5
.
1
.
10
Heisenberg picture [Descriptive]
5
.
2
Conservation Laws and Symmetries
5
.
3
Unsteady Perturbations of Systems
5
.
3
.
1
Schrödinger equation for a two-state system
5
.
3
.
2
Spontaneous and stimulated emission
5
.
3
.
3
Effect of a single wave
5
.
3
.
3
.
1
The wave
5
.
3
.
3
.
2
The Hamiltonian coefficients
5
.
3
.
4
Forbidden transitions
5
.
3
.
5
Selection rules
5
.
3
.
6
Angular momentum conservation
5
.
3
.
7
Parity
5
.
3
.
8
Absorption of a single weak wave
5
.
3
.
9
Absorption of incoherent radiation
5
.
3
.
10
Spontaneous emission of radiation
5
.
4
Position and Linear Momentum
5
.
4
.
1
The position eigenfunction
5
.
4
.
2
The linear momentum eigenfunction
5
.
5
Wave Packets in Free Space
5
.
5
.
1
Solution of the Schrödinger equation.
5
.
5
.
2
Component wave solutions
5
.
5
.
3
Wave packets
5
.
5
.
4
Group velocity
5
.
6
Almost Classical Motion [Descriptive]
5
.
6
.
1
Motion through free space
5
.
6
.
2
Accelerated motion
5
.
6
.
3
Decelerated motion
5
.
6
.
4
The harmonic oscillator
5
.
7
WKB Theory of Nearly Classical Motion
5
.
7
.
1
Solution
5
.
7
.
2
Solution
5
.
8
Scattering
5
.
8
.
1
Partial reflection
5
.
8
.
2
Tunneling
5
.
9
Reflection and Transmission Coefficients
6
. Numerical Procedures
6
.
1
The Variational Method
6
.
1
.
1
Basic variational statement
6
.
1
.
2
Differential form of the statement
6
.
1
.
3
Example application using Lagrangian multipliers
6
.
2
The Born-Oppenheimer Approximation
6
.
2
.
1
The Hamiltonian
6
.
2
.
2
The basic Born-Oppenheimer approximation
6
.
2
.
3
Going one better
6
.
3
The Hartree-Fock Approximation
6
.
3
.
1
Wave function approximation
6
.
3
.
2
The Hamiltonian
6
.
3
.
3
The expectation value of energy
6
.
3
.
4
The canonical Hartree-Fock equations
6
.
3
.
5
Additional points
6
.
3
.
5
.
1
Meaning of the orbital energies
6
.
3
.
5
.
2
Asymptotic behavior
6
.
3
.
5
.
3
Hartree-Fock limit
6
.
3
.
5
.
4
Configuration interaction
7
. Solids
7
.
1
Molecular Solids [Descriptive]
7
.
2
Ionic Solids [Descriptive]
7
.
3
Introduction to Band Structure [Descriptive]
7
.
4
Metals [Descriptive]
7
.
4
.
1
Lithium
7
.
4
.
2
One-dimensional crystals
7
.
4
.
3
Wave functions of one-dimensional crystals
7
.
4
.
4
Analysis of the wave functions
7
.
4
.
5
Floquet (Bloch) theory
7
.
4
.
6
Fourier analysis
7
.
4
.
7
The reciprocal lattice
7
.
4
.
8
The energy levels
7
.
4
.
9
Electrical conduction
7
.
4
.
10
Merging and splitting bands
7
.
4
.
11
Three-dimensional metals
7
.
5
Covalent Materials [Descriptive]
7
.
6
Confined Free Electrons
7
.
6
.
1
The Hamiltonian eigenvalue problem
7
.
6
.
2
Solution by separation of variables
7
.
6
.
3
Discussion of the solution
7
.
6
.
4
A numerical example
7
.
6
.
5
The density of states and confinement
7
.
6
.
6
Relation to Bloch functions
7
.
7
Free Electrons in a Lattice
7
.
7
.
1
The lattice structure
7
.
7
.
2
Occupied states and Brillouin zones
7
.
8
Nearly-Free Electrons
7
.
8
.
1
Energy changes due to a weak lattice potential
7
.
8
.
2
Discussion of the energy changes
7
.
9
Quantum Statistical Mechanics
7
.
