# Glossary of elementary quantum mechanics

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This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

## Contents

Cautions:

• Different authors may have different definitions for the same term.
• The discussions are restricted to Schrödinger picture and non-relativistic quantum mechanics.
• Notation:
• ${\displaystyle |x\rangle }$ - position eigenstate
• ${\displaystyle |\alpha \rangle ,|\beta \rangle ,|\gamma \rangle ...}$ - wave function of the state of the system
• ${\displaystyle \Psi }$ - total wave function of a system
• ${\displaystyle \psi }$ - wave function of a system (maybe a particle)
• ${\displaystyle \psi _{\alpha }(x,t)}$ - wave function of a particle in position representation, equal to ${\displaystyle \langle x|\alpha \rangle }$

## 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. ${\displaystyle \langle \alpha |=(|\alpha \rangle )^{\dagger }}$. 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, ${\displaystyle |\alpha \rangle }$ and ${\displaystyle |\alpha \rangle \langle \beta |}$.
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 ${\displaystyle |\alpha \rangle }$ is ${\displaystyle |\alpha \rangle \langle \alpha |}$.
Mathematically, a density matrix has to satisfy the following conditions:
• ${\displaystyle \operatorname {Tr} (\rho )=1}$
• ${\displaystyle \rho ^{\dagger }=\rho }$
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. [nb 1]
Ket
A wave function expressed in the form ${\displaystyle |a\rangle }$ is called a ket. See "bra–ket notation".
Mixed state
A mixed state is a statistical ensemble of pure state.
criterion:
Pure state: ${\displaystyle \operatorname {Tr} (\rho ^{2})=1}$
Mixed state: ${\displaystyle \operatorname {Tr} (\rho ^{2})<1}$
Normalizable wave function
A wave function ${\displaystyle |\alpha '\rangle }$ is said to be normalizable if ${\displaystyle \langle \alpha '|\alpha '\rangle <\infty }$. A normalizable wave function can be made to be normalized by ${\displaystyle |a'\rangle \to \alpha ={\frac {|\alpha '\rangle }{\sqrt {\langle \alpha '|\alpha '\rangle }}}}$.
Normalized wave function
A wave function ${\displaystyle |a\rangle }$ is said to be normalized if ${\displaystyle \langle a|a\rangle =1}$.
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: ${\displaystyle (n,l,m,s)}$, which corresponds to the eigenstate of observables ${\displaystyle H}$ (in terms of ${\displaystyle r}$), ${\displaystyle L}$ (magnitude of angular momentum), ${\displaystyle L_{z}}$ (angular momentum in ${\displaystyle z}$-direction), and ${\displaystyle S_{z}}$.
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 ${\displaystyle H_{A}\otimes H_{B}}$, if the Hilbert space of the wave functions for A and B are ${\displaystyle H_{A}}$ and ${\displaystyle H_{B}}$ respectively.
Total wave function of a particle
For single-particle system, the total wave function ${\displaystyle \Psi }$ 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:
1. A vector in Hilbert space which can represent a state; synonymous to "ket" or "state vector".
2. The state vector in a specific basis. It can be seen as a covariant vector in this case.
3. The state vector in position representation, e.g. ${\displaystyle \psi _{\alpha }(x_{0})=\langle x_{0}|\alpha \rangle }$, where ${\displaystyle |x_{0}\rangle }$ 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 ${\displaystyle {\hat {H}}}$
The operator represents the total energy of the system.
Schrödinger equation
