Glossary of elementary quantum mechanics
This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.
Cautions:
 Different authors may have different definitions for the same term.
 The discussions are restricted to Schrödinger picture and nonrelativistic quantum mechanics.
 Notation:
  position eigenstate
  wave function of the state of the system
  total wave function of a system
  wave function of a system (maybe a particle)
  wave function of a particle in position representation, equal to
Formalism
Kinematical postulates
 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. . 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, and .
 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 is .
 Mathematically, a density matrix has to satisfy the following conditions:
 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.^{[nb 1]}
 Ket
 A wave function expressed in the form is called a ket. See "bra–ket notation".
 Mixed state
 A mixed state is a statistical ensemble of pure state.
 criterion:
 Pure state:
 Mixed state:
 Normalizable wave function
 A wave function is said to be normalizable if . A normalizable wave function can be made to be normalized by .
 Normalized wave function
 A wave function is said to be normalized if .
 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: , which corresponds to the eigenstate of observables (in terms of ), (magnitude of angular momentum), (angular momentum in direction), and .
 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 , if the Hilbert space of the wave functions for A and B are and respectively.
 Total wave function of a particle
 For singleparticle system, the total wave function 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. , where is the position eigenstate.
Dynamics
 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 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
 The operator represents the total energy of the system.
 Schrödinger equation
  (1)
 (1) is sometimes called "TimeDependent Schrödinger equation" (TDSE).
 TimeIndependent Schrödinger Equation (TISE)
 A modification of the TimeDependent Schrödinger equation as an eigenvalue problem. The solutions are energy eigenstate of the system.
  (2)
Dynamics related to single particle in a potential / other spatial properties
 In this situation, the SE is given by the form
 It can be derived from (1) by considering and
 In this situation, the SE is given by the form
 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, when , for all .
 There is a criterion in terms of energy:
 Let be the expectation energy of the state. It is a bound state iff .
 Position representation and momentum representation
 Position representation of a wave function: ,
 momentum representation of a wave function: ;
 where is the position eigenstate and the momentum eigenstate respectively.
 The two representations are linked by Fourier transform.
 Probability amplitude
 A probability amplitude is of the form .
 Probability current
 Having the metaphor of probability density as mass density, then probability current
is the current:
 The probability current and probability density together satisfy the continuity equation:
 Probability density
 Given the wave function of a particle, is the probability density at position and time . means the probability of finding the particle near .
 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 be the expectation energy of the state. It is a scattering state iff .
 Squareintegrable
 Squareintegrable 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
of a state vector of a wave function, squareintegrable 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:^{[nb 2]}
 an eigenstate of the Hamiltonian operator
 an eigenfunction of TimeIndependent Schrödinger Equation
 a state of definite energy
 a state which "every expectation value is constant in time"
 a state whose probability density ( ) does not change with respect to time, i.e.
Measurement postulates
 Born's rule
 The probability of the state collapse to an eigenstate of an observable is given by .
 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 is a vector satisfied the eigenvalue equation: , where 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 of the observable M with respect to a state is the average outcome of measuring with respect to an ensemble of state .

can be calculated by:
 .
 If the state is given by a density matrix , .
 Hermitian operator
 An operator satisfying .
 Equivalently, for all allowable wave function .
 Observable
 Mathematically, it is represented by a Hermitian operator.
Indistinguishable particles
 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 halfinteger 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.
 Antisymmetrization of wave functions
 Symmetrization of wave functions
Quantum statistical mechanics
 Bose–Einstein distribution
 Bose–Einstein condensation
 Bose–Einstein condensation state (BEC state)
 Fermi energy
 Fermi–Dirac distribution
 Slater determinant
Nonlocality
Rotation: spin/angular momentum
Approximation methods
Historical Terms / semiclassical treatment
 Ehrenfest theorem
 A theorem connecting the classical mechanics and result dervied from Schrödinger equation.
 first quantization
 wave–particle duality
Uncategorized terms
See also
 Mathematical formulations of quantum mechanics
 List of mathematical topics in quantum theory
 List of quantummechanical potentials
 Introduction to quantum mechanics
Notes
References
 Elementary textbooks
 Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.
 Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.
 Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN 0306447908.
 Claude CohenTannoudji; Bernard Diu; Frank Laloë (2006). Quantum Mechanics. WileyInterscience. ISBN 9780471569527.
 Graduate textook
 Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison Wesley. ISBN 0201539292.
 Other
 Greenberger, Daniel; Hentschel, Klaus; Weinert, Friedel (Eds.) (2009). Compendium of Quantum Physics  Concepts, Experiments, History and Philosophy. Springer. ISBN 9783540706229.
 d'Espagnat, Bernard (2003). Veiled Reality: An Analysis of Quantum Mechanical Concepts (1st ed.). US: Westview Press.
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