Glossary of elementary quantum mechanics

; a complete set of wave functions

: A basis of the Hilbert space of wave functions with respect to a system.

; bra

: The Hermitian conjugate of a ket is called a bra. $\backslash langle\; \backslash alpha|\; =\; (|\backslash alpha\; \backslash rangle)^\backslash dagger$. See "bra–ket notation".

; Bra–ket notation

: The bra–ket notation is a way to represent the states and operators of a system by angle brackets and vertical bars, for example, $|\; \backslash alpha\; \backslash rangle$ and $|\; \backslash alpha\; \backslash rangle\; \backslash langle\; \backslash beta|$.

; Density matrix

: Physically, the density matrix is a way to represent pure states and mixed states. The density matrix of pure state whose ket is $|\backslash alpha\; \backslash rangle$ is $|\backslash alpha\; \backslash rangle\; \backslash langle\; \backslash alpha|$.

: Mathematically, a density matrix has to satisfy the following conditions:

:* $\backslash operatorname\{Tr\}(\backslash rho)\; =\; 1$

:* $\backslash rho^\backslash dagger\; =\; \backslash rho$

; Density operator

: Synonymous to "density matrix".

; Dirac notation

: Synonymous to "bra–ket notation".

; Hilbert space

: Given a system, the possible pure state can be represented as a vector in a Hilbert space. Each ray (vectors differ by phase and magnitude only) in the corresponding Hilbert space represent a state.Exception: superselection rules

; Ket

: A wave function expressed in the form $|a\backslash rangle$ is called a ket. See "bra–ket notation".

; Mixed state

: A mixed state is a statistical ensemble of pure state.

: criterion: {{unbulleted list | style = padding-left: 1.5em;

| Pure state: $\backslash operatorname\{Tr\}(\backslash rho^2)\; =\; 1$

| Mixed state:

}}

; Normalizable wave function

: A wave function $|\backslash alpha\text{'}\; \backslash rangle$ is said to be normalizable if . A normalizable wave function can be made to be normalized by $|a\text{'}\; \backslash rangle\; \backslash to\; \backslash alpha\; =\; \backslash frac\{|\backslash alpha\text{'}\; \backslash rangle\}\{\backslash sqrt\{\backslash langle\; \backslash alpha\text{'}|\backslash alpha\text{'}\; \backslash rangle\}\}$.

; Normalized wave function

: A wave function $|\; a\; \backslash rangle$ is said to be normalized if $\backslash langle\; a|\; a\; \backslash rangle\; =\; 1$.

; Pure state

: A state which can be represented as a wave function / ket in Hilbert space / solution of Schrödinger equation is called pure state. See "mixed state".

; Quantum numbers

: a way of representing a state by several numbers, which corresponds to a complete set of commuting observables.

: A common example of quantum numbers is the possible state of an electron in a central potential: $(n,\; \backslash ell,\; m,\; s)$, which corresponds to the eigenstate of observables $H$ (in terms of $r$), $L$ (magnitude of angular momentum), $L\_z$ (angular momentum in $z$-direction), and $S\_z$.

; Spin wave function

: Part of a wave function of particle(s). See "total wave function of a particle".

; Spinor

: Synonymous to "spin wave function".

; Spatial wave function

: Part of a wave function of particle(s). See "total wave function of a particle".

; State

: A state is a complete description of the observable properties of a physical system.

: Sometimes the word is used as a synonym of "wave function" or "pure state".

; State vector

: synonymous to "wave function".

; Statistical ensemble

: A large number of copies of a system.

; System

: A sufficiently isolated part in the universe for investigation.

; Tensor product of Hilbert space

: When we are considering the total system as a composite system of two subsystems A and B, the wave functions of the composite system are in a Hilbert space $H\_A\; \backslash otimes\; H\_B$, if the Hilbert space of the wave functions for A and B are $H\_A$ and $H\_B$ respectively.

; Total wave function of a particle

: For single-particle system, the total wave function $\backslash Psi$ of a particle can be expressed as a product of spatial wave function and the spinor. The total wave functions are in the tensor product space of the Hilbert space of the spatial part (which is spanned by the position eigenstates) and the Hilbert space for the spin.

; Wave function

: The word "wave function" could mean one of following:

:# A vector in Hilbert space which can represent a state; synonymous to "ket" or "state vector".

