The first part of this website on quantum mechanics arose from reading the book "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind. It covers the basics of quantum mechanics up to the harmonic oscillator, including mathematical basics.

Susskind's website is at

The second part of this website deals with the basics of quantum computing. It is under construction and constantly being updated.

…errors, broken links? … please give me a hint ...

You can reach me via the email-address dieter.kriesell (at)

Dieter Kriesell


Quantum mechanics:

2D Harmonic Oscillator

The 2D-harmonic oscillator in cartesian and polar coordinates.

Angular momentum

We calculate angular momentum in spherical coordinates both classic and quantum.

Chain rule

Two examples for the chain rule for partial derivatives.

Commutator uncertainty

Two observables are simultaneously measurable if and only if the respective matrices commutate.

Commutators with functions of operators

How to calculate commutators that contain a function of one operator.


Solving integrals containing sin and cos by use of complex functions.

Continuous functions as vectors

This paper describes how a continuous function can be quantized and how differentiation and integration are expressed explicitly with matrices.

Differential equations

We solve differential equations of first and second order with emphasis to complex solutions.

Dirac delta-function

The Dirac -function and the position operator .

Discrete probability and probability density

This paper shows the transformation process from discrete probability to continuous probability density. It is a kind of reverse process compared with the paper above.


A translation of the original paper with some comments.

Ehrenfest Theorem

From classical physics to quantum mechanics via the Ehrenfest Theorem.

Exponential Pauli

Power series of exponentials containing Pauli matrices.


How to handle exponentials with Pauli matrices.

Free particle

The free particle and its wave function.

Fourier series

Basics for understanding Fourier series

Gauss Integral

Proofs for some Gauss Integrals.

Hermitian – diagonal matrix

An example that shows that a 2x2 Hermitian matrix can be transformed in diagonal form.

Hermite polynomials

We become acquainted with Hermite polynomials and proof orthogonality.

Hilbert space

This paper starts with a finite dimensional Hilbert space and ends up with the spin of an electron.

Lagrange generalized coordinates

We check the derivation rules with cartesian and polar coordinates.


This short paper describes the Lorentz transformation in the simple case two systems moving away from each other at constant speed on the x-axis.

Mach-Zehnder Interferometer

This is more or less an exercise in complex arithmetic.

Mathematics needed

A compilation of mathematics you need if you want to start with quantum mechanics.

Matrix change of basis

Two ways to write a matrix in a new basis.

Momentum – Position

States with position and momentum will be described with either position wave functions or momentum wave functions. This paper presents both descriptions parallel.


Uncertainty of momentum and position Operator in momentum and position representation.

Number operator

We take a close look at the harmonic oscillator and number operator, raising operator and lowering operator.

Painter’s horse

An illustration of the spin problem and the tensor product.


Partial derivatives in cartesian and polar representation.

Probability current density

We work with the 1D step barrier problem and use the probability current to develop reflection and transmission probability


Time development and Schroedinger equation for constant and  commutating time dependent Hamiltonians.


Here you find a compilation of keywords for quantum mechanics on a basic level.

Quantum-abc, exercises

These are the exercises of the book “Quantum Mechanics, The Theoretical Minimum” of Leonard Susskind & Art Friedman.

Quantum harmonic oscillator

A traditional access to the quantum mechanic oscillator and the Hermite polynomials.

Ramsauer-Townsend effect

We calculate the ratio of incoming to outgoing wave.


Time-independent vs. time-dependent operators.


Two ways to get time-dependent Heisenberg operators out of time-independent Schroedinger operators.

Spin in magnetic field

We look at a spin ˝ in a magnetic field:

-          constant in direction and strength,

-          constant in direction but slowly changing its strength,

-          rotating (NMR).

Tensor Products

Tensor products, correlation, and superposition.

Trigonometric Identities

The trigonometric identities resolved in exponential terms.

Triple jump to Schroedinger

We start from classic physics, use de Broglie and arrive at the Schroedinger equation.

Twice infinite potential well

The infinite potential well is either presented by use of trigonometric or complex exponential functions. This paper works simultaneously through both and shows the difference – in the end all works fine.

Unitary Matrix

Unitarian matrices don’t change the dot product, map orthonormal bases to orthonormal bases and play an important role in the time development of quantum states.

Virial theorem (quantum)

We calculate the speed of the electron of the hydrogen atom by use of the quantum mechanical Viral theorem.

Wave functions

Wave functions (normalized polynomials) and matrices.


Quantum computing:

Bell states

Four Bell states in detail.

Quantum Fourier

The quantum Fourier transformation explicit for 1, 2, 3 and 4 qubits.


Application of Simon’s problem to a four-bit and a two-bit function.


An alternative access to the Deutsch-Jozsa algorithm for a 3-qubit input.

Quantum parallelism

An explicit and detailed work-through the Deutsch algorithm.

Grover’s algorithm

This is a detailed example of applicating Grover’s algorithm to a SAT problem.


List of combinations of  states with the Kronecker-product (tensor-product).

Partial scalar product

Arithmetic with bras and kets of different dimension

Spin states

A short description of spin states, the Hadamard Matrix, Bloch sphere and rotation.

Reversible gates

The Toffoli and the Fredkin quantum gates.

Basic vs. conceptual access

We examine a quantum circuit on conceptual level and basic level.

Constructing CNOT

We construct the CNOT-gate and its inverse.

Bell states

We work through a qubit-exercise, measuring Bell-states on both conceptual and basic level.


We work through an example of quantum teleportation on both conceptual and basic level.

Last actualization May 2024