# Index
[[Analytically solvable systems in quantum mechanics]]
[[Anti-commutators in quantum mechanics]]
[[Approximation methods in quantum mechanics]]
[[Closed quantum systems]]
[[Composite quantum systems]]
[[Commutators in quantum mechanics]]
[[Continuous eigenstate quantum systems]]
[Correspondance Principle](Correspondence%20Principle)
[[density matrix]]
[[Discrete eigenstate quantum systems]]
[[Ensembles of quantum systems]]
[Entanglement](Entanglement)
[[Groenewold-van Hove no-go theorem]]
[Hilbert space dimension in quantum mechanics](Hilbert%20space%20dimension%20in%20quantum%20mechanics.md)
[Hilbert Spaces in Quantum Mechanics](Hilbert%20Spaces%20in%20Quantum%20Mechanics)
[[inner products between state vectors]]
[[Linear transformations in quantum mechanics]]
[Mathematical Formulations of Quantum Mechanics](Mathematical%20Formulations%20of%20Quantum%20Mechanics)
[[matter waves]]
[mixed state](mixed%20state.md)
[[n-level system]]
[Open quantum systems](Open%20quantum%20systems.md)
[[Orthonormal bases in quantum mechanics]]
[[overlap]]
[[Postulates of Quantum Mechanics]]
[Pure state](Pure%20state.md)
[Quantization](Quantization.md)
[[Quantum harmonic oscillators]]
[Quantum mechanical particles](Quantum%20mechanical%20particles.md)
[[Quantum perturbation theory]]
[Quantum superposition](Quantum%20superposition.md)
[[Quantum systems]]
[[Schrödinger–HJW theorem]]
[[Separable states]]
[[State vector]]
[[state vector normalization]]
[[Three-Level Systems]]
[Two-Level Systems](Two-Level%20Systems.md)
[[unitary transformations in quantum mechanics]]
[[Wavefunction]]
## Sub-indices
[[Experiments in Quantum Mechanics (index)]]
[[Interpretations and Foundations of Quantum Mechanics (index)]]
[[Multi-particle quantum systems (index)]]
[[Quantum Dynamics (index)]]
[[Quantum Field Theory (Index)]]
[[Quantum measurement (index)]]
[[Stationary Quantum Systems (Index)]]
### Applications of Quantum Mechanics
[Atomic Physics (Index)](Atomic%20Physics%20(Index).md)
[[Elementary Particles (index)]]
[Quantum Information (index)](Quantum%20Information%20(index).md)
[[Quantum Optics (index)]]
---
# Basic Concepts
On this page we provide some introductory comments that contextualize the role of quantum physics and provide an overview of the structure of quantum physics in order to help guide the reader through its various topics.
On this page we also introduce a few key elements of ["old" quantum Theory](Quantum%20Mechanics%20(index).md#Old%20Quantum%20Theory) (that is, key principles and results that precede what we describe as [modern quantum theory](Quantum%20Mechanics%20(index).md#General%20structure%20of%20modern%20quantum%20theory)) along with some concepts many students going into an introductory course in quantum mechanics may be familiar with from other introductory science classes. As needed, similar references to old quantum theory will be discussed in sections included in the [sub-indices](Quantum%20Mechanics%20(index).md#Sub-indices) above.
## Old Quantum Theory
%%Here we don't go into all the results of old quantum theory but list out some of the still physically relevant models. Here many of the models will link to the index on atomic physics.%%
## General structure of modern quantum theory
There's no one way to organize topics in quantum mechanics, but I generally follow categorizations typical of what many introductory textbooks use.
In this top level [Index](Quantum%20Mechanics%20(index).md#Index) I list topics that I consider most fundamental towards understanding non-relativistic quantum theory and that most closely follow from or more importantly elaborate on the [postulates of quantum Mechanics.](Postulates%20of%20Quantum%20Mechanics.md) By ignoring relativity in the top level index, this lays bare the ideas that are unique and specific to quantum physics as being central to this section of [The Quantum Well](Welcome%20to%20The%20Quantum%20Well!.md).
### [Stationary Quantum Systems](Stationary%20Quantum%20Systems%20(Index).md) and [Quantum Dynamics](Quantum%20Dynamics%20(index).md)
.md#^afe0e2)
These freeze frames are then animated in the section dubbed [Quantum Dynamics.](Quantum%20Dynamics%20(index).md) Here we account for the passage of time in the evolution of state vectors as well as show how two quantum systems interact with each other under both semi-classical and fully quantum mechanical models.
