Quantum mechanics/Course

Subject classification: this is a physics resource.

Type classification: this resource is a course.

Mathematical background

To be a working quantum physicist, you will need a working knowledge of all of calculus; PDE's(partial differential equations) and ODE's(ordinary differential equations); and linear algebra. Students who have a stronger mathematical background usually appreciate the subject more easily, but this should not discourage anyone from learning the material.

How to understand quantum mechanics

Some of the results in quantum mechanics are not immediately intuitive - in this study guide we introduce many relevant analogies to more familiar things so that the concepts can be conveyed clearly. There are, however, many results which are incomprehensible and irrational to such an extent that possible analogies are considered inadequate. Often quantum mechanics is more about accepting than understanding. A highly regarded expert in the field, Richard Feynman, compressed his view on quantum mechanics simply: "I think I can safely say that nobody understands quantum mechanics."

Mathematical prerequisites

In order to study elementary quantum mechanics you must ideally have an understanding of the following mathematical ideas:

Complex numbers
Partial and Ordinary differential equations
Integral calculus I-III
linear algebra
fourier analysis
probability theory

This lesson series will aim to provide a brief overview of the underlying mathematical techniques required as the course progresses.

Background of Quantum Theory

The genesis of Quantum Theory and Quantum Mechanics came around 1900 when Max Planck came to tackle the phenomenon of thermal oscillation in matter. Planck hypothesised that the emitting bodies as modelled by statistical mechanics should only be allowed to absorb and emit energy in discrete 'packets' with only a finite range of allowed values. It is generally accepted that Planck did not believe this method had any fundamental meaning, and merely considered it a workaround or mathematical convenience. However, Planck's method was used a few years later by Einstein to resolve the so-called 'ultraviolet catastrophe'. This phenomenon refers to the failure of the Rayleigh-Jeans Law of black body radiation at high frequencies; the relationship tends to infinity at infinite frequency and thus predicts all blackbody sources in equilibrium should emit with infinite power at high frequency; which is clearly not the case. Einstein suggested that Planck's hypothesis in fact did accurately describe the nature of the system being studied.

The Wavefunction

The basis of quantum mechanics is a mathematical construct called the wavefunction. The wavefunction is a (almost always complex-valued) function of position ${\hat {r}}$ and describes, after some manipulation, a particle's behavior in space and time. The wavefunction is usually denoted by $\psi ({\hat {r}})$ . Understanding wavefunction is essential for understanding quantum mechanics.

Mathematical definitions

The wavefunction is defined such that its modulus squared, $|\psi ({\hat {r}})|^{2}=\psi ({\hat {r}})^{*}\psi ({\hat {r}})$ is equal to the particle's probability to be in a position ${\hat {r}}$ . The probability defines the statistical likelihood of the particle existing in that precise location when a measurement is made. It follows then, that the integral over a region of space, $\int _{a}^{b}|\psi (x)|^{2}\,dx$ , is the probability of finding a single particle in that region, and the integral over all space, $\int _{-\infty }^{\infty }|\psi (x)|^{2}\,dx$ is equal to 1 (probability distributions, by definition, must sum to 1 over all space).
The wavefunction can have an Operator applied to it to yield scalar values which describe measurable physical properties of the particle being studied. An Operator is a mathematical gadget which takes a function to another function. A generalized operator ${\hat {X}}$ is defined such that: