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Wave function (quantum state and probability amplitude)

A wave function is the complex-valued mathematical object that encodes a quantum system's probability amplitudes. Its modulus squared gives measurable probabilities and its evolution follows the Schrödinger equation.

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The wave function is the central mathematical description of the state of a quantum system. In basic formulations it is written as the symbol ψ or Ψ and assigns a complex number — a probability amplitude — to each possible configuration of the system. In the context of quantum mechanics, the wave function provides all information that can be used to predict measurement outcomes, though the link between the function and observed results requires an additional rule known as the Born rule.

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Mathematical character and properties

Formally the wave function is a complex-valued function of the coordinates of the system and time, ψ(x,t) for a particle in one spatial dimension. Physical predictions are obtained from the product ψ*(x,t)ψ(x,t), the absolute square of the amplitude, which gives a probability density for finding the particle at position x. The wave function is typically normalized so that the integral of this density over all space equals one. Global phase factors multiply ψ by a common complex phase without changing observable predictions, while relative phases between components can produce interference effects.

Evolution and history

The time evolution of nonrelativistic wave functions is governed by the Schrödinger equation. The equation was introduced in the 1920s and remains the standard dynamical law for many quantum systems: it determines how amplitudes change in time given the system's energy and interactions. Stationary states, solutions of the time-independent Schrödinger equation, appear when energy is well defined and lead to standing-wave-like spatial patterns.

Measurement, interpretation and multi-particle states

When a measurement is performed the wave function is used to compute probabilities; interpretations differ about what happens to the function itself. The Copenhagen view speaks of a collapse to an eigenstate, while other interpretations treat collapse differently or not at all. For many particles the wave function depends on all particle coordinates and may be symmetric or antisymmetric under exchange, a property that underlies the distinction between bosons and fermions. Examples include ψ describing an electron localized near a nucleus or extended molecular orbitals.

Applications and notable facts

The wave function underlies atomic and molecular structure, tunneling phenomena, quantum interference, and the design of devices such as transistors and quantum bits. Operators representing observables act on the wave function to produce expectation values via integrals of the form ∫ψ*Âψ. The complex nature of ψ means it is built from complex numbers, and alternative representations such as momentum-space wave functions are related by Fourier transforms. The concept ties to earlier ideas of a matter wave introduced by de Broglie and is formalized in the Schrödinger equation.

  • Key roles: encodes amplitudes, yields probabilities, predicts dynamics.
  • Limitations: not directly observable; physical interpretation depends on measurement theory.
  • Extensions: spinors and field-theory states generalize the single-particle wave function.

Quantum particle as a wave

Since the equations of motion are defined in complex space, they require for their general solution a function whose function values also lie in complex space. Therefore, the wave function is not real but complex-valued. One reflection of this is that ψ {\displaystyle \psi ({\vec {r}},t)}not necessarily have real physical meaning. It is usually not measurable, but only serves as a mathematical description of the quantum mechanical state of a physical system. However, it can be used to calculate the expected result of a measurement by complex conjugation.

For comparison: The electric field strength of a{\vec {E}}({\vec {r}},t) radio wave is also the solution of a (classical) electrodynamic wave equation. However, the electric field strength can be measured, for example, by an antenna and a radio receiver.

Particles with internal properties (such as the spin of a bound electron or the angular momentum of a photon) are described by wave functions with several components. Depending on the transformation behaviour of the wave functions in Lorentz transformations, one distinguishes in relativistic quantum field theory between scalar, tensorial and spinorial wave functions or fields.

Definition

Evolution coefficients of the state vector

Formally, the wave functions are the evolution coefficients of the quantum mechanical state vector in the spatial or momentum space. It is in Dirac notation

{\displaystyle {\begin{aligned}\psi ({\vec {x}},t)&=\langle x|\psi (t)\rangle \\{\tilde {\psi }}({\vec {p}},t)&=\langle p|\psi (t)\rangle \end{aligned}}}

with

  • the state vector |\psi \rangle
  • the locus eigenstates ⟨ \langle x|
  • the pulse eigenstates ⟨ {\displaystyle \langle p|}

so that:

{\displaystyle |\psi \rangle =\int \mathrm {d} ^{3}{\vec {x}}\,|x\rangle \langle x|\psi \rangle =\int \mathrm {d} ^{3}{\vec {x}}\,|x\rangle \psi ({\vec {x}})}

