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Contents
Quantum Mechanics I
Niels Walet, Fall 1998
Date:
July 10, 1998
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
8.1.7 Eigenfunctions of
8.2 Expectation value of
and
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
and
.
9.3 Eigenfunctions of
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
and
11.4 The hydrogen atom
11.5 Now where does the probability peak?
11.6 Spherical harmonics
11.7 General solutions
© 1998 Niels Walet, UMIST
Email
Niels.Walet@umist.ac.uk
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and
for the harmonic oscillator
and
.
through ladder operations
and 