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Mathematical formulation of quantum mechanics

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Mathematical structures that allow quantum mechanics to be explained
Part of a series of articles about
Quantum mechanics
iℏddt|Ψ⟩=H^|Ψ⟩{\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }

Themathematical formulations of quantum mechanics are thosemathematical formalisms that permit arigorous description ofquantum mechanics. This mathematical formalism uses mainly a part offunctional analysis, especiallyHilbert spaces, which are a kind oflinear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensionalHilbert spaces (L2 space mainly), andoperators on these spaces. In brief, values of physicalobservables such asenergy andmomentum were no longer considered as values offunctions onphase space, but aseigenvalues; more precisely asspectral values of linearoperators in Hilbert space.[1]

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas ofquantum state andquantum observables, which are radically different from those used in previousmodels of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated byHeisenberg through athought experiment, and is represented mathematically in the new formalism by thenon-commutativity of operators representing quantum observables.

Prior to the development of quantum mechanics as a separatetheory, the mathematics used in physics consisted mainly of formalmathematical analysis, beginning withcalculus, and increasing in complexity up todifferential geometry andpartial differential equations.Probability theory was used instatistical mechanics. Geometric intuition played a strong role in the first two and, accordingly,theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now calledclassical physics, and in particular within the same mathematical structures. The most sophisticated example of this is theSommerfeld–Wilson–Ishiwara quantization rule, which was formulated entirely on the classicalphase space.

History of the formalism

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The "old quantum theory" and the need for new mathematics

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Main article:Old quantum theory

In the 1890s,Planck was able to derive theblackbody spectrum, which was later used to avoid the classicalultraviolet catastrophe by making the unorthodox assumption that, in the interaction ofelectromagnetic radiation withmatter, energy could only be exchanged in discrete units which he calledquanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant,h, is now called thePlanck constant in his honor.

In 1905,Einstein explained certain features of thephotoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbedphotons.

light at the right frequency
light at the right frequency

All of these developments werephenomenological and challenged the theoretical physics of the time.Bohr and Sommerfeld went on to modifyclassical mechanics in an attempt to deduce theBohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of the Planck constant were actually allowed. The most sophisticated version of this formalism was the so-calledSommerfeld–Wilson–Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923,de Broglie proposed thatwave–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 ofErwin Schrödinger,Werner Heisenberg,Max Born,Pascual Jordan, and the foundational work ofJohn von Neumann,Hermann Weyl andPaul 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 afterWerner Heisenberg discovered the uncertainty relations andNiels Bohr introduced the idea ofcomplementarity.

The "new quantum theory"

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Werner Heisenberg'smatrix mechanics was the first successful attempt at replicating the observed quantization ofatomic spectra. Later in the same year, Schrödinger created hiswave mechanics. Schrödinger's formalism was considered easier to understand, visualize and calculate as it led todifferential equations, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.

Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that theabsolute square of the wave function of anelectron should be interpreted as thecharge density of an object smeared out over an extended, possibly infinite, volume of space. It wasMax Born who introduced the interpretation of theabsolute square of the wave function as the probability distribution of the position of apointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of theCopenhagen interpretation of quantum mechanics. Schrödinger'swave function can be seen to be closely related to the classicalHamilton–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 theHeisenberg 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 throughPoisson brackets, a procedure now known ascanonical quantization.

Already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented hismatrix 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 classicThe 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 thebra–ket notation, together with an abstract formulation in terms of theHilbert 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 theDirac–von Neumann axioms, is generally credited toJohn von Neumann's 1932 bookMathematical Foundations of Quantum Mechanics, althoughHermann Weyl had already referred to Hilbert spaces (which he calledunitary spaces) in his 1927 classic paper and1928 book. It was developed in parallel with a new approach to the mathematicalspectral theory based on linear operators rather than thequadratic forms that wereDavid Hilbert's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions aboutinterpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.

Later developments

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The application of the new quantum theory to electromagnetism resulted inquantum 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.

