Wave function

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The wave function ψ \psi ({\vec {x}},t)or ψ {\displaystyle {\tilde {\psi }}({\vec {p}},t)}describes the quantum mechanical state of an elementary particle or of a system of elementary particles in the spatial or in the momentum space. The basis of the description is the wave mechanics of Erwin Schrödinger. Its magnitude square determines the probability density for the location or the momentum of the particle. According to the Copenhagen interpretation of quantum mechanics, the wave function contains a description of all information of an entity or an entire system.

A wave function is the function that solves the quantum mechanical equation of motion, i.e. the Schrödinger, Klein-Gordon or Dirac equation, in place space or momentum space. Solutions to these wave equations can describe bound particles (such as electrons in the shells of an atom) or free particles (such as an α- or β-particle as a wave packet). The wave function is usually a complex function.

If a system with internal degrees of freedom, for example spin, is described by a wave function, the wave function is vector-valued. The non-relativistic wave function describing an electron therefore has two components; one for the configuration "spin up" and one for "spin down".

For particle systems (e.g. with several indistinguishable particles), such a solution is called a many-particle wave function. However, because of the interaction of the particles with each other, these solutions can usually no longer be calculated without the more modern methodology of quantum field theory.

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

  • the angular frequency ω \omega and
  • the wave vector {\vec {k}}

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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