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 the necessary math.

Susskind's website is at https://theoreticalminimum.com/

You find a dictionary of quantum mechanics at beginners’ level here: quantum-abc.

A philosophical paper dealing with (quantum)physics and the “real world” you find here.

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) gmx.de

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.
Bloch sphere	… with various basis vectors …
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.
Complexifying	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	A bunch of differential equations in physical context.
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.
Ehrenfest	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.
Exponential-spin-operator	How to handle exponentials with Pauli matrices.
FAQ 1	How to solve the time-dependent Schrödinger equation (no potential)?
FAQ 2	How to solve a wave packet in one dimension without potential by superposition of plane waves?
FAQ 3	We have a particle in a 3D-box. Within the box we have potential zero and infinity outside. How to calculate the energy eigenvalues and the eigenfunctions?
FAQ 4	How to Calculate energy values and eigenfunctions for the one-dimensional potential well?
FAQ 5	How to calculate energy values and eigenfunctions for the one-dimensional undamped harmonic oscillator?
FAQ 6	How to calculate the bouncing ball with quantum mechanics?
FAQ 7	Is there an uncertainty relation in the classical harmonic oscillator?
FAQ 8	How to perform a spin flip?
Free particle	The free particle and its wave function.
Fourier series	Basics for understanding Fourier series
Fourier series examples	Two worked through examples for Fourier series.
Gauss Integral	Proofs for some Gauss Integrals.
Hadamard matrices	About the use of Hadamard matrices in error correction.
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	We start with a finite dimensional Hilbert space and end up with the spin of an electron.
Hilbert space II	This paper shows the way from a (real) function to a function in an infinite dimensional Hilbert space suitable for the inner product and the Dirac delta.
Lagrange generalized coordinates	We check the derivation rules with cartesian and polar coordinates.
Lorentz	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.
Momentum-Position	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	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
Quantizing	Time development and Schroedinger equation for constant and commutating time dependent Hamiltonians.
Quantum-abc	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.
Schroedinger-Heisenberg	Time-independent vs. time-dependent operators.
Schroedinger-Heisenberg-twice	Two ways to get time-dependent Heisenberg operators out of time-independent Schroedinger operators.
Spin in magnetic field	We look at a spin ½ in a magnetic field: - constant in direction and strength, - constant in direction but slowly changing its strength, - rotating (NMR).
Spin and density matrix	The equations of motion by use of density matrices.
Spin and qubits	A precessing spin represents a qubit, we calculate probabilities for spin-up and spin-down.
Substituting variables	We substitute the variable in a polynomial and an exponential function and take a look at the derivatives.
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:

Basic vs. conceptual access	We examine a quantum circuit on conceptual level and basic level.
Bell states	Four Bell states in detail.
Bell states	We work through a qubit-exercise, measuring Bell-states on both conceptual and basic level.
Constructing CNOT	We construct the CNOT-gate and its inverse.
Deutsch-Jozsa	An alternative access to the Deutsch-Jozsa algorithm for a 3-qubit input.
Grover example	A numerical example for applying Grover’s algorithm.
Grover’s algorithm	This is a detailed example of applicating Grover’s algorithm to a SAT problem.
Grover’s algorithm Berti	This is the same example as above, solved by Alessandro Berti. The circuit used is different, the result the same.
Kronecker_product	List of combinations of |0>, |1>, |+>, |->, |y+>, |y-> states with the Kronecker-product (tensor-product).
No cloning theorem	A proof of the no cloning theorem
Partial scalar product	Arithmetic with bras and kets of different dimension
Quantum Fourier	The quantum Fourier transformation explicit for 1, 2, 3 and 4 qubits.
Quantum parallelism	An explicit and detailed work-through the Deutsch algorithm.
Reversible gates	The Toffoli and the Fredkin quantum gates.
Simon’s problem 2	Practical application of Simon’s problem from a programmer’s point of view.
Simon’s_problem	Application of Simon’s problem to a four-bit and a two-bit function, more mathematical.
Spin flip	If you are interested in how a spin flip physically can be performed, you will find a description here.
Spin states	A short description of spin states, the Hadamard Matrix, Bloch sphere and rotation.
Teleportation 1	We work through an example of quantum teleportation on both conceptual and basic level.
Teleportation 2	A second example with Alice and Bob

Last actualization March 2025