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 $|\alpha \rangle \langle \beta |$.
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:
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 $|a\rangle$ is called a ket. See "bra–ket notation".
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 wave function
A wave function $|\alpha '\rangle$ is said to be normalizable if $\langle \alpha '|\alpha '\rangle <\infty$. A normalizable wave function can be made to be normalized by $|a'\rangle \to \alpha ={\frac {|\alpha '\rangle }{\sqrt {\langle \alpha '|\alpha '\rangle ))))$.
Normalized wave function
A wave function $|a\rangle$ is said to be normalized if $\langle a|a\rangle =1$.
A common example of quantum numbers is the possible state of an electron in a central potential: $(n,\ell ,m,s)$, 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".
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 $H_{A}\otimes H_{B))$, 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 $\Psi$ 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. $\psi _{\alpha }(x_{0})=\langle x_{0}|\alpha \rangle$, where $|x_{0}\rangle$ is the position eigenstate.
The Schrödinger equation relates the Hamiltonian operator acting on a wave function to its time evolution (Equation 1): $i\hbar {\frac {\partial }{\partial t))|\alpha \rangle ={\hat {H))|\alpha \rangle$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 2): $E|\alpha \rangle ={\hat {H))|\alpha \rangle$
Dynamics related to single particle in a potential / other spatial properties
In this situation, the SE is given by the form $i\hbar {\frac {\partial }{\partial t))\Psi _{\alpha }(\mathbf {r} ,\,t)={\hat {H))\Psi _{\alpha }(\mathbf {r} ,\,t)=\left(-{\frac {\hbar ^{2)){2m))\nabla ^{2}+V(\mathbf {r} )\right)\Psi _{\alpha }(\mathbf {r} ,\,t)=-{\frac {\hbar ^{2)){2m))\nabla ^{2}\Psi _{\alpha }(\mathbf {r} ,\,t)+V(\mathbf {r} )\Psi _{\alpha }(\mathbf {r} ,\,t)$ 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 if and only if $E<\operatorname {min} \{V(r\to -\infty ),V(r\to +\infty )\))$.
Position representation and momentum representation
Having the metaphor of probability density as mass density, then probability current $J$ is the current: $J(x,t)={\frac {i\hbar }{2m))\left(\psi {\frac {\partial \psi ^{*)){\partial x))-{\frac {\partial \psi }{\partial x))\psi \right)$ The probability current and probability density together satisfy the continuity equation: ${\frac {\partial }{\partial t))|\psi (x,t)|^{2}+\nabla \cdot \mathbf {J} (x,t)=0$
Given the wave function 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 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>\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 $\Psi (x,t)$ of a state vector of a wave function, square-integrable means:
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. ${\frac {d}{dt))|\Psi (x,t)|^{2}=0$
The expectation value $\langle M\rangle$ of the observable M with respect to a state $|\alpha$ is the average outcome of measuring $M$ with respect to an ensemble of state $|\alpha$.
$\langle M\rangle$ can be calculated by: $\langle M\rangle =\langle \alpha |M|\alpha \rangle .$
If the state is given by a density matrix $\rho$, $\langle M\rangle =\operatorname {Tr} (M\rho )$.
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.
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 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.
The canonical commutation relations are the commutators between canonically conjugate variables. For example, position ${\hat {x))$ and momentum ${\hat {p))$: $[{\hat {x)),{\hat {p))]={\hat {x)){\hat {p))-{\hat {p)){\hat {x))=i\hbar$