# Mathematics for Quantum Physics#

*Mathematics for Quantum Physics* gives you a compact introduction
to the basic mathematical tools commonly used in quantum mechanics.
Throughout the course, we provide examples and applications, but at
the core, this is still a mathematics course.

The exercises at the end of each section are an important part of the material. We recommend doing as many of these as possible, as this is crucial for being able to apply these mathematical tools in practice.

Learning goals

After following this course, you will understand the following concepts and be able to apply them:

Analysis

Taylor series

Fourier series and Fourier transforms

Differential equations

Linear algebra

Matrix operations

Eigenvalues and eigenvectors

Diagonalisation

Hilbert spaces

Probability theory

Random variables

Probability distributions

Central limit theorem

As Griffiths writes in the preface to [Gri95]:

Is all this baggage really necessary? Perhaps not, but physics is like carpentry: Using the right tool makes the job

easier, not more difficult, and teaching quantum mechanics without the appropriate mathematical equipment is like asking the student to dig a foundation with a screwdriver.

## Prerequisites#

We assume that you are familiar with the following topics:

differential and integral calculus

infinite series

complex numbers and complex functions, at the level of Chapter 1 of the lecture notes also titled Mathematics for Quantum Physics

basic arithmetic with matrices, in particular the topics covered in the appendix (Basic matrix arithmetic).

## Notational conventions#

We write \(\bar z\) for the complex conjugate of a complex number \(z\), following the convention in mathematics. In physics and other applied fields, the complex conjugate is often denoted by \(z^*\).

Warning

In some web browsers (notably Firefox), the horizontal bar for complex conjugation is invisible for some formulas and zoom levels. In case the formulas \(\overline{A}\) and \(A\) look identical, try zooming in or out!

In the section on Hilbert spaces and operators, we adopt the “bra-ket” notation from physics for inner products on Hilbert spaces. We also use the notation \(A^\dagger\) for the Hermitean adjoint of an operator \(A\) or the conjugate transpose of a matrix \(A\).

For Fourier series and Fourier transforms, we have chosen a normalisation convention with a factor of \(2\pi\) inside complex exponentials. This is natural in the context of periodic functions with period 1, and avoids normalising factors.

## Contributing#

Whether you are a student taking this course or an instructor reusing the materials, we welcome all contributions. The source code is available in a TU Delft GitLab repository. Please contact the author if you have any suggestions or corrections!

## Acknowledgements#

Chapter 4 was adapted from the notes for the course Mathematics for Quantum Physics (part of the Quantum Science and Quantum Information minor at TU Delft) by Sonia Conesa Boj, Michael Wimmer and others.

I would like to thank Sarah Arpin for various additions and corrections to these notes.

These notes were built using Jupyter Book. Parts of the technical “infrastructure” were adapted from the Jupyter Book demo by Timon Idema.

## References#

David J. Griffiths. *Introduction to Quantum Mechanics*. Prentice Hall, 1995.

Gilbert Strang. *Calculus*. Wellesley-Cambridge Press, 2017. URL: https://ocw.mit.edu/courses/res-18-001-calculus-online-textbook-spring-2005/pages/textbook/.

Larry Wasserman. *All of Statistics*. Springer, 2004. URL: https://link.springer.com/book/10.1007/978-0-387-21736-9.