A physical system is generally described by three basic ingredients:
states;
observables; and
dynamics (or law of
time evolution) or, more generally, a
group of physical symmetries. A classical description can be given in a fairly direct way by a phase space
model of mechanics: states are points in a phase space formulated by
symplectic manifold, observables are real-valued functions on it, time evolution is given by a one-parameter
group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a
Hilbert space of states, observables are
self-adjoint operators on the space of states, time evolution is given by a
one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by
unitary transformations. (It is possible, to map this Hilbert-space picture to a
phase space formulation, invertibly. See below.) The following summary of the mathematical framework of quantum mechanics can be partly traced back to the
Dirac–von Neumann axioms.
Description of the state of a system Each isolated physical system is associated with a (topologically)
separable complex Hilbert space with
inner product .
Separability is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations. Quantum states can be identified with
equivalence classes in , where two vectors (of length 1) represent the same state if they differ only by a
phase factor: |\psi_k \rangle \sim |\psi_l\rangle \;\; \Leftrightarrow \;\; |\psi_k \rangle = e^{i\alpha} |\psi_l\rangle, \quad\ \alpha\in\mathbb{R}. As such, a quantum state is an element of a
projective Hilbert space, conventionally termed a
"ray". Accompanying Postulate I is the composite system postulate: In the presence of
quantum entanglement, the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or
superposition, of tensor products of states of component subsystems. A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a
density operator; Such quantum state is known as a
mixed state. The
density operator of a mixed state is a
trace class, nonnegative (
positive semi-definite)
self-adjoint operator \rho normalized to be of
trace 1. In turn, any
density operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see
purification theorem). In the absence of quantum entanglement, the quantum state of the composite system is called a
separable state. The density matrix of a bipartite system in a separable state can be expressed as \rho=\sum_k p_k \rho_1^k \otimes \rho_2^k , where \; \sum_k p_k = 1 . If there is only a single non-zero p_k, then the state can be expressed just as \rho = \rho_1 \otimes \rho_2 , and is called simply separable or product state.
Measurement on a system Description of physical quantities Physical observables are represented by
Hermitian matrices on . Since these operators are Hermitian, their
eigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is
discrete, then the possible results are
quantized.
Results of measurement By spectral theory, we can associate a
probability measure to the values of in any state . We can also show that the possible values of the observable in any state must belong to the
spectrum of . The
expectation value (in the sense of probability theory) of the observable for the system in state represented by the unit vector ∈
H is \langle\psi|A|\psi\rangle. If we represent the state in the basis formed by the eigenvectors of , then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue. {{Quote box \begin{alignat}{3}\mathbb{P}(a_n)&= |\langle a_n|\psi \rangle|^2 &&\,\,\text{(Discrete, nondegenerate spectrum)} \\ \mathbb{P}(a_n) &= \sum_i^{g_n} |\langle a_n^i|\psi \rangle|^2 &&\,\, \text{(Discrete, degenerate spectrum)} \\ d\mathbb{P}(\alpha) &= |\langle \alpha | \psi \rangle |^2 d\alpha &&\,\, \text{(Continuous, nondegenerate spectrum)} \end{alignat} }} For a mixed state , the expected value of in the state is \operatorname{tr}(A\rho), and the probability of obtaining an eigenvalue a_n in a discrete, nondegenerate spectrum of the corresponding observable A is given by \mathbb P(a_n)=\operatorname{tr}(|a_n\rangle\langle a_n|\rho)=\langle a_n|\rho|a_n\rangle . If the eigenvalue a_n has
degenerate, orthonormal eigenvectors \{|a_{n1}\rangle,|a_{n2}\rangle, \dots , |a_{nm}\rangle\} , then the
projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace: P_n=|a_{n1}\rangle\langle a_{n1}|+|a_{n2}\rangle\langle a_{n2}| + \dots + |a_{nm}\rangle\langle a_{nm}|, and then \mathbb P(a_n)=\operatorname{tr}(P_n\rho) . Postulates II.a and II.b are collectively known as the
Born rule of quantum mechanics.
Effect of measurement on the state When a measurement is performed, only one result is obtained (according to some
interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore, the state vector must change as a result of measurement, and
collapse onto the eigensubspace associated with the eigenvalue measured. For a mixed state , after obtaining an eigenvalue a_n in a discrete, nondegenerate spectrum of the corresponding observable A , the updated state is given by \rho'=\frac{P_n\rho P_n^\dagger}{\operatorname{tr}(P_n\rho P_n^\dagger)} . If the eigenvalue a_n has degenerate, orthonormal eigenvectors \{|a_{n1}\rangle,|a_{n2}\rangle, \dots ,|a_{nm}\rangle\} , then the
projection operator onto the eigensubspace is P_n=|a_{n1}\rangle\langle a_{n1}|+|a_{n2}\rangle\langle a_{n2}| + \dots + |a_{nm}\rangle\langle a_{nm}| . Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of
measurements, and are sometimes collectively called the measurement postulate(s). Note that the
projection-valued measures (PVM) described in the measurement postulate(s) can be generalized to
positive operator-valued measures (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see
Naimark's dilation theorem).
Time evolution of a system The Schrödinger equation describes how a state vector evolves in time. Depending on the text, it may be derived from some other assumptions, motivated on heuristic grounds, or asserted as a postulate. Derivations include using the
de Broglie relation between wavelength and momentum or
path integrals. Equivalently, the time evolution postulate can be stated as: For a closed system in a mixed state , the time evolution is \rho(t)=U(t;t_0)\rho(t_0) U^\dagger(t;t_0). The evolution of an
open quantum system can be described by
quantum operations (in an
operator sum formalism) and
quantum instruments, and generally does not have to be unitary.
