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 math necessary.
Susskind's
website is at https://theoreticalminimum.com/
A dictionary of quantum mechanics at beginners’
level you find here: quantum-abc.
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:
The 2D-harmonic
oscillator in cartesian and polar coordinates. |
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We calculate angular
momentum in spherical coordinates both classic and quantum. |
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Two examples for the
chain rule for partial derivatives. |
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Two observables are
simultaneously measurable if and only if the respective matrices commutate. |
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How to calculate
commutators that contain a function of one operator. |
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Solving integrals
containing sin and cos by use of complex functions. |
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This paper describes
how a continuous function can be quantized and how differentiation and
integration are expressed explicitly with matrices. |
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We solve
differential equations of first and second order with emphasis to complex
solutions. |
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The Dirac |
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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. |
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A translation of the
original paper with some comments. |
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From classical
physics to quantum mechanics via the Ehrenfest
Theorem. |
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Power series of
exponentials containing Pauli matrices. |
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How to handle
exponentials with Pauli matrices. |
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How to
solve the time-dependent Schrödinger equation (no potential). Show that there
exist solutions representing plane waves and discuss their physical meaning. |
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How to
solve a wave packet in one dimension without potential by superposition of
plane waves. |
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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. |
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How to Calculate
energy values and eigenfunctions for the one-dimensional potential well. |
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How to
calculate energy values and eigenfunctions for the one-dimensional undamped
harmonic oscillator. |
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The free particle
and its wave function. |
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Basics for
understanding Fourier series |
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Proofs for some
Gauss Integrals. |
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An example that
shows that a 2x2 Hermitian matrix can be transformed in diagonal form. |
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We become acquainted
with Hermite polynomials and proof orthogonality. |
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This paper starts
with a finite dimensional Hilbert space and ends up with the spin of an
electron. |
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We check the
derivation rules with cartesian and polar coordinates. |
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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. |
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This is more or less
an exercise in complex arithmetic. |
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A compilation of
mathematics you need if you want to start with quantum mechanics. |
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Two ways to write a
matrix in a new basis. |
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States with position
and momentum will be described with either position wave functions or
momentum wave functions. This paper presents both descriptions parallel. |
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Uncertainty of
momentum and position Operator in momentum and position representation. |
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We take a close look
at the harmonic oscillator and number operator, raising operator and lowering
operator. |
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An illustration of
the spin problem and the tensor product. |
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Partial derivatives
in cartesian and polar representation. |
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We work with the 1D
step barrier problem and use the probability current to develop reflection
and transmission probability |
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Time development and
Schroedinger equation for constant and commutating
time dependent Hamiltonians. |
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Here you find a
compilation of keywords for quantum mechanics on a basic level. |
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These are the
exercises of the book “Quantum Mechanics, The Theoretical Minimum” of Leonard
Susskind & Art Friedman. |
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A traditional access
to the quantum mechanic oscillator and the Hermite polynomials. |
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We calculate the
ratio of incoming to outgoing wave. |
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Time-independent vs.
time-dependent operators. |
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Two ways to get
time-dependent Heisenberg operators out of time-independent Schroedinger operators. |
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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). |
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We substitute the variable
in a polynomial and an exponential function and take a look at the
derivatives. |
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Tensor products,
correlation, and superposition. |
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The trigonometric
identities resolved in exponential terms. |
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We start from
classic physics, use de Broglie and arrive at the Schroedinger
equation. |
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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. |
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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. |
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We calculate the
speed of the electron of the hydrogen atom by use of the quantum mechanical
Viral theorem. |
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Wave functions
(normalized polynomials) and matrices. |
Quantum
computing:
Four Bell states in
detail. |
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The quantum Fourier
transformation explicit for 1, 2, 3 and 4 qubits. |
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Application of
Simon’s problem to a four-bit and a two-bit function, more mathematical. |
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Practical
application of Simon’s problem from a programmer’s point of view. |
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An alternative
access to the Deutsch-Jozsa algorithm for a 3-qubit input. |
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An explicit and
detailed work-through the Deutsch algorithm. |
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This is a detailed
example of applicating Grover’s algorithm to a SAT problem. |
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List of combinations
of |0>, |1>, |+>, |->, |y+>, |y-> states with the Kronecker-product (tensor-product). |
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Arithmetic with bras
and kets of different dimension |
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A short description
of spin states, the Hadamard Matrix, Bloch sphere and rotation. |
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The Toffoli and the Fredkin quantum gates. |
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We examine a quantum
circuit on conceptual level and basic level. |
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We construct the
CNOT-gate and its inverse. |
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We work through a
qubit-exercise, measuring Bell-states on both conceptual and basic level. |
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We work through an
example of quantum teleportation on both conceptual and basic level. |
Last
actualization July 2024