{\displaystyle |\psi \rangle =\int \mathrm {d} ^{3}{\vec {p}}\,|p\rangle \langle p|\psi \rangle =\int \mathrm {d} ^{3}{\vec {p}}\,|p\rangle {\tilde {\psi }}({\vec {p}})}

The location and momentum eigenstates are the eigenstates of the location operator {\hat {x}} and momentum operator respectively.} {\hat {p}}, for which {\displaystyle {\hat {x}}|x\rangle =x|x\rangle }and {\displaystyle {\hat {p}}|p\rangle =p|p\rangle }holds. From the definition, it is obvious that the wave function in the spatial as well as the momentum space follow a normalization condition, since the state vector is already normalized:

{\displaystyle 1=\langle \psi |\psi \rangle =\int \mathrm {d} ^{3}{\vec {x}}\,\psi ^{\dagger }({\vec {x}})\psi ({\vec {x}})=\int \mathrm {d} ^{3}{\vec {p}}\,{\tilde {\psi }}^{\dagger }({\vec {p}}){\tilde {\psi }}({\vec {p}})}

Solution of the equation of motion

Of more practical importance are the wave functions as solutions of the equations of motion in place or momentum space. Here one makes use of the fact that the location operator in the location basis is a multiplication operator and the momentum operator in the location basis is a differential operator. In momentum basis the roles are reversed, there the location operator is a differential operator and the momentum operator is a multiplication operator.

All equations of motion in quantum mechanics are wave equations. The Schrödinger equation is in the base-independent Dirac notation

{\displaystyle \mathrm {i} \hbar \partial _{t}|\psi \rangle ={\frac {{\hat {p}}^{2}}{2m}}|\psi \rangle +V({\hat {x}})|\psi \rangle }

and in the local area

{\displaystyle \mathrm {i} \hbar \partial _{t}\psi ({\vec {x}},t)={\frac {-\hbar ^{2}}{2m}}\Delta \psi ({\vec {x}},t)+V({\vec {x}})\psi ({\vec {x}},t)}

with

  • the reduced Planck quantum of action \hbar ,
  • the Laplace operator Δ \Delta ,
  • the mass of the particle mand
  • a location-dependent potential V(x);

all properties of the wave function (discussed in the context of this article) which solve the non-relativistic Schrödinger equation can be generalized to the relativistic case of the Klein-Gordon or the Dirac equation.

Although the Schrödinger equation, in contrast to its relativistic equivalents, is not a wave equation in the mathematically strict sense, a solution of the Schrödinger equation in local space at vanishing potential is a plane wave, represented by the function

{\displaystyle \psi ({\vec {x}},t)=\exp(\mathrm {i} (\omega t-{\vec {k}}\cdot {\vec {x}}))}.

Their dispersion relation is:

{\displaystyle \omega ({\vec {k}})={\frac {\hbar {\vec {k}}^{2}}{2m}}}

with

is given.

Since the equations of motion are linear, any superposition of solutions is again a solution.

wave function in momentum space

The wave function in momentum space ψ {\displaystyle {\tilde {\psi }}({\vec {p}})}is related to the wave function in location space ψ \psi ({\vec x})via a Fourier transform. It holds

{\displaystyle {\tilde {\psi }}({\vec {p}},t)=\int \mathrm {d} ^{3}{\vec {x}}\,\psi ({\vec {x}},t)e^{-\mathrm {i} {\vec {p}}\cdot {\vec {x}}}}

together with the substitution \vec p = \hbar \vec k . Due to Plancherel's theorem, the Fourier transform is compatible with normalization, so the wavefunction in momentum space is normalized in the same way as the wavefunction in place space.

Questions and answers

Q: What does the wave function represent in quantum mechanics?

A: The wave function describes the probability of finding an electron somewhere in its matter wave.

Q: How is the wave function usually represented?

A: The wave function is usually represented by Ψ or ψ.

Q: What does the square of the wave function give?

A: The square of the wave function gives the probability of finding the location of the electron in the given area.

Q: Why is the normal answer for the wave function usually a complex number?

A: The normal answer for the wave function is usually a complex number because it takes into account the wave-particle duality of electrons.

Q: What is the Schrödinger equation?

A: The Schrödinger equation is an equation in quantum mechanics that describes the evolution of a physical system over time.

Q: Who introduced the concept of the wave function?

A: The concept of the wave function was first introduced in the Schrödinger equation.

Q: How does the wave function concept relate to electrons?

A: The wave function concept describes the probability of finding an electron in a given area, taking into account the wave-particle duality of electrons.

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