### Quantum Mechanical systems
%%Here zero in on what makes something quantum%%
### [Quantum Measurement](Quantum%20measurement%20(index).md)
.md#^372dca)
### [Quantum Field Theory](Quantum%20Field%20Theory%20(Index).md)
## Planck's Constant
## Debroiglie Waves
The _Debroiglie wavelength_ of an object with mass $m$ is $\lambda=\frac{h}{mv}=\frac{h}{p}$ where $h$ is [Planck's constant](Quantum%20Mechanics%20(index).md#Planck's%20Constant) and here we also rewrite in terms of [momentum](Mechanics%20(index).md#momentum) [$p=mv.$](Mechanics%20(index)#^1fc95c) ^39bbae
The [Debroiglie relation above,](Quantum%20mechanics%20(index)#^39bbae) establishes the notion of _wave-particle duality,_ where $\lambda$ is the wavelength of a so called [matter wave.](matter%20waves.md)
It is believed that the [Debroiglie relation](Quantum%20mechanics%20(index)#^39bbae) applies to all matter regardless of scale. This notion is sometimes referred to as the _DeBroiglie hypothesis._
# Mathematical preliminaries
## Complex numbers in quantum mechanics
## Operators and vectors in quantum mechanics
### Bra-ket notation
Bra-ket notation refers to a way of writing [vectors](State%20vector.md) that represent [Quantum systems](Quantum%20systems.md) and products of those vectors in [quantum mechanics.](Quantum%20Mechanics%20(index).md)
### Operators in quantum mechanics
The notation convention we follow for operators in quantum mechanics is to write them with hats. E.g. $\hat{A},$ $\hat{x}.$ Like with how they are expressed in other areas, these operators are often capitalized, however, this is not often the case in quantum physics, which may be one reason to write hats. The hat is merely to distinguish operators from physical parameters and constants which may be either capitalized or lowercase in accordance with typical conventions. ^bff927
## "Non-relativistic" vs. "relativistic quantum mechanics"
%%# German-English Glossary of terms
* _der Zustand_ - state (quantum mechanics)
* _der Strahl_ - ray (geometry), beam (e.g. light beam)
* _der Dichteoperator_ - density matrix%%
---
# Bibliography
Altland A., Simons B., _Condensed Matter Field Theory_
.pdf)
[Ashcroft N. W., Mermin, N. D.,_Solid State Physics_, Harcourt College Publishers, 1976.](Ashcroft%20N.%20W.,%20Mermin,%20N.%20D.,%20Solid%20State%20Physics,%20Harcourt%20College%20Publishers,%201976..md)
[Born, M., Zur Quantenmechanik der Stoßvorgänge, _Zeitschrift für Physik_ 37, 863-867, 1926.](Born,%20M.,%20Zur%20Quantenmechanik%20der%20Stoßvorgänge,%20Zeitschrift%20für%20Physik%2037,%20863-867,%201926..md)
[Brillouin zones, from University of Cambridge -- DoITPoMS online resources, 2008.](Brillouin%20zones,%20from%20University%20of%20Cambridge%20--%20DoITPoMS%20online%20resources,%202008..md)
[Brunner, I., TM1 Quantum Mechanics II, Tutorial 5, Winter 2017-2018](Brunner,%20I.,%20TM1%20Quantum%20Mechanics%20II,%20Tutorial%205,%20Winter%202017-2018.md)
[Cage, M. E., Dziuba, R.F., Bruce, F. F., A Test of the quantum Hall Effect as a resistance standard, _IEEE Transactions on Instrumentation and Measurement_, 1985.](Cage,%20M.%20E.,%20Dziuba,%20R.F.,%20Bruce,%20F.%20F.,%20A%20Test%20of%20the%20quantum%20Hall%20Effect%20as%20a%20resistance%20standard,%20IEEE%20Transactions%20on%20Instrumentation%20and%20Measurement,%201985..md)
