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Wave function

From Wikipedia, the free encyclopedia

Not to be confused withWave equation.

Some trajectories of aharmonic oscillator(a ball attached to aspring)inclassical mechanics(AB) and

quantum mechanics(CH). In quantum mechanics (CH), the ball has a wave function, which is shown

withreal partin blue andimaginary partin red. The trajectories C,D,E,F, (but not G or H) are examples

ofstanding waves,(or "stationary states"). Each standing-wave frequency is proportional to a possible

energy levelof the oscillator. This "energy quantization" does not occur in classical physics, where the

oscillator can have anyenergy.

A wave functionor wavefunction(also named a state function) inquantum mechanics

describes thequantum stateof a particle and how it behaves. Typically, its values arecomplex

numbersand, for a single particle, it is afunctionof space and time. TheSchrdinger equationdescribes how the wave function evolves over time. The wave function behaves qualitatively like

otherwaves,likewater wavesor waves on a string, because the Schrdinger equation is

mathematically a type ofwave equation.This explains the name "wave function", and gives rise

towaveparticle duality.

The most common symbols for a wave function are or (lower-case and capitalpsi).

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Although values of are complex numbers, ||2isrealcorresponding byMax Born's proposal to

theprobability densityof finding a particle in a given place at a given time, if the particle's

position is to bemeasured.Louis de Broglie in his later years proposed a real-valued wave

function connected to the complex wave function by a proportionality constant and developed

thede BroglieBohm theory.

Theunit of measurementfor depends on the system. For one particle in three dimensions, its

units are [length]3/2

. These unusual units are required so that an integral of ||2over a region of

three-dimensional space is a unitless probability (i.e., the probability that the particle is in that

region). For different numbers of particles and/or dimensions, the units may be different and can

be found bydimensional analysis.[1]

The wave function is central to quantum mechanics as the most direct way to describe themotion

of aparticle.

Although the wavefunction contains information, it is acomplex-valuedquantity; only itsrelative phase and relative magnitude can be measured. It does not directly tell anything about

the magnitudes or directions of measurable observables. An operator extracts this information by

acting on the wavefunction . For details and examples on how quantum mechanical operators

act on the wave function, commutation of operators, and expectation values of operators; see

Operator (physics).

Historical background

Quantum mechanics

Introduction

GlossaryHistory

Background[show]

Fundamental concepts[hide]

Complementarity Decoherence Entanglement Energy level Nonlocality

Quantum state Superposition Tunnelling

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Uncertainty Wave function

Wave function collapse Symmetry

Measurement

Experiments[show]

Formulations[show]

Equations[show]

Interpretations[show]

Advanced topics[show]

Scientists[show]

v t e

In the 1920s and 1930s, quantum mechanics was developed usingcalculusandlinear algebra.Those who used the techniques of calculus includedLouis de Broglie,Erwin Schrdinger,and

others, developing "wave mechanics". Those who applied the methods of linear algebra included

Werner Heisenberg,Max Born,and others, developing "matrix mechanics". Schrdinger

subsequently showed that the two approaches were equivalent.[2]

In each case, the wavefunction

was at the centre of attention in two forms, giving quantum mechanics its unity.

In 1905 Planck postulated the proportionality between the frequency of a photon and its energy,

in thePlanckEinstein equation,E= hf. In 1925, De Broglie published the symmetric relationbetweenmomentumandwavelength,p= h/, now called theDe Broglie relation.These

equations representwaveparticle duality.In 1926, Schrdinger published the famous waveequation now named after him, indeed theSchrdinger equation,based onclassicalenergyconservationusingquantum operatorsand the de Broglie relations such that the solutions of the

equation are the wavefunctions for the quantum system. LaterPauliinvented thePauli equation

that adds a description of electron'sspinandmagnetic dipole.However, no one, even

Schrdinger and De Broglie, were clear on how to interpret it.[3]

Around 192427, Max Born,Heisenberg, Bohr and others provided the perspective ofprobability amplitude.

[4]This is the

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Copenhagen interpretationof quantum mechanics. There are many otherinterpretations of

quantum mechanics,but this is considered the most importantsince quantum calculationscanbe understood.