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-calledclassical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular,quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

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-calledhidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help ofquantum optics.

Postulates of quantum mechanics

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A physical system is generally described by three basic ingredients:states;observables; anddynamics (or law oftime evolution) or, more generally, agroup of physical symmetries. A classical description can be given in a fairly direct way by a phase spacemodel of mechanics: states are points in a phase space formulated bysymplectic manifold, observables are real-valued functions on it, time evolution is given by a one-parametergroup of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of aHilbert space of states, observables areself-adjoint operators on the space of states, time evolution is given by aone-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized byunitary transformations. (It is possible, to map this Hilbert-space picture to aphase space formulation, invertibly. See below.)

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

Description of the state of a system

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Each isolated physical system is associated with a (topologically)separablecomplexHilbert spaceH withinner product⟨φ|ψ⟩.

Postulate I

The state of an isolated physical system is represented, at a fixed timet{\displaystyle t}, by a state vector|ψ⟩{\displaystyle |\psi \rangle } belonging to a Hilbert spaceH{\displaystyle {\mathcal {H}}} called thestate space.

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 withequivalence classes inH, where two vectors (of length 1) represent the same state if they differ only by aphase factor:[4][5]|ψk⟩∼|ψl⟩⇔|ψk⟩=eiα|ψl⟩, α∈R.{\displaystyle |\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 aprojective Hilbert space, conventionally termed a"ray".[6][7]

Accompanying Postulate I is the composite system postulate:[8]

Composite system postulate

The Hilbert space of a composite system is the Hilbert spacetensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

In the presence ofquantum 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, orsuperposition, 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 adensity operator; Such quantum state is known as amixed state. Thedensity operator of a mixed state is atrace class, nonnegative (positive semi-definite)self-adjoint operatorρ{\displaystyle \rho } normalized to be oftrace 1. In turn, anydensity operator of a mixed state can be represented as a subsystem of a larger composite system in a pure state (seepurification theorem).

In the absence of quantum entanglement, the quantum state of the composite system is called aseparable state. The density matrix of a bipartite system in a separable state can be expressed asρ=∑kpkρ1k⊗ρ2k{\displaystyle \rho =\sum _{k}p_{k}\rho _{1}^{k}\otimes \rho _{2}^{k}}, where∑kpk=1{\displaystyle \;\sum _{k}p_{k}=1}. If there is only a single non-zeropk{\displaystyle p_{k}}, then the state can be expressed just asρ=ρ1⊗ρ2,{\textstyle \rho =\rho _{1}\otimes \rho _{2},} and is called simply separable or product state.

Measurement on a system

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Description of physical quantities

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Physical observables are represented byHermitian matrices onH. Since these operators are Hermitian, theireigenvalues are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable isdiscrete, then the possible results arequantized.

Postulate II.a

Every measurable physical quantityA{\displaystyle {\mathcal {A}}} is described by a Hermitian operatorA{\displaystyle A} acting in the state spaceH{\displaystyle {\mathcal {H}}}. This operator is an observable, meaning that itseigenvectors form a basis forH{\displaystyle {\mathcal {H}}}. The result of measuring a physical quantityA{\displaystyle {\mathcal {A}}} must be one of the eigenvalues of the corresponding observableA{\displaystyle A}.

Results of measurement

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By spectral theory, we can associate aprobability measure to the values ofA in any stateψ. We can also show that the possible values of the observableA in any state must belong to thespectrum ofA. Theexpectation value (in the sense of probability theory) of the observableA for the system in state represented by the unit vectorψ ∈H is⟨ψ|A|ψ⟩{\displaystyle \langle \psi |A|\psi \rangle }. If we represent the stateψ in the basis formed by the eigenvectors ofA, 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 quantityA{\displaystyle {\mathcal {A}}} is measured on a system in a normalized state|ψ⟩{\displaystyle |\psi \rangle }, the probability of obtaining an eigenvalue (denotedan{\displaystyle a_{n}} for discrete spectra andα{\displaystyle \alpha } for continuous spectra) of the corresponding observableA{\displaystyle A} is given by theamplitude squared of the appropriate wave function (projection onto corresponding eigenspace).