Other implications of the postulates • Physical symmetries act on the Hilbert space of quantum states
unitarily or
antiunitarily due to
Wigner's theorem (
supersymmetry is another matter entirely). • Density operators are those that are in the closure of the
convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are
extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors
pure states and other density operators
mixed states. • One can in this formalism state Heisenberg's
uncertainty principle and
prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article. Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of
spin and Pauli's
exclusion principle, see below.
Spin In addition to their other properties, all particles possess a quantity called
spin, an
intrinsic angular momentum. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position and time as continuous variables, . For spin wavefunctions the spin is an additional discrete variable: , where takes the values; \sigma = -S \hbar , -(S-1) \hbar , \dots, 0, \dots ,+(S-1) \hbar ,+S \hbar \,. That is, the state of a single particle with spin is represented by a -component
spinor of complex-valued wave functions. Two classes of particles with
very different behaviour are
bosons which have integer spin (), and
fermions possessing half-integer spin ().
Symmetrization postulate In quantum mechanics, two particles can be distinguished from one another using two methods. By performing a measurement of intrinsic properties of each particle, particles of different types can be distinguished. Otherwise, if the particles are identical, their trajectories can be tracked which distinguishes the particles based on the locality of each particle. While the second method is permitted in classical mechanics, (i.e. all classical particles are treated with distinguishability), the same cannot be said for quantum mechanical particles since the process is infeasible due to the fundamental uncertainty principles that govern small scales. Hence the requirement of indistinguishability of quantum particles is presented by the symmetrization postulate. The postulate is applicable to a system of bosons or fermions, for example, in predicting the spectra of
helium atom. The postulate, explained in the following sections, can be stated as follows: Exceptions can occur when the particles are constrained to two spatial dimensions where existence of particles known as
anyons are possible which are said to have a continuum of statistical properties spanning the range between fermions and bosons. The connection between behaviour of identical particles and their spin is given by
spin statistics theorem. It can be shown that two particles localized in different regions of space can still be represented using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the same result. Hence the symmetrization postulate is applicable in the general case of a system of identical particles.
Exchange Degeneracy In a system of identical particles, let
P be known as exchange operator that acts on the wavefunction as: : P \bigg(\cdots|\psi\rang |\phi\rang \cdots\bigg) \equiv \cdots |\phi\rang |\psi\rang \cdots If a physical system of identical particles is given, wavefunction of all particles can be well known from observation but these cannot be labelled to each particle. Thus, the above exchanged wavefunction represents the same physical state as the original state which implies that the wavefunction is not unique. This is known as exchange degeneracy. More generally, consider a linear combination of such states, |\Psi\rangle . For the best representation of the physical system, we expect this to be an eigenvector of
P since exchange operator is not excepted to give completely different vectors in projective Hilbert space. Since P^2 = 1, the possible eigenvalues of
P are +1 and −1. The |\Psi\rangle states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows: : P|\cdots n_i,n_j \cdots; S\rang = + |\cdots n_i,n_j \cdots; S\rang : P|\cdots n_i, n_j \cdots; A\rang = - |\cdots n_i, n_j \cdots; A\rang The explicit symmetric/antisymmetric form of |\Psi\rangle is
constructed using a symmetrizer or
antisymmetrizer operator. Particles that form symmetric states are called
bosons and those that form antisymmetric states are called as fermions. The relation of spin with this classification is given from
spin statistics theorem which shows that integer spin particles are bosons and half integer spin particles are fermions.
Pauli exclusion principle The property of spin relates to another basic property concerning systems of identical particles: the
Pauli exclusion principle, which is a consequence of the following permutation behaviour of an -particle wave function; again in the position representation one must postulate that for the transposition of any two of the particles one always should have i.e., on
transposition of the arguments of any two particles the wavefunction should
reproduce, apart from a prefactor which is for bosons, but () for
fermions. Electrons are fermions with ; quanta of light are bosons with . Due to the form of anti-symmetrized wavefunction: : \Psi^{(A)}_{n_1 \cdots n_N} (x_1, \ldots, x_N) = \frac{1}{\sqrt{N!}} \left| \begin{matrix} \psi_{n_1}(x_1) & \psi_{n_1}(x_2) & \cdots & \psi_{n_1}(x_N) \\ \psi_{n_2}(x_1) & \psi_{n_2}(x_2) & \cdots & \psi_{n_2}(x_N) \\ \vdots & \vdots & \ddots & \vdots \\ \psi_{n_N}(x_1) & \psi_{n_N}(x_2) & \cdots & \psi_{n_N}(x_N) \\ \end{matrix} \right| if the wavefunction of each particle is completely determined by a set of quantum numbers, then two fermions cannot share the same set of quantum numbers since the resulting function cannot be anti-symmetrized (i.e. above formula gives zero). The same cannot be said of Bosons since their wavefunction is: : |x_1 x_2 \cdots x_N; S \rangle = \frac{\prod_j n_j!}{N!} \sum_p \left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle where n_j is the number of particles with same wavefunction.
Exceptions for symmetrization postulate In nonrelativistic quantum mechanics all particles are either bosons or
fermions; in relativistic quantum theories also
"supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension can one construct entities where is replaced by an arbitrary complex number with magnitude 1, called
anyons. In relativistic quantum mechanics, spin statistic theorem can prove that under certain set of assumptions that the integer spins particles are classified as bosons and half spin particles are classified as
fermions. Anyons which form neither symmetric nor antisymmetric states are said to have fractional spin. Although
spin and the
Pauli principle can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the
periodic system of chemistry, are consequences of the two properties. == Mathematical structure of quantum mechanics ==