In 1927,HartreeandFockmade the first step in an attempt to solve theN-bodywave function,

and developed theself-consistency cycle: aniterativealgorithmto approximate the solution. Nowit is also known as theHartreeFock method.[5]TheSlaterdeterminantandpermanent(of amatrix)was part of the method, provided byJohn C. Slater.

Interestingly, Schrdinger did encounter an equation for which the wave function satisfied

relativisticenergy conservation beforehe published the non-relativistic one, but it led to

unacceptable consequences; negativeprobabilitiesand negativeenergies,so he discarded it.[6]:3

In 1927,Klein,Gordonand Fock also found it, but taking a step further: incorporated the

electromagneticinteractioninto it and proved it wasLorentz-invariant.De Broglie also arrived at

exactly the same equation in 1928. This relativistic wave equation is now known most

commonly as theKleinGordon equation.[7]

In 1927,Pauliphenomenologically found a non-relativistic equation to describe spin-1/2

particles in electromagnetic fields, now called thePauli equation.Pauli found the wavefunction

was not a single complex number, but two complex numbers, which correspond to the spin +1/2

and 1/2 states of the fermion. Soon after in 1928,Diracfound an equation from the firstsuccessful unification ofspecial relativityand quantum mechanics applied to theelectronnowcalled theDirac equation.He found the wavefunction for this equation could not be a single

complex number, but afour-componentspinor.[5]

Spinautomatically entered into the properties

of the wavefunction. Later other wave equations were developed: seerelativistic wave equations

for further information.

Wave functions and function spaces

Functional analysisis commonly used to formulate the wavefunction with a necessary

mathematical precision; usually they arequadratically integrable functions(at least locally)

because it is compatible with the Hilbert space formalism mentioned below. The set on which

theirfunction spaceis defined is theconfiguration spaceof the system. In many situations it is an

Euclidean space,that implies that wavefunctions arefunctions of several real variables.

Superficially, this formalism is simple to understand for the following reasons.

If the wavefunction is to change throughout space and time, one would expect the wavefunction to be a function of the position and time coordinates. It is solved from theSchrdinger

equation(or otherrelativistic wave equations), alinear partial differential equation:

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_functionhttp://en.wikipedia.org/wiki/Quadratically_integrable_functionhttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Functions_of_several_real_variableshttp://en.wikipedia.org/wiki/Functions_of_several_real_variableshttp://en.wikipedia.org/wiki/Functions_of_several_real_variableshttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Relativistic_wave_equationshttp://en.wikipedia.org/wiki/Relativistic_wave_equationshttp://en.wikipedia.org/wiki/Relativistic_wave_equationshttp://en.wikipedia.org/wiki/Linear_partial_differential_equationhttp://en.wikipedia.org/wiki/Linear_partial_differential_equationhttp://en.wikipedia.org/wiki/Linear_partial_differential_equationhttp://en.wikipedia.org/wiki/Linear_partial_differential_equationhttp://en.wikipedia.org/wiki/Relativistic_wave_equationshttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Schr%C3%B6dinger_equationhttp://en.wikipedia.org/wiki/Functions_of_several_real_variableshttp://en.wikipedia.org/wiki/Euclidean_spacehttp://en.wikipedia.org/wiki/Configuration_spacehttp://en.wikipedia.org/wiki/Function_spacehttp://en.wikipedia.org/wiki/Quadratically_integrable_functionhttp://en.wikipedia.org/wiki/Functional_analysishttp://en.wikipedia.org/wiki/Relativistic_wave_equationshttp://en.wikipedia.org/wiki/Spin_%28physics%29http://en.wikipedia.org/wiki/Wave_function#cite_note-Quanta_1974-5http://en.wikipedia.org/wiki/Dirac_spinorhttp://en.wikipedia.org/wiki/Dirac_equationhttp://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Special_relativityhttp://en.wikipedia.org/wiki/Paul_Dirachttp://en.wikipedia.org/wiki/Pauli_equationhttp://en.wikipedia.org/wiki/Wolfgang_Paulihttp://en.wikipedia.org/wiki/Wave_function#cite_note-7http://en.wikipedia.org/wiki/Klein%E2%80%93Gordon_equationhttp://en.wikipedia.org/wiki/Lorentz_covariancehttp://en.wikipedia.org/wiki/Interactionhttp://en.wikipedia.org/wiki/Electromagnetic_forcehttp://en.wikipedia.org/wiki/Walter_Gordon_%28physicist%29http://en.wikipedia.org/wiki/Oskar_Kleinhttp://en.wikipedia.org/wiki/Wave_function#cite_note-6http://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Probabilityhttp://en.wikipedia.org/wiki/Theory_of_relativityhttp://en.wikipedia.org/wiki/John_C._Slaterhttp://en.wikipedia.org/wiki/Matrix_%28mathematics%29http://en.wikipedia.org/wiki/Permanenthttp://en.wikipedia.org/wiki/Determinanthttp://en.wikipedia.org/wiki/Slater_determinanthttp://en.wikipedia.org/wiki/Wave_function#cite_note-Quanta_1974-5http://en.wikipedia.org/wiki/Hartree%E2%80%93Fock_methodhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Iterationhttp://en.wikipedia.org/wiki/Many-body_problemhttp://en.wikipedia.org/wiki/Vladimir_Fockhttp://en.wikipedia.org/wiki/Douglas_Hartreehttp://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanicshttp://en.wikipedia.org/wiki/Copenhagen_interpretation

