# 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:

## 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.

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