Commins, E.D. _Quantum Mechanics: And Experimentalist Approach_
.pdf)
Fox, M., _Quantum Optics: An Introduction_
.pdf)
Schulman L.S., _Techniques and Applications of Path Integration_
%20(2005,%20Dover%20Publications)%20-%20libgen.lc.pdf)
Metcalf, H. J., van der Straten P., _Laser cooling and trapping_
%20-%20libgen.lc.pdf)
R. P. Feynman, Rev. of Mod. Phys., 20, 367
Preskill J. _Lecture Notes for Physics 229: Quantum Information and Computation_ (1998)
[Schollwöck, U. _Grundlegende Konzepte_ 1-2, T2, _Quantenmechanik_, Lecture Notes, Winter 2019-2020.](Schollwöck,%20U.%20Grundlegende%20Konzepte%201-2,%20T2,%20Quantenmechanik,%20Lecture%20Notes,%20Winter%202019-2020..md)
Schollwöck, U. _Grundlegende Konzepte 2_, T2: Quantenmechanik Lecture Notes, Winter 2019/2020 (German)
Schollwöck, U. _Grundlegende Formulismus 1_, T2: Quantenmechanik Lecture Notes, Winter 2019/2020 (German)
Wheeler N., _Remarks Concerning the status & some ramifications of Ehrenfest's Theorem_
Zee A. _Quantum Field Theory in a Nutshell_
%20-%20libgen.lc.pdf)
Mukhanov V., Homework 2, Quantum Mechanics 2 (2019/2020)
Mukhanov V., Homework 10, Quantum Mechanics 2 (2019/2020)
Dirac, P. A. M., _The Principles of Quantum Mechanics_, Oxford Univ Press, Revised 4th edition (1978)
%20-%20libgen.lc.pdf)
Lamb., W.E. _Antiphoton_ Appl. Phys. B 60, 77-84 (1995)
Loudon R., _Quantum Theory of Light_
.pdf)
Borenstein, M., Lamb, W.E. _Classical Laser_ Phys. Rev. A 5, 1298
Scully M. O., Zubairy M. S. _Quantum Optics_
.pdf)
Allen L., Eberly J.H., _Optical Resonance and Two-Level Atoms_
%20-%20libgen.lc.pdf)
Everett H., _The Theory of the Universal Wave Function_
%201973%20version.pdf)
Hoffmann S., _Exercises on Quantum Mechanics, Problem sheet 8_, Quantum Mechanics II (TM1/TV), 2020/2021
[Baym, G., _Lectures on Quantum Mechanics_, Westview Press, 1990](Baym,%20G.,%20Lectures%20on%20Quantum%20Mechanics,%20Westview%20Press,%201990.md)
[Franklin, J., Harmonic Oscillator Physics, Lecture 9, Physics 342, _Quantum Mechanics I_, Lecture Notes, Spring 2010](Franklin,%20J.,%20Harmonic%20Oscillator%20Physics,%20Lecture%209,%20Physics%20342,%20Quantum%20Mechanics%201,%20Lecture%20Notes,%20Spring%202010.md)
[Griffiths D. J., _Introduction to Quantum Mechanics_, Pearson Prentice Hall, 2nd edition, 2005.](Griffiths%20D.%20J.,%20Introduction%20to%20Quantum%20Mechanics,%20Pearson%20Prentice%20Hall,%202nd%20edition,%202005..md)
[Groenwold, H.J., _On the principles of elementary quantum mechanics_, Physica, vol. 12., 1946.](Groenwold,%20H.J.,%20On%20the%20principles%20of%20elementary%20quantum%20mechanics,%20Physica,%20vol.%2012.,%201946..md)
[Grushin A., _Introduction to topological phases in condensed matter_, 2022.](Grushin%20A.,%20Introduction%20to%20topological%20phases%20in%20condensed%20matter,%20Les%20Houches%20doctoral%20school,%20Frontiers%20of%20Condensed%20Matter,%202022..md)
[Hanbury Brown R., Twiss, _A test of a new type of type of stellar interferometer on Sirius_, Nature 178, 1956.](Hanbury%20Brown%20R.,%20Twiss,%20A%20test%20of%20a%20new%20type%20of%20type%20of%20stellar%20interferometer%20on%20Sirius,%20Nature%20178,%201956..md)
[Hughston L. P., A complete classification of quantum ensembles having a given density matrix, _Phys Lett A 183_, 1993.](Hughston%20L.%20P.,%20A%20complete%20classification%20of%20quantum%20ensembles%20having%20a%20given%20density%20matrix,%20Phys%20Lett%20A%20183.%201993..md)