${\displaystyle i\hbar {\frac {\partial }{\partial t}}|\alpha \rangle ={\hat {H}}|\alpha \rangle }$ -- (1)
(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 eigenstate of the system.
${\displaystyle E\alpha \rangle ={\hat {H}}|\alpha \rangle }$ -- (2)
In this situation, the SE is given by the form
${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi _{\alpha }(\mathbf {r} ,\,t)={\hat {H}}\Psi =\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V(\mathbf {r} )\right)\Psi _{\alpha }(\mathbf {r} ,\,t)=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\Psi _{\alpha }(\mathbf {r} ,\,t)+V(\mathbf {r} )\Psi _{\alpha }(\mathbf {r} ,\,t)}$
It can be derived from (1) by considering ${\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }$ and ${\displaystyle {\hat {H}}:=-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\hat {V}}}$
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, ${\displaystyle |\psi (\mathbf {r} ,t)|^{2}\to 0}$ when ${\displaystyle |\mathbf {r} |\to +\infty }$, for all ${\displaystyle t>0}$.
There is a criterion in terms of energy:
Let ${\displaystyle E}$ be the expectation energy of the state. It is a bound state iff ${\displaystyle E<\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}$.
Position representation and momentum representation
Position representation of a wave function: ${\displaystyle \Psi _{\alpha }(x,t):=\langle x|\alpha \rangle }$,
momentum representation of a wave function: ${\displaystyle {\tilde {\Psi }}_{\alpha }(p,t):=\langle p|\alpha \rangle }$ ;
where ${\displaystyle |x\rangle }$ is the position eigenstate and ${\displaystyle |p\rangle }$ the momentum eigenstate respectively.
The two representations are linked by Fourier transform.
Probability amplitude
A probability amplitude is of the form ${\displaystyle \langle \alpha |\psi \rangle }$.
Probability current
Having the metaphor of probability density as mass density, then probability current ${\displaystyle J}$ is the current:
${\displaystyle J(x,t)={\frac {i\hbar }{2m}}(\psi {\frac {\partial \psi ^{*}}{\partial x}}-{\frac {\partial \psi }{\partial x}}\psi )}$
The probability current and probability density together satisfy the continuity equation:
${\displaystyle {\frac {\partial }{\partial t}}|\psi (x,t)|^{2}+\nabla \cdot \mathbf {J(x,t)} =0}$
Probability density
Given the wave function of a particle, ${\displaystyle |\psi (x,t)|^{2}}$ is the probability density at position ${\displaystyle x}$ and time ${\displaystyle t}$. ${\displaystyle |\psi (x_{0},t)|^{2}\,dx}$ means the probability of finding the particle near ${\displaystyle x_{0}}$.
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 ${\displaystyle E}$ be the expectation energy of the state. It is a scattering state iff ${\displaystyle E>\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\}}$.
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 ${\displaystyle \Psi (x,t)}$ of a state vector of a wave function, square-integrable means:
1D case: ${\displaystyle \int _{-\infty }^{+\infty }|\Psi (x,t)|^{2}\,dx<+\infty }$.
3D case: ${\displaystyle \int _{V}|\Psi (\mathbf {r} ,t)|^{2}\,dV<+\infty }$.
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 Time-Independent Schrödinger Equation
• a state of definite energy
• a state which "every expectation value is constant in time"
• a state whose probability density (${\displaystyle |\psi (x,t)|^{2}}$) does not change with respect to time, i.e. ${\displaystyle {\frac {d}{dt}}|\Psi (x,t)|^{2}=0}$