P(an)=|⟨an|ψ⟩|2(Discrete, nondegenerate spectrum)P(an)=∑ign|⟨ani|ψ⟩|2(Discrete, degenerate spectrum)dP(α)=|⟨α|ψ⟩|2dα(Continuous, nondegenerate spectrum){\displaystyle {\begin{alignedat}{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{alignedat}}}

For a mixed stateρ, the expected value ofA in the stateρ istr⁡(Aρ){\displaystyle \operatorname {tr} (A\rho )}, and the probability of obtaining an eigenvaluean{\displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observableA{\displaystyle A} is given byP(an)=tr⁡(|an⟩⟨an|ρ)=⟨an|ρ|an⟩{\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (|a_{n}\rangle \langle a_{n}|\rho )=\langle a_{n}|\rho |a_{n}\rangle }.

If the eigenvaluean{\displaystyle a_{n}} hasdegenerate, orthonormal eigenvectors{|an1⟩,|an2⟩,…,|anm⟩}{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}, then theprojection operator onto the eigensubspace can be defined as the identity operator in the eigensubspace:Pn=|an1⟩⟨an1|+|an2⟩⟨an2|+⋯+|anm⟩⟨anm|,{\displaystyle P_{n}=|a_{n1}\rangle \langle a_{n1}|+|a_{n2}\rangle \langle a_{n2}|+\dots +|a_{nm}\rangle \langle a_{nm}|,}and thenP(an)=tr⁡(Pnρ){\displaystyle \mathbb {P} (a_{n})=\operatorname {tr} (P_{n}\rho )}.

Postulates II.a and II.b are collectively known as theBorn rule of quantum mechanics.

Effect of measurement on the state

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When a measurement is performed, only one result is obtained (according to someinterpretations 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, andcollapse onto the eigensubspace associated with the eigenvalue measured.

Postulate II.c

If the measurement of the physical quantityA{\displaystyle {\mathcal {A}}} on the system in the state|ψ⟩{\displaystyle |\psi \rangle } gives the resultan{\displaystyle a_{n}}, then the state of the system immediately after the measurement is the normalized projection of|ψ⟩{\displaystyle |\psi \rangle } onto the eigensubspace associated withan{\displaystyle a_{n}}

|ψ⟩⟹anPn|ψ⟩⟨ψ|Pn|ψ⟩{\displaystyle |\psi \rangle \quad {\overset {a_{n}}{\Longrightarrow }}\quad {\frac {P_{n}|\psi \rangle }{\sqrt {\langle \psi |P_{n}|\psi \rangle }}}}

For a mixed stateρ, after obtaining an eigenvaluean{\displaystyle a_{n}} in a discrete, nondegenerate spectrum of the corresponding observableA{\displaystyle A}, the updated state is given byρ′=PnρPn†tr⁡(PnρPn†){\textstyle \rho '={\frac {P_{n}\rho P_{n}^{\dagger }}{\operatorname {tr} (P_{n}\rho P_{n}^{\dagger })}}}. If the eigenvaluean{\displaystyle a_{n}} has degenerate, orthonormal eigenvectors{|an1⟩,|an2⟩,…,|anm⟩}{\displaystyle \{|a_{n1}\rangle ,|a_{n2}\rangle ,\dots ,|a_{nm}\rangle \}}, then theprojection operator onto the eigensubspace isPn=|an1⟩⟨an1|+|an2⟩⟨an2|+⋯+|anm⟩⟨anm|{\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 ofmeasurements, and are sometimes collectively called the measurement postulate(s).

Note that theprojection-valued measures (PVM) described in the measurement postulate(s) can be generalized topositive 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 (seeNaimark's dilation theorem).

Time evolution of a system

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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 thede Broglie relation between wavelength and momentum orpath integrals.