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Functions can easily describewave-like motion, usingperiodic functions,andFourier analysiscan be readily done.

Functions are easy to produce, visualize, and interpret, because of the pictorial nature of thegraph of a function.One can plotcurves,surfaces,contour lines,more generally anylevel sets.If

the situation is in a high number of dimensionsone can analyze the function in a lower

dimensional slice to see the behavior of the function within that confined region.

For concreteness and simplicity, in this article, whencoordinatesare needed we useCartesian

coordinatesso that ris short for (x,y,z), althoughspherical polar coordinatesand other

orthogonal coordinatesare often useful to solve the Schrdinger equation for potentials with

certain geometric symmetries, in which case the position and wavefunction is expressed in these

coordinates.

One does not have to define wavefunctions necessarily onreal spaces:appropriate function

spaces can be defined wherever ameasurecan provide integration.Operator theoryandlinear

algebra,as shown next, can deal with situations where thereal analysisis not applicable.

Requirements

Continuity of the wavefunction and its first spatial derivative (in thexdirection, yand zcoordinates not

shown), at some time t.

The following constraints on the wavefunction are formulated for the calculations and physical

interpretation to make sense:[8][9]

It must everywhere be acontinuous function,andcontinuously differentiable.

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It must everywhere satisfy the relevant normalization condition, because the particle or systemof particles exists somewhere with 100% certainty. For this to be so, the wavefunction must be

square integrable.

A requirement less restrictive is that the wavefunction must belong to theSobolev spaceW1,2

. It

means that it is differentiable in the sense ofdistributions,and itsgradientissquare-integrable.This relaxation is necessary for potentials that are not functions but are distributions, such as the

dirac delta function.

If these requirements are not met, it is not possible to interpret the wavefunction as a probability

amplitude.[10]

Definitions (spin-0 particles)

One spin-0 particle in one spatial dimension

Standing wavesfor aparticle in a box,examples ofstationary states.

Travelling waves of a free particle.

Thereal partsof position and momentum wave functions (x) and (p), and corresponding probability

densities |(x)|2and |(p)|

2, for one spin-0 particle in onexorpdimension. The wavefunctions shown

are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds

to the probability density (notthe wavefunction) of finding the particle at positionxor momentump.

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For now, consider the simple case of a single particle, withoutspin,in one spatial dimension.

(More general cases are discussed below).

Position-space wavefunction

The state of such a particle is completely described by its wave function:

,

wherexis position and tis time. This function iscomplex-valued,meaning that (x, t) is acomplex number.

Interpreted as a probability amplitude, if the particle's position ismeasured,its location is not

deterministic, but is described by aprobability distribution.The probability that its positionx

will be in the interval [a, b] (meaning ax b) is:

where tis the time at which the particle was measured. In other words, |(x, t)|2is theprobabilitydensitythat the particle is atx, rather than some other location; seeprobability amplitudefor

details. This leads to the normalization condition:

because if the particle is measured, there is 100% probability that it will besomewhere.

Theinner productof two wave functions 1(x, t) and 2(x, t) is useful and important for anumber of reasons, and can be defined as the complex number (at time t):

In theCopenhagen interpretation,the modulus squared of this complex number gives a real

number, |1, 2|2, which is interpreted as the probability of the wavefunction 2"collapsing"to the new wavefunction 1.

