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# Glossary of elementary quantum mechanics

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### 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 non-relativistic quantum mechanics.
• Notation:
• | x \rangle - position eigenstate
• | \alpha \rangle, | \beta \rangle, | \gamma \rangle ... - wavefunction of the state of the system
• \Psi - total wavefunction of a system
• \psi - wavefunction of a system (maybe a particle)
• \psi_\alpha(x,t) - wavefunction of a particle in position representation, equal to \langle x | \alpha \rangle

## Formalism

### Kinematical postulates

a complete set of wavefunctions
A basis of the Hilbert space of wavefunctions with respect to a system.
bra
The Hermitian conjugate of a ket is called a bra. \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, | \alpha \rangle and | \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 |\alpha \rangle is |\alpha \rangle \langle \alpha|.
Mathematically, a density matrix has to satisfy the following conditions:
• \operatorname{Tr}(\rho) = 1
• \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 wavefunction expressed in the form |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: \operatorname{Tr}(\rho^2) = 1
Mixed state: \operatorname{Tr}(\rho^2) < 1
Normalizable wavefunction
A wavefunction |\alpha' \rangle is said to be normalizable if \langle \alpha'| \alpha' \rangle < \infty. A normalizable wavefunction can be made to be normalized by |a' \rangle \to \alpha = \frac{|\alpha' \rangle}{\sqrt{\langle \alpha'|\alpha' \rangle}}.
Normalized wavefunction
A wavefunction | a \rangle is said to be normalized if \langle a| a \rangle = 1.
Pure state
A state which can be represented as a wavefunction / 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, l, 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 wavefunction

Part of a wavefunction of particle(s). See "total wavefunction of a particle".

Spinor

Synonymous to "spin wavefunction".

Spatial wavefunction

Part of a wavefunction of particle(s). See "total wavefunction 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 "wavefunction" or "pure state".
State vector
synonymous to "wavefunction".
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 wavefunctions of the composite system are in a Hilbert space H_A \otimes H_B, if the Hilbert space of the wavefunctions for A and B are H_A and H_B respectively.
Total wavefunction of a particle
For single-particle system, the total wavefunction \Psi of a particle can be expressed as a product of spatial wavefunction and the spinor. The total wavefunctions 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.
Wavefunction
The word "wavefunction" 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. \psi_\alpha(x_0) = \langle x_0 | \alpha \rangle, where | x_0 \rangle is the position eigenstate.

### Dynamics

Degeneracy
See "degenerate energy level".
Degenerate energy level
If the energy of different state (wavefunctions 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 \hat H
The operator represents the total energy of the system.
Schrödinger equation
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.
E \alpha \rangle = \hat H | \alpha \rangle -- (2)

#### Dynamics related to single particle in a potential / other spatial properties

In this situation, the SE is given by the form
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 \Psi_\alpha(x,t) := \langle x |\alpha\rangle and \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, | \psi( \mathbf{r}, t) |^2 \to 0 when |\mathbf{r}| \to +\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 iff E < \operatorname{min}\{ V( r \to - \infty ) , V( r \to + \infty ) \}.
Position representation and momentum representation
Position representation of a wavefunction: \Psi_\alpha(x,t) := \langle x |\alpha\rangle,
momentum representation of a wavefunction: \tilde{\Psi}_\alpha(p,t) := \langle p |\alpha\rangle ;
where | x \rangle is the position eigenstate and | p \rangle the momentum eigenstate respectively.
The two representations are linked by Fourier transform.
Probability amplitude
Synonymous to "probability density".
Probability current
Having the metaphor of probability density as mass density, then probability current J is the current:
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:
\frac{\partial }{\partial t}|\psi(x,t)|^2 + \nabla \cdot \mathbf{J(x,t)} = 0
Probability density
Given the wavefunction of a particle, |\psi(x,t)|^2 is the probability density at position x and time t. |\psi(x_0,t)|^2 \, dx means the probability of finding the particle near x_0.
Scattering state
The wavefunction 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 iff 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 wavefunction of a bound state of the system.
Given the position representation \Psi(x,t) of a state vector of a wavefunction, square-integrable means:
1D case: \int_{-\infty}^{+\infty} |\Psi(x,t)|^2 \, dx < +\infty.
3D case: \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 ( |\psi(x,t)|^2) does not change with respect to time, i.e. \frac{d}{dt}|\Psi(x,t)|^2 = 0

### Measurement postulates

Born's rule
The probability of the state | \alpha \rangle collapse to an eigenstate | k \rangle of an observable is given by |\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 A is a vector satisfied the eigenvalue equation: A |\alpha \rangle = c |\alpha \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 of the observable M with respect to a state | \alpha is the average outcome of measuring M with respect to an ensemble of state | \alpha .
can be calculated by:
= \langle \alpha | M | \alpha \rangle.
If the state is given by a density matrix \rho, = \operatorname{Tr}( M \rho).
Hermitian operator
An operator satisfying A = A^\dagger.
Equivalently, \langle \alpha | A|\alpha \rangle = \langle \alpha | A^\dagger |\alpha \rangle for all allowable wavefunction |\alpha\rangle.
Observable
Mathematically, it is represented by a Hermitian operator.
Quantum Zeno effect
The phenomenon that a frequent measurement leads to "freezing" of the state.

### 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 wavefunctions
Symmetrization of wavefunctions
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 dervied from Schrödinger equation.
first quantization
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.

## References

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