Postulate III

The time evolution of the state vector|ψ(t)⟩{\displaystyle |\psi (t)\rangle } is governed by the Schrödinger equation, whereH(t){\displaystyle H(t)} is the observable associated with the total energy of the system (called theHamiltonian)

iℏddt|ψ(t)⟩=H(t)|ψ(t)⟩{\displaystyle i\hbar {\frac {d}{dt}}|\psi (t)\rangle =H(t)|\psi (t)\rangle }

Equivalently, the time evolution postulate can be stated as:

Postulate III

The time evolution of aclosed system is described by aunitary transformation on the initial state.

|ψ(t)⟩=U(t;t0)|ψ(t0)⟩{\displaystyle |\psi (t)\rangle =U(t;t_{0})|\psi (t_{0})\rangle }

For a closed system in a mixed stateρ, the time evolution isρ(t)=U(t;t0)ρ(t0)U†(t;t0){\displaystyle \rho (t)=U(t;t_{0})\rho (t_{0})U^{\dagger }(t;t_{0})}.

The evolution of anopen quantum system can be described byquantum operations (in anoperator sum formalism) andquantum instruments, and generally does not have to be unitary.

Other implications of the postulates

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  • Physical symmetries act on the Hilbert space of quantum statesunitarily orantiunitarily due toWigner's theorem (supersymmetry is another matter entirely).
  • Density operators are those that are in the closure of theconvex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors areextreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectorspure states and other density operatorsmixed states.
  • One can in this formalism state Heisenberg'suncertainty principle andprove 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 ofspin and Pauli'sexclusion principle, see below.

Spin

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In addition to their other properties, all particles possess a quantity calledspin, anintrinsic 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 positionr and timet as continuous variables,ψ =ψ(r,t). For spin wavefunctions the spin is an additional discrete variable:ψ =ψ(r,t,σ), whereσ takes the values;σ=−Sℏ,−(S−1)ℏ,…,0,…,+(S−1)ℏ,+Sℏ.{\displaystyle \sigma =-S\hbar ,-(S-1)\hbar ,\dots ,0,\dots ,+(S-1)\hbar ,+S\hbar \,.}

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

Two classes of particles withvery different behaviour arebosons which have integer spin (S = 0, 1, 2, ...), andfermions possessing half-integer spin (S =1⁄2,3⁄2,5⁄2, ...).

Symmetrization postulate

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Main article:Identical particles

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 ofhelium atom. The postulate, explained in the following sections, can be stated as follows:

Symmetrization postulate[9]

The wavefunction of a system ofN identical particles (in 3D) is either totally symmetric (Bosons) or totally antisymmetric (Fermions) under interchange of any pair of particles.

Exceptions can occur when the particles are constrained to two spatial dimensions where existence of particles known asanyons 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 byspin 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

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In a system of identical particles, letP be known as exchange operator that acts on the wavefunction as:

P(⋯|ψ⟩|ϕ⟩⋯)≡⋯|ϕ⟩|ψ⟩⋯{\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,|Ψ⟩{\displaystyle |\Psi \rangle }. For the best representation of the physical system, we expect this to be an eigenvector ofP since exchange operator is not excepted to give completely different vectors in projective Hilbert space. SinceP2=1{\displaystyle P^{2}=1}, the possible eigenvalues ofP are +1 and −1. The|Ψ⟩{\displaystyle |\Psi \rangle } states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows:

P|⋯ni,nj⋯;S⟩=+|⋯ni,nj⋯;S⟩{\displaystyle P|\cdots n_{i},n_{j}\cdots ;S\rangle =+|\cdots n_{i},n_{j}\cdots ;S\rangle }
P|⋯ni,nj⋯;A⟩=−|⋯ni,nj⋯;A⟩{\displaystyle P|\cdots n_{i},n_{j}\cdots ;A\rangle =-|\cdots n_{i},n_{j}\cdots ;A\rangle }

The explicit symmetric/antisymmetric form of|Ψ⟩{\displaystyle |\Psi \rangle } isconstructed using a symmetrizer orantisymmetrizer operator. Particles that form symmetric states are calledbosons and those that form antisymmetric states are called as fermions. The relation of spin with this classification is given fromspin statistics theorem which shows that integer spin particles are bosons and half integer spin particles are fermions.