Although the inner product of two wavefunctions is a complex number, the inner product of a

wavefunction with itself, , , is alwaysa positive real number. A wavefunction isnormalized if that real number is 1: , = 1.

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The number is called thenormof the wavefunction. If the wavefunction is not normalized, then dividing by its norm will normalize it. Note that the complex modulus

and the norm are not the same.

A set of wavefunctions 1, 2, ... areorthonormalif they are each normalized (inner producteach wavefunction with itself is 1) and are allorthogonalto each other (inner product of any two

different wavefunctions is zero):

where mand neach take values 1, 2, ..., and mnis theKronecker delta.

Since the Schrdinger equation is linear, if any number of wavefunctons nfor n= 1, 2, ... are

solutions of the equation, then so is their sum, and theirscalar multiplesby complex numbers an

(taking scalar multiplication and addition together is known as alinear combination):

This is thesuperposition principle.Note that multiplying a wavefunction by any nonzero but

constant complex number c(also called a phase factor in this context) does not change any

information about the quantum system, because csatisfies exactly the same Schrdingerequation (the constant ccancels).

Since linear combinations of wavefunctions obtain more wavefunctions, the set of all

wavefunctions forms a complexvector spaceover the field of complex numbers (more details

are given later).

Momentum-space wavefunction

The particle also has a wave function inmomentum space:

wherepis themomentumin one dimension, which can be any value from to +, and tistime. Analogous to the position case, the inner product of two wave functions 1(p, t) and 2(p,t) can be defined as:

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Interpreted as a probability amplitude, if the particle's momentum ismeasured,the result is not

deterministic, but is described by a probability distribution:

,

and the normalization condition is similar:

Relation between wavefunctions

The position-space and momentum-space wave functions areFourier transformsof each other,

therefore both contain the same information, and either one alone is sufficient to calculate any

property of the particle. For one dimension:[11]

Sometimes thewave-vectorkis used in place ofmomentump, since they are related by thede

Broglie relation

and the equivalent space is referred to ask-space.Again it makes no difference which is used

sincepand kare equivalentup to a constant. In practice, the position-space wavefunction isused much more often than the momentum-space wavefunction.

Example of normalization

A particle is restricted to a 1D region betweenx= 0 andx=L; its wave function is:

.

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To normalize the wave function we need to find the value of the arbitrary constantA; solved

from

From , we have ||2=A2, so the integral becomes;

Solving this equation givesA= 1/L, so the normalized wave function in the box is;

One spin-0 particle in three spatial dimensions

The electron probability density for the first fewhydrogen atomelectronorbitalsshown as cross-

sections. These orbitals form anorthonormal basisfor the wave function of the electron. Different

orbitals are depicted with different scale.

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Interpreted as a probability amplitude, the probability of measuring the momentum vector in a

region of momentum spaceMis given by:

where, analogous to position space, d3p= dpxdpydpzis a differential 3-momentum volume

element in momentum space. The normalization condition is:

To get back to the position space wavefunction, we apply the inverse Fourier transform on the

momentum space wavefunction:

Many spin-0 particles in three spatial dimensions

Travelling waves of two free particles, with two of three dimensions suppressed. Top is position space

wavefunction, bottom is momentum space wavefunction, with corresponding probability densities.

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If there are many particles, in general there is only one wave function, not a separate wave

function for each particle. The fact that onewave function describes manyparticles is what

makesquantum entanglementand theEPR paradoxpossible. The position-space wave function

forNparticles is written:[5]

where riis the position of the ith particle in three-dimensional space, and tis time.

The inner product of two wave functions 1(r1, r2, ..., rN, t) and 2(r1, r2, ..., rN, t) can bedefined as the complex number:

(altogether, this is 3None-dimensional integrals).

Interpreted as a probability amplitude, if the particles' positions are all measured simultaneously

at time t, the probability that particle 1 is in regionR1andparticle 2 is in regionR2and so on is:

The normalization condition is:

Definitions (particles with spin)

One particle with spin in three dimensions

For a particle withspin,the wave function can be written in "positionspin space" as:

where ris a position in three-dimensional space, tis time, andszis thespin projection quantum

numberalong thezaxis. (Thezaxis is an arbitrary choice; other axes can be used instead if the

wave function is transformed appropriately, see below.) Theszparameter, unlike rand t, is a

discrete variable. For example, for aspin-1/2particle,szcan only be +1/2 or 1/2, and not anyother value. (In general, for spins,szcan bes,s 1, ... , s.)

