Mathematical formulation of quantum mechanics

In the 1890s,

In 1905,

All of these developments were

In 1923, de Broglie proposed that wave–particle duality applied not only to photons but to electrons and every other physical system.

The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.

The "new quantum theory"

Werner Heisenberg's

Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrödinger's wave function can be seen to be closely related to the classical Hamilton–Jacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac^[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

To be more precise, already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics–– the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.

Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the bra–ket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in many types of generalizations of the field.

The first complete mathematical formulation of this approach, known as the

Later developments

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases.

• Path integral formulation
• Phase-space formulation of quantum mechanics & geometric quantization
• quantum field theory in curved spacetime
• algebraic and constructive quantum field theory
• formalism
• Generalized statistical model of quantum mechanics

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called

Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics.

Postulates of quantum mechanics

A physical system is generally described by three basic ingredients:

The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Dirac–von Neumann axioms.^[3]

Description of the state of a system

Each isolated physical system is associated with a (topologically)

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 H, where two vectors (of length 1) represent the same state if they differ only by a phase factor.^[4] As such, quantum states form a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrödinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid.^[5]

Accompanying Postulate I is the composite system postulate:^[6]

In the presence of

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 ${\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}$, where ${\displaystyle \;\sum _{k}p_{k}=1}$. If there is only a single non-zero ${\displaystyle p_{k}}$, then the state can be expressed just as ${\textstyle \rho =\rho _{1}\otimes \rho _{2},}$ and is called simply separable or product state.

Measurement on a system

Description of physical quantities

Physical observables are represented by

Results of measurement

By spectral theory, we can associate a

(in the sense of probability theory) of the observable A for the system in state represented by the unit vector ψ ∈ H is ${\displaystyle \langle \psi |A|\psi \rangle }$. If we represent the state ψ in the basis formed by the eigenvectors of A, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.

Postulate II.b

When the physical quantity ${\displaystyle {\mathcal {A}}}$ is measured on a system in a normalized state ${\displaystyle |\psi \rangle }$, the probability of obtaining an eigenvalue (denoted ${\displaystyle a_{n}}$ for discrete spectra and ${\displaystyle \alpha }$ for continuous spectra) of the corresponding observable ${\displaystyle A}$ is given by the amplitude squared of the appropriate wave function (projection onto corresponding eigenvector).

${\displaystyle {\begin{aligned}\mathbb {P} (a_{n})&=|\langle a_{n}|\psi \rangle |^{2}&{\text{(Discrete, nondegenerate spectrum)}}\\\mathbb {P} (a_{n})&=\sum _{i}^{g_{n}}|\langle a_{n}^{i}|\psi \rangle |^{2}&{\text{(Discrete, degenerate spectrum)}}\\d\mathbb {P} (\alpha )&=|\langle \alpha |\psi \rangle |^{2}d\alpha &{\text{(Continuous, nondegenerate spectrum)}}\end{aligned}}}$

For a mixed state ρ, the expected value of A in the state ρ is ${\displaystyle \operatorname {tr} (A\rho )}$, and the probability of obtaining an eigenvalue ${\displaystyle a_{n}}$ in a discrete, nondegenerate spectrum of the corresponding observable ${\displaystyle A}$ is given by ${\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle }$.

If the eigenvalue ${\displaystyle a_{n}}$ has degenerate, orthonormal eigenvectors ${\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}$, then the projection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace:

$${\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,}$$

${\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )}$

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 ρ, after obtaining an eigenvalue ${\displaystyle a_{n}}$ in a discrete, nondegenerate spectrum of the corresponding observable ${\displaystyle A}$, the updated state is given by ${\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}}$. If the eigenvalue ${\displaystyle a_{n}}$ has degenerate, orthonormal eigenvectors ${\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}$, then the projection operator onto the eigensubspace is ${\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|}$.

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

Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the

Equivalently, the time evolution postulate can be stated as:

For a closed system in a mixed state ρ, the time evolution is ${\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})}$.

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
  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.
• Recent research has shown^[7] that the composite system postulate (tensor product postulate) can be derived from the state postulate (Postulate I) and the measurement postulates (Postulates II); Moreover, it has also been shown^[8] that the measurement postulates (Postulates II) can be derived from "unitary quantum mechanics", which includes only the state postulate (Postulate I), the composite system postulate (tensor product postulate) and the unitary evolution postulate (Postulate III).

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 r and time t as continuous variables, ψ = ψ(r, t). For spin wavefunctions the spin is an additional discrete variable: ψ = ψ(r, t, σ), where σ takes the values;

$${\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}$$

That is, the state of a single particle with spin S is represented by a (2S + 1)-component spinor of complex-valued wave functions.

Two classes of particles with very different behaviour are bosons which have integer spin (S = 0, 1, 2, ...), and fermions possessing half-integer spin (S = 1⁄2, 3⁄2, 5⁄2, ...).