Pauli exclusion principle

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

Pauli principle

ψ(…,ri,σi,…,rj,σj,…)=(−1)2S⋅ψ(…,rj,σj,…,ri,σi,…){\displaystyle \psi (\dots ,\,\mathbf {r} _{i},\sigma _{i},\,\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots )=(-1)^{2S}\cdot \psi (\dots ,\,\mathbf {r} _{j},\sigma _{j},\,\dots ,\mathbf {r} _{i},\sigma _{i},\,\dots )}

i.e., ontransposition of the arguments of any two particles the wavefunction shouldreproduce, apart from a prefactor(−1)2S which is+1 for bosons, but (−1) forfermions.Electrons are fermions withS = 1/2; quanta of light are bosons withS = 1.

Due to the form of anti-symmetrized wavefunction:

Ψn1⋯nN(A)(x1,…,xN)=1N!|ψn1(x1)ψn1(x2)⋯ψn1(xN)ψn2(x1)ψn2(x2)⋯ψn2(xN)⋮⋮⋱⋮ψnN(x1)ψnN(x2)⋯ψnN(xN)|{\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 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:

|x1x2⋯xN;S⟩=∏jnj!N!∑p|xp(1)⟩|xp(2)⟩⋯|xp(N)⟩{\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 }

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

Exceptions for symmetrization postulate

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In nonrelativistic quantum mechanics all particles are either bosons orfermions; in relativistic quantum theories also"supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimensiond = 2 can one construct entities where(−1)2S is replaced by an arbitrary complex number with magnitude 1, calledanyons. 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 asfermions. Anyons which form neither symmetric nor antisymmetric states are said to have fractional spin.

Althoughspin and thePauli 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. theperiodic system of chemistry, are consequences of the two properties.

Mathematical structure of quantum mechanics

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Pictures of dynamics

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Main article:Dynamical pictures

Summary:

Evolution of:Picture ()
Schrödinger (S)Heisenberg (H)Interaction (I)
Ket state|ψS(t)⟩=e−iHS t/ℏ|ψS(0)⟩{\displaystyle |\psi _{\rm {S}}(t)\rangle =e^{-iH_{\rm {S}}~t/\hbar }|\psi _{\rm {S}}(0)\rangle }constant|ψI(t)⟩=eiH0,S t/ℏ|ψS(t)⟩{\displaystyle |\psi _{\rm {I}}(t)\rangle =e^{iH_{0,\mathrm {S} }~t/\hbar }|\psi _{\rm {S}}(t)\rangle }
ObservableconstantAH(t)=eiHS t/ℏASe−iHS t/ℏ{\displaystyle A_{\rm {H}}(t)=e^{iH_{\rm {S}}~t/\hbar }A_{\rm {S}}e^{-iH_{\rm {S}}~t/\hbar }}AI(t)=eiH0,S t/ℏASe−iH0,S t/ℏ{\displaystyle A_{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }A_{\rm {S}}e^{-iH_{0,\mathrm {S} }~t/\hbar }}
Density matrixρS(t)=e−iHS t/ℏρS(0)eiHS t/ℏ{\displaystyle \rho _{\rm {S}}(t)=e^{-iH_{\rm {S}}~t/\hbar }\rho _{\rm {S}}(0)e^{iH_{\rm {S}}~t/\hbar }}constantρI(t)=eiH0,S t/ℏρS(t)e−iH0,S t/ℏ{\displaystyle \rho _{\rm {I}}(t)=e^{iH_{0,\mathrm {S} }~t/\hbar }\rho _{\rm {S}}(t)e^{-iH_{0,\mathrm {S} }~t/\hbar }}

Representations

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The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg'scanonical commutation relations. TheStone–von Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to thephase space formulation of quantum mechanics, which works in fullphase space instead ofHilbert space, so then with a more intuitive link to theclassical limit thereof. This picture also simplifies considerationsofquantization, the deformation extension from classical to quantum mechanics.