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The inner product of two wave functions 1(r1,sz, t) and 2(r1,sz, t) can be defined as thecomplex number:

Interpreted as a probability amplitude, if the particle's position and spin is measured

simultaneously at time t, the probability that its position is inR1andits spin projection quantum

number is a certain valuesz= mis:

The normalization condition is:

.

Since the spin quantum number has discrete values, it must be written as a sum rather than an

integral, taken over all possible values.

It is convenient to write the wavefunction as a column vector, in which there are as many entries

in the column vector as there are allowed values ofsz, and the entries are indexed by the spin

quantum number:[13]

and the normalization condition is equivalent to:

.

where the dagger denotes theHermitian conjugate(complex conjugatetransposeof thecolumn

vectorinto arow vector).

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Many particles with spin in three dimensions

Likewise, the wavefunction forNparticles each with spin is:

The inner product of two wave functions 1(r1, r2, ..., rN,sz1,sz2, ...,sz N, t) and 2(r1, r2, ..., rN,sz1,sz2, ...,sz N, t) can be defined as the complex number:

Now there are 3None-dimensional integrals followed byNsums.

The probability that particle 1 is in regionR1with spinsz1= m1andparticle 2 is in regionR2with

spinsz2= m2etc. reads:

The normalization condition is:

Distinguishable and identical particles

Further information:Identical particles#Wavefunction representation

In quantum mechanics there is a fundamental distinction betweenidentical particlesand

distinguishable particles. For example, any two electrons are fundamentally indistinguishable

from each other; the laws of physics make it impossible to "stamp an identification number" on a

certain electron to keep track of it.[14]

This translates to a requirement on the wavefunction: For

example, if particles 1 and 2 are indistinguishable, then:

wheresis thespin quantum numberof the particle:integerforbosons(s= 1, 2, 3, ...) and half-

integer forfermions(s= 1/2, 3/2, ...).

The wavefunction is said to besymmetric(no sign change) under boson interchange and

antisymmetric(sign changes) under fermion interchange. The latter feature of the wavefunction

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leads to thePauli principle.Generally, bosonic and fermionic symmetry requirements are the

manifestation ofparticle statisticsand are present in other quantum state formalisms.

ForNinteracting distinguishableparticles (all the particles are different, no two areidentical),

which do not interact mutually and move independently in a time-independentpotential, the

(spatial part of the) wavefunction can be separated into a product of separate wavefunctions foreach particle:

[15]

Thisseparation of variablesis a simple method for solving partial differential equations like the

Schrdinger equation. If the potential is time-dependent, then the wavefunction cannot be

separated into the separate wavefunctions of the particles.

For non-interacting distinguishable particles in a time-independent potential, the spatial part ofthe wavefunction is the product of separate wavefunctions for each particle:

Units of the wavefunction

Even though wavefunctions are complex numbers, both the real and imaginary parts each have

the same units (theimaginary unitiis a number without unit). The units of depend on thenumber of particles the wavefunction describes, and the number of spatial or momentum

dimensions of the system. In general, forNparticles with positions r1, r2, ..., rNin nspatial

dimensions, the normalization conditions require to have units of [length]Nn/2

. The square root

of the length unit is removed when one finds ||2, which has units of [length]

Nn.

In momentum space, length is replaced by momentum, and the units are [momentum]Nn/2

.

These results are true for particles with or without spin, since for particles with spin, the

summations are over dimensionless spin quantum numbers.

Wave functions as elements of an abstract vector space

Main article:Quantum state

The set of all possible wave functions (at any given time) forms an abstract mathematicalvector

space.Specifically, the entirewave function is treated as asingleabstract vector:

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where |is a "ket" (a vector) written inbraket notation.As always, the state vector for thesystem is solved from theSchrdinger equation(or otherdynamical pictures of quantum

mechanics):

This vector space is infinite-dimensional,because there is no finite set of functions which can be

added together in various combinations to create every possible function. It is aHilbert space,for

the following reasons.