:# The state vector in a specific basis. It can be seen as a covariant vector in this case.

:# The state vector in position representation, e.g. $\backslash psi\_\backslash alpha(x\_0)\; =\; \backslash langle\; x\_0\; |\; \backslash alpha\; \backslash rangle$, where $|\; x\_0\; \backslash rangle$ is the position eigenstate.

{{Main|Schrödinger equation}}

; Degeneracy

: See "degenerate energy level".

;Degenerate energy level

: If the energy of different state (wave functions which are not scalar multiple of each other) is the same, the energy level is called degenerate.

: There is no degeneracy in a 1D system.

; Energy spectrum

: The energy spectrum refers to the possible energy of a system.

: For bound system (bound states), the energy spectrum is discrete; for unbound system (scattering states), the energy spectrum is continuous.

:: related mathematical topics: Sturm–Liouville equation

; Hamiltonian $\backslash hat\; H$

: The operator represents the total energy of the system.

; Schrödinger equation

: The Schrödinger equation relates the Hamiltonian operator acting on a wave function to its time evolution (Equation {{EquationRef|1}}): $$i\backslash hbar\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash alpha\backslash rangle\; =\; \backslash hat\; H\; |\; \backslash alpha\; \backslash rangle$$ {{EquationNote|1|Equation (1)}} is sometimes called "Time-Dependent Schrödinger equation" (TDSE).

; Time-Independent Schrödinger Equation (TISE)

: A modification of the Time-Dependent Schrödinger equation as an eigenvalue problem. The solutions are energy eigenstates of the system (Equation {{EquationRef|2}}): $$E\; |\backslash alpha\; \backslash rangle\; =\; \backslash hat\; H\; |\; \backslash alpha\; \backslash rangle$$

In this situation, the SE is given by the form $$i\; \backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash Psi\_\backslash alpha(\backslash mathbf\{r\},\backslash ,t)\; =\; \backslash hat\; H\; \backslash Psi\_\backslash alpha(\backslash mathbf\{r\},\backslash ,t)\; =\; \backslash left(-\backslash frac\{\backslash hbar^2\}\{2m\}\backslash nabla^2\; +\; V(\backslash mathbf\{r\})\backslash right)\backslash Psi\_\backslash alpha(\backslash mathbf\{r\},\backslash ,t)\; =\; -\backslash frac\{\backslash hbar^2\}\{2m\}\backslash nabla^2\backslash Psi\_\backslash alpha(\backslash mathbf\{r\},\backslash ,t)\; +\; V(\backslash mathbf\{r\})\backslash Psi\_\backslash alpha(\backslash mathbf\{r\},\backslash ,t)$$ It can be derived from (1) by considering $\backslash Psi\_\backslash alpha(x,t)\; :=\; \backslash langle\; x\; |\backslash alpha\backslash rangle$ and $\backslash hat\; H\; :=\; -\backslash frac\{\backslash hbar^2\}\{2m\}\; \backslash nabla^2\; +\; \backslash hat\; V$

; Bound state

: A state is called bound state if its position probability density at infinite tends to zero for all the time. Roughly speaking, we can expect to find the particle(s) in a finite size region with certain probability. More precisely, $|\; \backslash psi(\; \backslash mathbf\{r\},\; t)\; |^2\; \backslash to\; 0$ when $|\backslash mathbf\{r\}|\; \backslash to\; +\backslash infty$, for all $t\; >0$.

: There is a criterion in terms of energy:

:: Let $E$ be the expectation energy of the state. It is a bound state if and only if .

; Position representation and momentum representation

:; Position representation of a wave function: $\backslash Psi\_\backslash alpha(x,t)\; :=\; \backslash langle\; x\; |\backslash alpha\backslash rangle$,

:; momentum representation of a wave function: $\backslash tilde\{\backslash Psi\}\_\backslash alpha(p,t)\; :=\; \backslash langle\; p\; |\backslash alpha\backslash rangle$ ;

: where $|\; x\; \backslash rangle$ is the position eigenstate and $|\; p\; \backslash rangle$ the momentum eigenstate respectively.

: The two representations are linked by Fourier transform.

; Probability amplitude

: A probability amplitude is of the form $\backslash langle\; \backslash alpha\; |\backslash psi\backslash rangle$.