[Kittel, C., _Introduction to Solid State Physics_, Wiley & Sons, 8th edition, 2005](Kittel,%20C.,%20Introduction%20to%20Solid%20State%20Physics,%20Wiley%20&%20Sons,%208th%20edition,%202005..md)
[Kronig R. de L., Penney W. G., Quantum mechanics of electrons in crystal lattices, _Procedings of the Royal Society of London A_ 130, 499-513, 1931.](Kronig%20R.%20de%20L.,%20Penney%20W.%20G.,%20Quantum%20mechanics%20of%20electrons%20in%20crystal%20lattices,%20Procedings%20of%20the%20Royal%20Society%20of%20London%20A%20130,%20499-513,%201931..md)
[Likharev, K., _Essential Graduate Physics-Quantum Mechanics_, LibreTexts, 2024](Likharev,%20K.,%20Essential%20Graduate%20Physics-Quantum%20Mechanics,%20LibreTexts,%202024.md)
[McGreevy, John. A., Physics 212A Lecture Notes, Fall 2015.](McGreevy,%20John.%20A.,%20Physics%20212A%20Lecture%20Notes,%20Fall%202015..md)
[Mandel L., Wolf E., _Optical Coherence and Quantum Optics_, Cambridge University Press, 1995.](Mandel%20L.,%20Wolf%20E.,%20Optical%20Coherence%20and%20Quantum%20Optics,%20Cambridge%20University%20Press,%201995..md)
[Moyal J. E., _Quantum mechanics as a statistical theory_, Mathematical Proceedings of the Cambridge Philosophical Society, 1949.](Moyal%20J.%20E.,%20Quantum%20mechanics%20as%20a%20statistical%20theory,%20Mathematical%20Proceedings%20of%20the%20Cambridge%20Philosophical%20Society,%201949..md)
[Nielson, M. A., Chuang, I. L. _Quantum Computation and Quantum Information_, Cambridge University Press, 2010](Nielsen,%20M.%20A.,%20Chuang,%20I.%20L.%20Quantum%20Computation%20and%20Quantum%20Information,%20Cambridge%20University%20Press,%202010.md)
[Peres, A., _Quantum Theory Concepts and Methods_, Kluwer Academic Publishers, 2002](Peres,%20A.,%20Quantum%20Theory%20Concepts%20and%20Methods,%20Kluwer%20Academic%20Publishers,%202002.md)
[Sakurai, J. J., Napolitano J., Modern Quantum Mechanics, Addison-Wesley, 2nd Edition, 2011](Sakurai,%20J.%20J.,%20Napolitano%20J.,%20Modern%20Quantum%20Mechanics,%20Addison-Wesley,%202nd%20Edition,%202011.md)
[Schollwöck, U. Grundlegende Formulismus 1-2, T2, _Quantenmechanik_, Lecture Notes, Winter 2019-2020.](Schollwöck,%20U.%20Grundlegende%20Formulismus%201-2,%20T2,%20Quantenmechanik,%20Lecture%20Notes,%20Winter%202019-2020..md)
[Schollwöck, U. Homework 0, _Quantum Mechanics 1_ (German) (2019-2020)](Schollwöck,%20U.%20Homework%200,%20Quantum%20Mechanics%201%20(German)%20(2019-2020).md)
[Schollwöck, U, J. Homework 2, Quantum Mechanics 1 (German), (2019-2020)](Schollwöck,%20U,%20J.%20Homework%202,%20Quantum%20Mechanics%201%20(German)%20(2019,2020).md)
[Shankar, R., _Principles of Quantum Mechanics_, Plenum Press, 2nd edition, 1994.](Shankar,%20R.,%20Principles%20of%20Quantum%20Mechanics,%20Plenum%20Press,%202nd%20edition,%201994..md)
[Söderholm, J., Björk, G., Trifonov, Unpolarized light in quantum optics, arXivquant-ph0007099, 2000](Söderholm,%20J.,%20Björk,%20G.,%20Trifonov,%20Unpolarized%20light%20in%20quantum%20optics,%20arXivquant-ph0007099,%202000.md)
[Steck, D. A., _Quantum and Atom Optics_, 2007 (last revised in September 2021)](Steck,%20D.%20A.,Quantum%20and%20Atom%20Optics,%202007%20(last%20revised%20September%202021).md)