The statement that "wave functions form an abstract vector space" means that it is possiblemultiplywave functions by complex numbers and add together different wave functions in a

coherent superposition.If |and |are two states in the Hilbert space, and aand bare two

complex numbers, then thelinear combination

(subject to normalization, see below) is also in the Hilbert space. Thedual vectorsare denoted

as "bras", |, which do not live in the same space as|, but instead thedual space:

where * denotescomplex conjugate.

The inner product of wave functions can be defined.See thequantum statearticle for more explanation of the Hilbert space formalism and its

consequences to quantum physics.

There are several advantages to understanding wave functions as elements of an abstract vector

space:

All the powerful tools oflinear algebracan be used to manipulate and understand wavefunctions. For example:

o Linear algebra explains how a vector space can be given abasis,and then any vector inthe vector space can be expressed in this basis. This explains the relationship between awave function in position space and a wave function in momentum space, and suggests

that there are other possibilities too.

o Braket notationcan be used to manipulate wave functions. The idea thatquantum statesare vectors in an abstract vector space (technically, a complex

projectiveHilbert space)is completely general in all aspects of quantum mechanics and

quantum field theory,whereas the idea that quantum states are complex-valued "wave"

functions of space is only true in certain situations.

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Following is a breakdown of thebraketformalism. Kets are analogous to the more elementaryEuclidean vectors,although the components are complex-valued.

Discrete and continuous bases

Discrete componentsAkof a complex vector |A= kAk|ek, which belongs to a countably infinite-

dimensional Hilbert space; there are countably infinitely many kvalues and basis vectors |ek.

Continuous components(x) of a complex vector |= dx(x)|, which belongs to an uncountably

infinite-dimensionalHilbert space;there are uncountably infinitely manyxvalues and basis vectors |x.

Components of complex vectors plotted against index number; discrete kand continuousx. Two

probability amplitudes out of infinitely many are highlighted.

A Hilbert space with a discrete basis |ifor i= 1, 2...nisorthonormalif the inner product of all

pairs of basis kets are given by theKronecker delta:

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Orthonormal bases are convenient to work with because the inner product of two vectors have

simple expressions. A wave function |expressed in this discrete basis of the Hilbert space, and

the corresponding bra in the dual space, are respectively given by:

where the complex numbers ci= i|are the components of the vector. Thecolumn vectoris a

useful representation in terms of matrices. The entire vector |is independent of the basis, but

the components depend on the basis. If achange of basisis made, the components of the vector

must also change to compensate.

A Hilbert space with a continuous basis { |} is orthonormal if the inner product of all pairs of

basis kets are given by theDirac delta function:

As with the discrete bases, a symbol is used in the basis states, two common notations are |

and sometimes |. A particular basis ket may be subscripted |0 |0or primed | |.

While discrete basis vectors are summed over a discrete index, continuous basis vectors are

integrated over a continuous index (a variable of a function). In what follows, all integrals are

with respect to thereal-valued basis variable (not complex-valued), over the required range.

Usually this is just thereal lineorsubsetsof it. The state |in the continuous basis of the

Hilbert space, with the corresponding bra in the dual space, are respectively given by:[16][17]

where the components are the complex-valued functions () = |of a real variable .

Completeness conditions

Thecompleteness conditions(also called closure relations) are

for the discrete and continuous orthonormal bases, respectively. An orthonormal set of kets form

bases if and only if they satisfy these relations.[17]

In each case, the equality to unity means this is

anidentity operator;its action on any state leaves it unchanged. Multiplying any state on the

right of these gives the representation of the state |in the basis. The inner product of a first

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state |with a second |can also be obtained by multiplying | on the left and |on the right

of the relevant completeness condition.

Inner product

Physically, the nature of the inner product is dependent on the basis in use, because the basis ischosen to reflect the quantum state of the system.

If |is a state in the above basis with components c1, c2, ..., cnand |is another state in the

same basis with componentsz1,z2, ...,zn, the inner product is the complex number:

If |is a state in the above continuous basis with components (), and |is another state in the

same basis with components (), the inner product is the complex number:

where the integrals are taken over all and .