; Probability current

: Having the metaphor of probability density as mass density, then probability current $J$ is the current: $J(x,t)\; =\; \backslash frac\{i\; \backslash hbar\}\{2m\}\; \backslash left(\; \backslash psi\; \backslash frac\{\backslash partial\; \backslash psi^*\}\{\backslash partial\; x\}\; -\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; x\}\; \backslash psi\; \backslash right)$ The probability current and probability density together satisfy the continuity equation: $$\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |\backslash psi(x,t)|^2\; +\; \backslash nabla\; \backslash cdot\; \backslash mathbf\{J\}(x,t)\; =\; 0$$

; Probability density

: Given the wave function of a particle, $|\backslash psi(x,t)|^2$ is the probability density at position $x$ and time $t$. $|\backslash psi(x\_0,t)|^2\; \backslash ,\; dx$ means the probability of finding the particle near $x\_0$.

; Scattering state

: The wave function of scattering state can be understood as a propagating wave. See also "bound state".

: There is a criterion in terms of energy:

:: Let $E$ be the expectation energy of the state. It is a scattering state if and only if $E\; >\; \backslash operatorname\{min\}\backslash \{\; V(\; r\; \backslash to\; -\; \backslash infty\; )\; ,\; V(\; r\; \backslash to\; +\; \backslash infty\; )\; \backslash \}$.

; Square-integrable

: Square-integrable is a necessary condition for a function being the position/momentum representation of a wave function of a bound state of the system.

: Given the position representation $\backslash Psi(x,t)$ of a state vector of a wave function, square-integrable means:

:* 1D case: .

:* 3D case: .

; Stationary state

: A stationary state of a bound system is an eigenstate of Hamiltonian operator. Classically, it corresponds to standing wave. It is equivalent to the following things:Some textbooks (e.g. Cohen Tannoudji, Liboff) define "stationary state" as "an eigenstate of a Hamiltonian" without specific to bound states.

:* an eigenstate of the Hamiltonian operator

:* an eigenfunction of Time-Independent Schrödinger Equation

:* a state of definite energy

:* a state which "every expectation value is constant in time"

:* a state whose probability density ($|\backslash psi(x,t)|^2$) does not change with respect to time, i.e. $\backslash frac\{d\}\{dt\}\; |\backslash Psi(x,t)|^2\; =\; 0$

{{Main|Measurement in quantum mechanics}}

; Born's rule

: The probability of the state $|\; \backslash alpha\; \backslash rangle$ collapse to an eigenstate $|\; k\; \backslash rangle$ of an observable is given by $|\backslash langle\; k\; |\; \backslash alpha\; \backslash rangle|^2$.

; Collapse

: "Collapse" means the sudden process which the state of the system will "suddenly" change to an eigenstate of the observable during measurement.

; Eigenstates

: An eigenstate of an operator $A$ is a vector satisfied the eigenvalue equation: $A\; |\backslash alpha\; \backslash rangle\; =\; c\; |\backslash alpha\; \backslash rangle$, where $c$ is a scalar.

: Usually, in bra–ket notation, the eigenstate will be represented by its corresponding eigenvalue if the corresponding observable is understood.

; Expectation value

: The expectation value $\backslash langle\; M\; \backslash rangle$ of the observable M with respect to a state $|\; \backslash alpha$ is the average outcome of measuring $M$ with respect to an ensemble of state $|\; \backslash alpha$.

: $\backslash langle\; M\; \backslash rangle$ can be calculated by: $$\backslash langle\; M\; \backslash rangle\; =\; \backslash langle\; \backslash alpha\; |\; M\; |\; \backslash alpha\; \backslash rangle.$$

: If the state is given by a density matrix $\backslash rho$, $\backslash langle\; M\; \backslash rangle\; =\; \backslash operatorname\{Tr\}(\; M\; \backslash rho)$.

; Hermitian operator

: An operator satisfying $A\; =\; A^\backslash dagger$.

: Equivalently, $\backslash langle\; \backslash alpha\; |\; A|\backslash alpha\; \backslash rangle\; =\; \backslash langle\; \backslash alpha\; |\; A^\backslash dagger\; |\backslash alpha\; \backslash rangle$ for all allowable wave function $|\backslash alpha\backslash rangle$.

; Observable

: Mathematically, it is represented by a Hermitian operator.