[von Neumann J., _Mathematical Foundations of Quantum Mechanics_. Translated by Robert T. Beyer. Princeton University Press, 2018.](von%20Neumann%20J.,%20Mathematical%20Foundations%20of%20Quantum%20Mechanics.%20Translated%20by%20Robert%20T.%20Beyer.%20Princeton%20University%20Press,%202018..md)
[Woit, Peter. _Quantum Theory, Groups and Representations: An Introduction_. Springer. 2017.](Woit,%20Peter.%20Quantum%20Theory,%20Groups%20and%20Representations%20An%20Introduction,%20Springer,%202017.md)
[Wheeler, J. A., Zurek, W. H., _Quantum Theory and Measurement_, Princeton University Press,1983.](Wheeler,%20J.%20A.,%20Zurek,%20W.%20H.,%20Quantum%20Theory%20and%20Measurement,%20Princeton%20University%20Press,1983..md)
[Wooters, W. K., Zurek W. H., _A single quantum cannot be cloned_, Nature, 299, 1982](Wooters,%20W.%20K.,%20Zurek%20W.%20H.,%20A%20single%20quantum%20cannot%20be%20cloned,%20Nature,%20299,%201982.md)
#QuantumMechanics/TwoLevelSystems
#QuantumMechanics/ApproximationMethods
#QuantumMechanics/ApproximationMethods/QuantumPerturbationTheory
#QuantumMechanics/FoundationsOfQuantumMechanics
#QuantumMechanics/MathematicalFoundations
#QuantumMechanics/Scattering
#QuantumMechanics/QuantumStateRepresentations/StateVectors
#QuantumMechanics/QuantumStateRepresentations/Wavefunctions
#QuantumMechanics/QuantumStateRepresentations/DensityMatrices
#QuantumMechanics/StationaryStateQuantumSystems
#QuantumMechanics/StationaryStateQuantumSystems/QuantizedAngularMomentum
#QuantumMechanics/StationaryStateQuantumSystems/QuantumHarmonicOscillators
#QuantumMechanics/QuantumOptics
#PhysicalExamples/PhysicalExamplesInQuantumMechanics/PhysicalExamplesInQuantumOptics
#PhysicalExamples/PhysicalExamplesInQuantumMechanics
#ExperimentalTools/QuantumMechanicsExperimentalTools/QuantumOpticsExperimentalTools
#PhysicalExamples/PhysicalExamplesInQuantumMechanics/PhysicalExamplesInQuantumOptics
#ExperimentalTools/QuantumMechanicsExperimentalTools/QuantumOpticsExperimentalTools
#QuantumMechanics/QuantumMeasurement
#QuantumMechanics/QuantumMeasurement/QuantumObservables
#QuantumMechanics/QuantumInformation
#QuantumMechanics/QuantumInformation/Entanglement
#QuantumMechanics/QuantumInformation/QuantumComputation
#QuantumMechanics/QuantumInformation/QuantumComputation/QuantumErrorCorrection
#QuantumMechanics/MathematicalFoundations/QuantumInformation
#QuantumMechanics/RelativisticQuantumMechanics
#QuantumMechanics/QuantumFieldTheory
#QuantumMechanics/QuantumDynamics
#QuantumMechanics/QuantumDynamics/PathIntegrals
#QuantumMechanics/QuantumDynamics/TwoLevelSystems
#QuantumMechanics/QuantumDynamics/QuantumHarmonicOscillators
#QuantumMechanics/MultiParticleQuantumSystems
#QuantumMechanics/MultiParticleQuantumSystems/SolidStatePhysics
#QuantumMechanics/MultiParticleQuantumSystems/SolidStatePhysics/lattices
#QuantumMechanics/MultiParticleQuantumSystems/Quasiparticles
#PhysicalExamples/PhysicalExamplesInQuantumMechanics/PhysicalExamplesInSolidStatePhysics
#QuantumMechanics/InterpretationsOfQuantumMechanics
#QuantumMechanics/RelativisticQuantumMechanics/ElementaryParticles
#QuantumMechanics/AtomicPhysics
#QuantumMechanics/MultiParticleQuantumSystems/QuantumStatisticalMechanics
#QuantumMechanics/MultiParticleQuantumSystems/QuantumThermodynamics
#Bibliography
#index