### Measurement postulates

Born's rule
The probability of the state ${\displaystyle |\alpha \rangle }$ collapse to an eigenstate ${\displaystyle |k\rangle }$ of an observable is given by ${\displaystyle |\langle k|\alpha \rangle |^{2}}$.
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 ${\displaystyle A}$ is a vector satisfied the eigenvalue equation: ${\displaystyle A|\alpha \rangle =c|\alpha \rangle }$, where ${\displaystyle c}$ 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 ${\displaystyle }$ of the observable M with respect to a state ${\displaystyle |\alpha }$ is the average outcome of measuring ${\displaystyle M}$ with respect to an ensemble of state ${\displaystyle |\alpha }$ .
${\displaystyle }$ can be calculated by:
${\displaystyle =\langle \alpha |M|\alpha \rangle }$.
If the state is given by a density matrix ${\displaystyle \rho }$, ${\displaystyle =\operatorname {Tr} (M\rho )}$.
Hermitian operator
An operator satisfying ${\displaystyle A=A^{\dagger }}$.
Equivalently, ${\displaystyle \langle \alpha |A|\alpha \rangle =\langle \alpha |A^{\dagger }|\alpha \rangle }$ for all allowable wave function ${\displaystyle |\alpha \rangle }$.
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 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

### Quantum statistical mechanics

Bose–Einstein distribution
Bose–Einstein condensation
Bose–Einstein condensation state (BEC state)
Fermi energy
Fermi–Dirac distribution
Slater determinant

## Nonlocality

Entanglement
Bell's inequality
Entangled state
separable state
no cloning theorem

## Rotation: spin/angular momentum

Spin
angular momentum
Clebsch–Gordan coefficients
singlet state and triplet state

## Approximation methods

Born–Oppenheimer approximation
WKB approximation
time-dependent perturbation theory
time-independent perturbation theory

## Historical Terms / semi-classical treatment

Ehrenfest theorem
A theorem connecting the classical mechanics and result derived from Schrödinger equation.
first quantization
${\displaystyle x\to {\hat {x}},\,p\to i\hbar {\frac {\partial }{\partial x}}}$
wave–particle duality

## Uncategorized terms

uncertainty principle
Canonical commutation relations
Path integral
wavenumber

## Notes

1. Exception: superselection rules
2. Some textbooks (e.g. Cohen Tannoudji, Liboff) define "stationary state" as "an eigenstate of a Hamiltonian" without specific to bound states.

## Related Research Articles

In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets". A ket looks like "". Mathematically it denotes a vector, , in an abstract (complex) vector space , and physically it represents a state of some quantum system. A bra looks like "", and mathematically it denotes a linear functional , i.e. a linear map that maps each vector in to a number in the complex plane . Letting the linear functional act on a vector is written as .

In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's energy spectrum or its set of energy eigenvalues, is the set of possible outcomes obtainable from a measurement of the system's total energy. Due to its close relation to the energy spectrum and time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, x, and momentum, p, can be predicted from initial conditions. Such variable pairs are known as complementary variables or canonically conjugate variables, and, depending on interpretation, the uncertainty principle limits to what extent such conjugate properties maintain their approximate meaning, as the mathematical framework of quantum physics does not support the notion of simultaneously well-defined conjugate properties expressed by a single value. The uncertainty principle implies that it is in general not possible to predict the value of a quantity with arbitrary certainty, even if all initial conditions are specified.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability or probability density.

In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one. A creation operator increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. In many subfields of physics and chemistry, the use of these operators instead of wavefunctions is known as second quantization.

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point in space.

In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In many-body theory, the term Green's function is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.

In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.

The eigenstate thermalization hypothesis is a set of ideas which purports to explain when and why an isolated quantum mechanical system can be accurately described using equilibrium statistical mechanics. In particular, it is devoted to understanding how systems which are initially prepared in far-from-equilibrium states can evolve in time to a state which appears to be in thermal equilibrium. The phrase "eigenstate thermalization" was first coined by Mark Srednicki in 1994, after similar ideas had been introduced by Josh Deutsch in 1991. The principal philosophy underlying the eigenstate thermalization hypothesis is that instead of explaining the ergodicity of a thermodynamic system through the mechanism of dynamical chaos, as is done in classical mechanics, one should instead examine the properties of matrix elements of observable quantities in individual energy eigenstates of the system.

## References

• Elementary textbooks
• Griffiths, David J. (2004). (2nd ed.). Prentice Hall. ISBN   0-13-805326-X.
• Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN   0-8053-8714-5.
• Shankar, R. (1994). Principles of Quantum Mechanics. Springer. ISBN   0-306-44790-8.
• Claude Cohen-Tannoudji; Bernard Diu; Frank Laloë (2006). Quantum Mechanics. Wiley-Interscience. ISBN   978-0-471-56952-7.