10
Additional Points [Descriptive]
7
.
10
.
1
Thermal properties
7
.
10
.
2
Ferromagnetism
7
.
10
.
3
X-ray diffraction
8
. Basic and Quantum Thermodynamics
8
.
1
Temperature
8
.
2
Single-Particle and System Eigenfunctions
8
.
3
How Many System Eigenfunctions?
8
.
4
Particle-Energy Distribution Functions
8
.
5
The Canonical Probability Distribution
8
.
6
Low Temperature Behavior
8
.
7
The Basic Thermodynamic Variables
8
.
8
Introduction to the Second Law
8
.
9
The Reversible Ideal
8
.
10
Entropy
8
.
11
The Big Lie of Distinguishable Particles
8
.
12
The New Variables
8
.
13
Microscopic Meaning of the Variables
8
.
14
Application to Particles in a Box
8
.
14
.
1
Bose-Einstein condensation
8
.
14
.
2
Fermions at low temperatures
8
.
14
.
3
A generalized ideal gas law
8
.
14
.
4
The ideal gas
8
.
14
.
5
Blackbody radiation
8
.
14
.
6
The Debye model
9
. Electromagnetism
9
.
1
All About Angular Momentum
9
.
1
.
1
The fundamental commutation relations
9
.
1
.
2
Ladders
9
.
1
.
3
Possible values of angular momentum
9
.
1
.
4
A warning about angular momentum
9
.
1
.
5
Triplet and singlet states
9
.
1
.
6
Clebsch-Gordan coefficients
9
.
1
.
7
Some important results
9
.
1
.
8
Momentum of partially filled shells
9
.
1
.
9
Pauli spin matrices
9
.
1
.
10
General spin matrices
9
.
2
The Relativistic Dirac Equation
9
.
3
The Electromagnetic Hamiltonian
9
.
4
Maxwell’s Equations [Descriptive]
9
.
5
Example Static Electromagnetic Fields
9
.
5
.
1
Point charge at the origin
9
.
5
.
2
Dipoles
9
.
5
.
3
Arbitrary charge distributions
9
.
5
.
4
Solution of the Poisson equation
9
.
5
.
5
Currents
9
.
5
.
6
Principle of the electric motor
9
.
6
Particles in Magnetic Fields
9
.
7
Stern-Gerlach Apparatus [Descriptive]
9
.
8
Nuclear Magnetic Resonance
9
.
8
.
1
Description of the method
9
.
8
.
2
The Hamiltonian
9
.
8
.
3
The unperturbed system
9
.
8
.
4
Effect of the perturbation
10
. Some Additional Topics
10
.
1
Perturbation Theory
10
.
1
.
1
Basic perturbation theory
10
.
1
.
2
Ionization energy of helium
10
.
1
.
3
Degenerate perturbation theory
10
.
1
.
4
The Zeeman effect
10
.
1
.
5
The Stark effect
10
.
1
.
6
The hydrogen atom fine structure
10
.
1
.
6
.
1
Fine structure
10
.
1
.
6
.
2
Weak and intermediate Zeeman effect
10
.
1
.
6
.
3
Lamb shift
10
.
1
.
6
.
4
Hyperfine splitting
10
.
2
Quantum Field Theory in a Nanoshell
10
.
2
.
1
Occupation numbers
10
.
2
.
2
Annihilation and creation operators
10
.
2
.
2
.
1
Definition
10
.
2
.
2
.
2
The caHermitians
10
.
2
.
2
.
3
Examples
10
.
2
.
2
.
4
More single particle states
10
.
2
.
3
Quantization of radiation
10
.
2
.
3
.
1
Classical energy
10
.
2
.
3
.
2
Quantization
10
.
2
.
3
.
3
Photon spin
10
.
2
.
3
.
4
Traveling waves
10
.
2
.
4
Spontaneous emission
10
.
2
.
5
Field operators
10
.
2
.
6
An example using field operators
11
. The Interpretation of Quantum Mechanics
11
.
1
Schrödinger’s Cat
11
.
2
Instantaneous Interactions
11
.
3
Global Symmetrization
11
.
4
Failure of the Schrödinger Equation?
11
.
5
The Many-Worlds Interpretation
11
.
6
The Arrow of Time
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