Thequantum 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 theSegal–Bargmann (Fock-space or coherent state) representation (named afterIrving Segal andValentine Bargmann). All four are unitarily equivalent.

Time as an operator

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The framework presented so far singles out time asthe 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 parameters, and in that case the timet becomes an additional generalized coordinate of the physical system. At the quantum level, translations ins would be generated by a "Hamiltonian"H −E, whereE is the energy operator andH is the "ordinary" Hamiltonian. However, sinces is an unphysical parameter,physical states must be left invariant by "s-evolution", and so the physical state space is the kernel ofH −E (this requires the use of arigged Hilbert space and a renormalization of the norm).

This is related to thequantization of constrained systems andquantization of gauge theories. Itis also possible to formulate a quantum theory of "events" where time becomes an observable.[12]

Problem of measurement

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Main article:Measurement in quantum mechanics

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 ofmeasurement.[13] The von Neumann description of quantum measurement of an observableA, when the system is prepared in a pure stateψ 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 theCompton–Simon experiment; it is not applicable to most present-day measurements within the quantum domain):

For example, suppose the state space is then-dimensional complex Hilbert spaceCn andA is a Hermitian matrix with eigenvaluesλi, with corresponding eigenvectorsψi. The projection-valued measure associated withA,EA, is thenEA⁡(B)=|ψi⟩⟨ψi|,{\displaystyle \operatorname {E} _{A}(B)=|\psi _{i}\rangle \langle \psi _{i}|,}whereB is a Borel set containing only the single eigenvalueλi. If the system is prepared in state|ψ⟩{\displaystyle |\psi \rangle }Then the probability of a measurement returning the valueλi can be calculated by integrating the spectral measure⟨ψ∣EA⁡ψ⟩{\displaystyle \langle \psi \mid \operatorname {E} _{A}\psi \rangle }overBi. This gives trivially⟨ψ|ψi⟩⟨ψi∣ψ⟩=|⟨ψ∣ψi⟩|2.{\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 theprojection 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|ψi⟩⟨ψi|{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|}by a finite set of positive operatorsFiFi∗{\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 ...λn} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome isλi. Instead of collapsing to the (unnormalized) state|ψi⟩⟨ψi|ψ⟩{\displaystyle |\psi _{i}\rangle \langle \psi _{i}|\psi \rangle }after the measurement, the system now will be in the stateFi|ψ⟩.{\displaystyle F_{i}|\psi \rangle .}

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

The same formulation applies to generalmixed 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 otherquantum operations, which are described bycompletely positive maps which do not increase the trace.

List of mathematical tools

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Part of the folklore of the subject concerns themathematical physics textbookMethods of Mathematical Physics put together byRichard Courant fromDavid Hilbert'sGöttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.

The main tools include:

See also

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Notes

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  1. ^Byron & Fuller 1992, p. 277.
  2. ^Dirac 1925.
  3. ^Cohen-Tannoudji, Diu & Laloë 2020.
  4. ^Bäuerle & de Kerf 1990, p. 330.
  5. ^Solem & Biedenharn 1993.
  6. ^Bäuerle & de Kerf 1990, p. 341.
  7. ^Weyl 1950, pp. 180–181.
  8. ^Jauch, Wigner & Yanase 1997.
  9. ^abSakurai & Napolitano 2021, p. 443.
  10. ^Sakurai & Napolitano 2021, p. 434-437.
  11. ^Cohen-Tannoudji, Diu & Laloë 2020, p. 1375–1377.
  12. ^Edwards 1979.
  13. ^Greenstein & Zajonc 2006, p. 215.

References

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Further reading

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