; Exchange

; Intrinsically identical particles

: If the intrinsic properties (properties that can be measured but are independent of the quantum state, e.g. charge, total spin, mass) of two particles are the same, they are said to be (intrinsically) identical.

; Indistinguishable particles

: If a system shows measurable differences when one of its particles is replaced by another particle, these two particles are called distinguishable.

; Bosons

:Bosons are particles with integer spin (s = 0, 1, 2, ... ). They can either be elementary (like photons) or composite (such as mesons, nuclei or even atoms). There are five known elementary bosons: the four force carrying gauge bosons γ (photon), g (gluon), Z (Z boson) and W (W boson), as well as the Higgs boson.

; Fermions

:Fermions are particles with half-integer spin (s = 1/2, 3/2, 5/2, ... ). Like bosons, they can be elementary or composite particles. There are two types of elementary fermions: quarks and leptons, which are the main constituents of ordinary matter.

; Anti-symmetrization of wave functions

; Symmetrization of wave functions

; Pauli exclusion principle

; Bose–Einstein distribution

; Bose–Einstein condensation

; Bose–Einstein condensation state (BEC state)

; Fermi energy

; Fermi–Dirac distribution

; Slater determinant

; Entanglement

; Bell's inequality

; Entangled state

; separable state

; no-cloning theorem

; Spin

; angular momentum

; Clebsch–Gordan coefficients

; singlet state and triplet state

; adiabatic approximation

; Born–Oppenheimer approximation

; WKB approximation

; time-dependent perturbation theory

; time-independent perturbation theory

; Ehrenfest theorem

: A theorem connecting the classical mechanics and result derived from Schrödinger equation.

; first quantization

: $x\; \backslash to\; \backslash hat\; x\; ,\; \backslash ,\; p\; \backslash to\; i\; \backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}$

; wave–particle duality

; uncertainty principle

; Canonical commutation relations

: The canonical commutation relations are the commutators between canonically conjugate variables. For example, position $\backslash hat\; x$ and momentum $\backslash hat\; p$: $$[\backslash hat\; x,\; \backslash hat\; p]\; =\; \backslash hat\; x\; \backslash hat\; p\; -\; \backslash hat\; p\; \backslash hat\; x\; =\; i\; \backslash hbar$$

; Path integral

; wavenumber

• Mathematical formulations of quantum mechanics
• List of mathematical topics in quantum theory
• List of quantum-mechanical potentials
• Introduction to quantum mechanics

{{Reflist}}

• Elementary textbooks
• {{cite book | author=Griffiths, David J. | author-link=David J. Griffiths | title=Introduction to Quantum Mechanics | edition=2nd | publisher=Prentice Hall | year=2004 | isbn=0-13-805326-X | url-access=registration | url=https://archive.org/details/introductiontoel00grif_0 }}
• {{cite book | author=Liboff, Richard L. | author-link=Liboff, Richard L. | title=Introductory Quantum Mechanics | publisher=Addison-Wesley | year=2002 | isbn=0-8053-8714-5}}
• {{cite book | author=Shankar, R. | title=Principles of Quantum Mechanics | publisher=Springer | year=1994| isbn=0-306-44790-8}}
• {{cite book

|author1=Claude Cohen-Tannoudji |author2=Bernard Diu |author3=Frank Laloë |title=Quantum Mechanics

|publisher=Wiley-Interscience

|year=2006

|isbn=978-0-471-56952-7}}

• Graduate textook
• {{cite book | author=Sakurai, J. J. | author-link=J. J. Sakurai | title=Modern Quantum Mechanics| publisher=Addison Wesley | year=1994 | isbn=0-201-53929-2}}
• Other
• {{cite book

|editor=Greenberger, Daniel

|editor2=Hentschel, Klaus

|editor3=Weinert, Friedel

|title=Compendium of Quantum Physics - Concepts, Experiments, History and Philosophy

|publisher=Springer

|year=2009

|isbn=978-3-540-70622-9}}

• {{Cite book

| first = Bernard | last = d'Espagnat

| year = 2003

| title = Veiled Reality: An Analysis of Quantum Mechanical Concepts

| edition = 1st

| location = US

| publisher = Westview Press

}}

{{Quantum mechanics topics}}

{{DEFAULTSORT:Glossary Of Elementary Quantum Mechanics}}

Quantum Mechanics, Glossary Of Elementary

Category:Wikipedia glossaries using description lists