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Contents
1
Introduction
1.1
Black-body radiation
1.2
Photo-electric effect
1.3
Hydrogen atom
1.4
Wave particle duality
1.5
Uncertainty
1.6
Tunneling
2
Concepts from classical mechanics
2.1
Conservative fields
2.2
Energy function
2.3
Simple example
3
The Schrödinger equation
3.1
The state of a quantum system
3.2
Operators
3.3
Analysis of the wave equation
4
Bound states of the square well
4.1
B
2
=
0
4.2
A
2
=
0
4.3
Some consequences
4.4
Lessons from the square well
4.5
A physical system (approximately) described by a square well
5
Infinite well
5.1
Zero of energy is arbitrary
5.2
Solution
6
Scattering from potential steps and square barriers, etc.
6.1
Non-normalisable wave functions
6.2
Potential step
6.3
Square barrier
7
The Harmonic oscillator
7.1
Dimensionless coordinates
7.2
Behaviour for large
|
y
|
7.3
Taylor series solution
7.4
A few solutions
7.5
Quantum-Classical Correspondence
8
The formalism underlying QM
8.1
Key postulates
8.1.1
Wavefunction
8.1.2
Observables
8.1.3
Hermitean operators
8.1.4
Eigenvalues of Hermitean operators
8.1.5
Outcome of a single experiment
8.1.6
Eigenfunctions of
ˆ
x
8.1.7
Eigenfunctions of
ˆ
p
8.2
Expectation value of
ˆ
x
2
and
ˆ
p
2
for the harmonic oscillator
8.3
The measurement process
8.3.1
Repeated measurements
9
Ladder operators
9.1
Harmonic oscillators
9.2
The operators
ˆ
a
and
ˆ
a
†
.
9.3
Eigenfunctions of
ˆ
H
through ladder operations
9.4
Normalisation and Hermitean conjugates
10
Time dependent wave functions
10.1
correspondence between time-dependent and time-independent solutions
10.2
Superposition of time-dependent solutions
10.3
Completeness and time-dependence
10.4
Simple example
10.5
Wave packets (states of minimal uncertainty)
10.6
computer demonstration
11
3D Schrödinger equation
11.1
The momentum operator as a vector
11.2
Spherical coordinates
11.3
Solutions independent of
𝜃
and
φ
11.4
The hydrogen atom
11.5
Now where does the probability peak?
11.6
Spherical harmonics
11.7
General solutions
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