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.^[9] 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.^[10] 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:

${\displaystyle P{\bigg (}\cdots |\psi \rangle |\phi \rangle \cdots {\bigg )}\equiv \cdots |\phi \rangle |\psi \rangle \cdots }$

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.^[11]

More generally, consider a linear combination of such states, ${\displaystyle |\Psi \rangle }$. 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 ${\displaystyle P^{2}=1}$, the possible eigenvalues of P are +1 and −1. The ${\displaystyle |\Psi \rangle }$ states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows:

${\displaystyle P|\cdots n_{i},n_{j}\cdots ;S\rangle =+|\cdots n_{i},n_{j}\cdots ;S\rangle }$

${\displaystyle P|\cdots n_{i},n_{j}\cdots ;A\rangle =-|\cdots n_{i},n_{j}\cdots ;A\rangle }$

The explicit symmetric/antisymmetric form of ${\displaystyle |\Psi \rangle }$ is

The property of spin relates to another basic property concerning systems of N identical particles: the Pauli exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have

i.e., on

Due to the form of anti-symmetrized wavefunction:

${\displaystyle \Psi _{n_{1}\cdots n_{N}}^{(A)}(x_{1},\ldots ,x_{N})={\frac {1}{\sqrt {N!}}}\left|{\begin{matrix}\psi _{n_{1}}(x_{1})&\psi _{n_{1}}(x_{2})&\cdots &\psi _{n_{1}}(x_{N})\\\psi _{n_{2}}(x_{1})&\psi _{n_{2}}(x_{2})&\cdots &\psi _{n_{2}}(x_{N})\\\vdots &\vdots &\ddots &\vdots \\\psi _{n_{N}}(x_{1})&\psi _{n_{N}}(x_{2})&\cdots &\psi _{n_{N}}(x_{N})\\\end{matrix}}\right|}$

if the wavefunction of each particle is completely determined by a set of quantum number, 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:

${\displaystyle |x_{1}x_{2}\cdots x_{N};S\rangle ={\frac {\prod _{j}n_{j}!}{N!}}\sum _{p}\left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle }$

where ${\displaystyle n_{j}}$ is the number of particles with same wavefunction.

Exceptions for symmetrization postulate

In nonrelativistic quantum mechanics all particles are either bosons or

Summary:

Evolution of:	Picture ( v t e )
	Schrödinger (S)	Heisenberg (H)	Interaction (I)
Ket state	${\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }$	constant	${\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }$
Observable	constant	${\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}$	${\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}$
Density matrix	${\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}$	constant	${\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}$

Representations

The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's

The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the Segal–Bargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.

Time as an operator

The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H − E, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H − E (this requires the use of a rigged Hilbert space and a renormalization of the norm).

This is related to the

The problem of measurement

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.^[13] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the Compton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain):

• Let A have spectral resolution
  $${\displaystyle A=\int \lambda \,d\operatorname {E} _{A}(\lambda ),}$$

  
  where E_A is the
  resolution of the identity (also called projection-valued measure
  ) associated with A. Then the probability of the measurement outcome lying in an interval B of R is |E_A(B) ψ|². In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure
  $${\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle .}$$

  
• If the measured value is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state E_A(B)ψ. If the measured value does not lie in B, replace B by its complement for the above state.

For example, suppose the state space is the n-dimensional complex Hilbert space Cⁿ and A is a Hermitian matrix with eigenvalues λ_i, with corresponding eigenvectors ψ_i. The projection-valued measure associated with A, E_A, is then

$${\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,}$$

where B is a Borel set containing only the single eigenvalue λ_i. If the system is prepared in state

$${\displaystyle |\psi \rangle }$$

Then the probability of a measurement returning the value λ_i can be calculated by integrating the spectral measure

$${\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle }$$

over B_i. This gives trivially

$${\displaystyle \langle \psi |\psi _{i}\rangle \langle \psi _{i}\mid \psi \rangle =|\langle \psi \mid \psi _{i}\rangle |^{2}.}$$

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

$${\displaystyle |\psi _{i}\rangle \langle \psi _{i}|}$$

by a finite set of positive operators

$${\displaystyle F_{i}F_{i}^{*}}$$

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {λ₁ ... λ_n} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is λ_i. Instead of collapsing to the (unnormalized) state

$${\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle }$$

after the measurement, the system now will be in the state

$${\displaystyle F_{i}|\psi \rangle .}$$

Since the F_i F_i* operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general

mixed states

.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).

List of mathematical tools

Part of the folklore of the subject concerns the

The main tools include:

• eigenvalues
• linear operators, spectral theory
• ordinary differential equations, Sturm–Liouville theory, eigenfunctions
• harmonic analysis: Fourier transforms

See also

• List of mathematical topics in quantum theory
• Symmetry in quantum mechanics

Notes

References

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