# Wave function

**wavefunctionwave functionsnormalizedwavefunctionsnormalizationnormalizablenormalized wavefunctionstateswave-functionbelow**

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.wikipedia

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### Quantum mechanics

**quantum physicsquantum mechanicalquantum theory**

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

In one of them, a mathematical function, the wave function, provides information about the probability amplitude of position, momentum, and other physical properties of a particle.

### Probability amplitude

**probability densityBorn probabilitytransition amplitude**

The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it.

Probability amplitudes provide a relationship between the wave function (or, more generally, of a quantum state vector) of a system and the results of observations of that system, a link first proposed by Max Born.

### Quantum state

**eigenstatepure stateeigenstates**

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.

If this Hilbert space, by choice of representation (essentially a choice of basis corresponding to a complete set of observables), is exhibited as a function space (a Hilbert space in its own right), then the functions representing elements of the Hilbert space are called wave functions.

### Schrödinger equation

**Schrödinger's equationwave mechanicsSchrödinger wave equation**

The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation.

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system.

### Max Born

**BornBorn, M.Born M**

In Born's statistical interpretation in non-relativistic quantum mechanics, the squared modulus of the wave function,

Born won the 1954 Nobel Prize in Physics for his "fundamental research in quantum mechanics, especially in the statistical interpretation of the wave function".

### Psi (Greek)

**psiΨψ**

(lower-case and capital psi, respectively).

The letter psi is commonly used in physics to represent wave functions in quantum mechanics, such as in the Schrödinger equation and bra–ket notation:.

### Measurement in quantum mechanics

**measurementquantum measurementmeasurements**

, is a real number interpreted as the probability density of measuring a particle's being detected at a given place – or having a given momentum – at a given time, and possibly having definite values for discrete degrees of freedom.

As an analogue, in quantum mechanics a system is described by its quantum state or wave function, which contains the probabilities of possible positions and momenta.

### Erwin Schrödinger

**SchrödingerErwin SchroedingerErwin Schrodinger**

Those who used the techniques of calculus included Louis de Broglie, Erwin Schrödinger, and others, developing "wave mechanics".

Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as ' or ', was a Nobel Prize-winning Austrian physicist who developed a number of fundamental results in the field of quantum theory: the Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time.

### Born rule

**Born's ruleBorn's lawrules**

The inner product between two wave functions is a measure of the overlap between the corresponding physical states, and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products.

In its simplest form it states that the probability density of finding the particle at a given point is proportional to the square of the magnitude of the particle's wavefunction at that point.

### Momentum

**conservation of momentumlinear momentummomenta**

In 1905, Einstein postulated the proportionality between the frequency f of a photon and its energy E, E = hf, and in 1916 the corresponding relation between photon's momentum p and wavelength \lambda, where h is the Planck constant.

The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function.

### Hartree–Fock method

**Hartree–Fockself-consistent fieldHartree–Fock theory**

Now it is also known as the Hartree–Fock method.

In computational physics and chemistry, the Hartree–Fock (HF) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

### Slater determinant

**Slater determinantsdeterminant**

The Slater determinant and permanent (of a matrix) was part of the method, provided by John C. Slater.

In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system.

### Many-body problem

**many-bodymany-body systemmany-particle system**

In 1927, Hartree and Fock made the first step in an attempt to solve the N-body wave function, and developed the self-consistency cycle: an iterative algorithm to approximate the solution.

As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible.

### Relativistic wave equations

**relativistic wave equationrelativistic quantum field equationsrelativistic**

Later, other relativistic wave equations were found.

(Greek psi), are referred to as "wave functions" in the context of RQM, and "fields" in the context of QFT.

### Fourier transform

**continuous Fourier transformFourierFourier transforms**

For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform.

In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant.

### Electron

**electronse − electron mass**

Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin. Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation.

In quantum mechanics, the wave-like property of one particle can be described mathematically as a complex-valued function, the wave function, commonly denoted by the Greek letter psi .

### Superposition principle

**superpositionlinear superpositionsuperpose**

According to the superposition principle of quantum mechanics, wave functions can be added together and multiplied by complex numbers to form new wave functions and form a Hilbert space.

The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation.

### Photon

**photonslight quantaincident photon**

Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin.

More generally, the normal concept of a Schrödinger probability wave function cannot be applied to photons.

### Dirac equation

**Dirac particleDiracDirac mass**

Soon after in 1928, Dirac found an equation from the first successful unification of special relativity and quantum mechanics applied to the electron, now called the Dirac equation.

is the wave function for the electron of rest mass

### Position and momentum space

**momentum spaceposition spaceposition**

For instance, it may be taken to be a function of all the position coordinates of the particles over position space, or the momenta of all the particles over momentum space; the two are related by a Fourier transform.

If one chooses the eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function \psi(r) in position space (our ordinary notion of space in terms of length).

### Relativistic quantum mechanics

**non-relativisticnonrelativisticrelativistic equations**

The branch of quantum mechanics where these equations are studied the same way as the Schrödinger equation, often called relativistic quantum mechanics, while very successful, has its limitations (see e.g. Lamb shift) and conceptual problems (see e.g. Dirac sea).

The solution is a complex-valued wavefunction

### Bargmann–Wigner equations

**Bargmann-Wigner equationsBargmann–Wigner program**

), and, more generally, the Bargmann–Wigner equations.

). The solutions to the equations are wavefunctions, mathematically in the form of multi-component spinor fields.

### Wave function collapse

**wavefunction collapsecollapsecollapse of the wave function**

"collapsing" to the new wave function

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world.

### Spin (physics)

**spinnuclear spinspins**

Some particles, like electrons and photons, have nonzero spin, and the wave function for such particles includes spin as an intrinsic, discrete degree of freedom; other discrete variables can also be included, such as isospin.

The corresponding normalized eigenvectors are:

### Wave–particle duality

**wave-particle dualityparticle theory of lightwave nature**

This explains the name "wave function", and gives rise to wave–particle duality.

In the formalism of the theory, all the information about a particle is encoded in its wave function, a complex-valued function roughly analogous to the amplitude of a wave at each point in space.