Normalization

The square of thenorm(magnitude)of the state vector |is given by the inner product of |

with itself, a real number:

for the discrete and continuous bases, respectively. Each say the projection of a complex

probability amplitude onto itself is real. If |is normalized, these expressions would be unity. If

the state is not normalized, then dividing by its magnitude normalizes the state to:

Normalized components and probabilities

For the discrete basis, projecting the normalized state |Nonto a particular state the system may

collapse to, |q, gives the complex number;

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so the modulus squared of this gives a real number;

In theCopenhagen interpretation,this is the probability of state |qoccurring.

In the continuous basis, the projection of the normalized state onto some particular basis |is acomplex-valued function;

so the squared modulus is a real-valued function

In theCopenhagen interpretation,this function is theprobability density functionof measuring

the observable , so integrating this with respect to between a bgives:

the probability of finding the system with between = aand = b.

Wave function collapse

The physical meaning of the components of |is given by the wave functioncollapse postulate

also known asWave function collapse.If the observable(s) (momentum and/or spin, position

and/or spin, etc.) corresponding to states |ihas distinct and definite values, i, and a

measurement of that variable is performed on a system in the state |then the probability of

measuringiis |i||2. If the measurement yields i, the system "collapses" to the state |i,irreversibly and instantaneously.

Time dependence

In theSchrdinger picture,the states evolve in time, so the time dependence is placed in |

according to:[18]

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In three dimensions, |can be expanded in terms of a continuum of states with definite position,

|r, also written |x,y,zor |r, corresponding to each r= (x,y,z).

If the particle is confined to a regionR(a subset of 3d space), the state is;

The closure relation is

leading to the inner product of |with itself leads to the normalization conditions in the three-

dimensional definitions above:

.

Projecting |onto a particular position state |r0, where r0is inR:

State space for many spin-0 particles in 3d

In three dimensions, |can be expanded in terms of a continuum of states for all the particles

each with definite position, |r1, r2, ..., rN, corresponding to each rj= (xj,yj,zj).

If particle 1 is in regionR1, particle 2 is in regionR2, and so on, the state in this position

representation is:

The closure relation is

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leading to the inner product of |with itself leads to the normalization conditions in the three-

dimensional definitions above:

.

Projecting |onto a particular position state |r1 0, r2 0, ..., rN0, where r1 0is inR1, r2 0is inR2,

etc., gives (r1 0, r2 0, ..., rN0).

Position and spin representations

State space for one particle with spin in 3d

In Dirac notation, for a particle with spins, in all three spatial dimensions, the basis states |r,sz

are a combination of the discrete variableszand the continuous variable r,[13]

more specifically a

tensor productof the spin basis |szand position basis |r, which exists in a new space from the

spin space and position space alone.[19]

Applying the above formalism, the state can be written:

and therefore the closure relation is:

Projecting onto a particular position-spin state |r0, m, where r0is inR:

where the joint orthogonality relation

has been employed. For more particles, there are sums over the allowed spins for each particle,

and integrals over the position vector for each particle.

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Ontology

Main article:Interpretations of quantum mechanics

Whether the wave function really exists, and what it represents, are major questions in the

interpretation of quantum mechanics.Many famous physicists of a previous generation puzzledover this problem, such asSchrdinger,EinsteinandBohr.Some advocate formulations or

variants of theCopenhagen interpretation(e.g. Bohr,Wignerandvon Neumann)while others,

such asWheelerorJaynes,take the more classical approach[20]

and regard the wave function as

representing information in the mind of the observer, i.e. a measure of our knowledge of reality.

Some, including Schrdinger, Einstein,BohmandEverettand others, argued that the wave

function must have an objective, physical existence. The latter argument is consistent with the

fact that whenever two observers both think that a system is in apure quantum state,they will

always agree on exactly what state it is in (but this may not be true if one or both of them thinks

the system is in amixed state).[21]

For more on this topic, seeInterpretations of quantum

mechanics.

Examples

One-dimensional quantum tunnelling

Main articles:Finite potential barrierandQuantum tunnelling

Scattering at a finite potential barrier of height . The amplitudes and direction of left and right

moving waves are indicated. In red, those waves used for the derivation of the reflection and

transmission amplitude. for this illustration.

One of most prominent features of the wave mechanics is a possibility for a particle to reach a

location with a prohibitive (in classical mechanics)force potential.In the one-dimensional case

of particles with energy less than in the square potential

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