Wave function

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The wave function or describes the quantum mechanical state of an elementary particle or a system of elementary particles in spatial or momentum space . The basis of the description is the wave mechanics by Erwin Schrödinger . Their square of absolute values determines the probability density for the position 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. ${\displaystyle \psi ({\vec {x}},t)}$ ${\displaystyle {\tilde {\psi }}({\vec {p}},t)}$

A wave function is the function that solves the quantum mechanical equation of motion , i.e. the Schrödinger , Klein-Gordon or Dirac equation , in spatial space or in momentum space. Solutions of these wave equations can describe both bound particles (such as electrons in the shells of an atom ) or free particles (e.g. 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 for describing an electron therefore has two components; one for the "spin up" configuration and one for "spin down".

In particle systems (e.g. with several indistinguishable particles ) such a solution is called a many-body wave function . Because of the interaction of the particles with one another, 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 a function for the general solution whose function values ​​are also in complex space. Therefore the wave function is not real, but complex-valued. This is reflected u. a. is reflected in the fact that it does not necessarily have a real physical meaning. As a rule, it cannot be measured , but only serves to mathematically describe the quantum mechanical state of a physical system. However, it can be used to calculate the expected result of a measurement using complex conjugation . ${\displaystyle \psi ({\vec {r}},t)}$

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

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 behavior of the wave functions in Lorentz transformations, a distinction is made in the relativistic quantum field theory between scalar , tensor and spinoral wave functions or fields.

definition

Expansion coefficients of the state vector

From a formal point of view, the wave functions are the expansion coefficients of the quantum mechanical state vector in space or momentum space. It's 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 ${\displaystyle |\psi \rangle }$
• the local eigenco-states ${\displaystyle \langle x|}$
• the momentum eigenco states ${\displaystyle \langle p|}$

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 position and momentum eigen-states are the eigen-states of the position operator or momentum operator for which and applies. From the definition it becomes obvious that the wave function in the position as well as in the momentum space follows a normalization condition, since the state vector is already normalized: ${\displaystyle {\hat {x}}}$ ${\displaystyle {\hat {p}}}$ ${\displaystyle {\hat {x}}|x\rangle =x|x\rangle }$ ${\displaystyle {\hat {p}}|p\rangle =p|p\rangle }$

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

Solution of the equation of motion

The wave functions are of more practical importance as a solution to the equations of motion in space or momentum space. One makes use of the fact that the position operator in the position basis is a multiplication operator and the momentum operator in the position basis is a differential operator. In the momentum basis the roles are reversed, there the position 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 local space

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

With

• the reduced Planck quantum of action , ${\displaystyle \hbar }$
• the Laplace operator , ${\displaystyle \Delta }$
• the mass of the particle and ${\displaystyle m}$
• a location-dependent potential ; ${\displaystyle V(x)}$

all properties of the wave function (discussed in this article) that 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, does not represent a wave equation in the strict mathematical sense, a solution of the Schrödinger equation in spatial space with 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 and ${\displaystyle \omega }$
• the wave vector ${\displaystyle {\vec {k}}}$

given is.

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

Wave function in momentum space

The wave function in momentum space is linked to the wave function in spatial space via a Fourier transformation . It applies ${\displaystyle {\tilde {\psi }}({\vec {p}})}$ ${\displaystyle \psi ({\vec {x}})}$

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

in addition to the replacement . Due to Plancherel's theorem , the Fourier transform is compatible with normalization, so that the wave function in momentum space is normalized just like the wave function in spatial space. ${\displaystyle {\vec {p}}=\hbar {\vec {k}}}$

Example: free particle

The wave function of a free particle can be represented as a Fourier series over plane waves: ${\displaystyle \psi ({\vec {x}},t)}$

${\displaystyle \psi ({\vec {x}},t)=\int {\frac {\mathrm {d} ^{3}{\vec {k}}}{(2\pi )^{3}}}A({\vec {k}})e^{\mathrm {i} (\omega t-{\vec {k}}\cdot {\vec {x}})}}$

With

• the position vector ${\displaystyle {\vec {x}}}$
• the wave vector that defines the direction and wavelength of the wave ${\displaystyle {\vec {k}}}$
• the complex-valued amplitudes dependent on the wave vector ${\displaystyle A({\vec {k}})}$
• the angular frequency , which describes the oscillation period of the wave and is linked to the wave vector via a dispersion relation . ${\displaystyle \omega ({\vec {k}})}$

The amplitudes must be chosen so that the normalization of the wave function is guaranteed. The square of the magnitude of the wave function is through

${\displaystyle {\begin{aligned}|\psi ({\vec {x}},t)|^{2}&=\psi ^{\dagger }({\vec {x}},t)\,\psi ({\vec {x}},t)\\&=\int {\frac {\mathrm {d} ^{3}{\vec {k}}}{(2\pi )^{3}}}\int {\frac {\mathrm {d} ^{3}{\vec {k}}'}{(2\pi )^{3}}}A^{\dagger }({\vec {k}})A({\vec {k}}')e^{\mathrm {i} ({\vec {k}}-{\vec {k}}')\cdot {\vec {x}}}e^{-\mathrm {i} (\omega -\omega ')t}\end{aligned}}}$

given. An integration over the entire volume results with the representation of the Dirac distribution : ${\displaystyle \textstyle \int \mathrm {d} ^{3}x\,e^{i{\vec {k}}\cdot {\vec {x}}}=\delta ^{(3)}({\vec {k}})}$

${\displaystyle \int \mathrm {d} ^{3}{\vec {x}}\,|\psi ({\vec {x}},t)|^{2}=\int {\frac {\mathrm {d} ^{3}{\vec {k}}}{(2\pi )^{3}}}A^{\dagger }({\vec {k}})A({\vec {k}})=1}$.

In practice, this can be done, for example, by a Gaussian envelope

${\displaystyle A({\vec {k}})=\left[{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {{\vec {k}}^{2}}{2\sigma ^{2}}}}\right]^{1/2}}$

will be realized. By choosing this envelope, a particle with minimal spatial impulse uncertainty and an expected value of the impulse at is described. is the width of the wave packet, which to a certain extent indicates how the uncertainty is distributed over the location and impulse expected value. ${\displaystyle {\vec {p}}_{0}=0}$ ${\displaystyle \sigma }$

Measurements in wave mechanics

A statement in the quantum mechanical measuring process is that during a measurement the wave function collapses instantaneously to an eigenvalue of the operator associated with the measurement. This eigenvalue is the result of the measurement. The probability of collapsing to one of these eigenvalues ​​is through in matrix mechanics

${\displaystyle P=\|\langle \phi |\psi \rangle \|^{2}}$

given, where the eigenstate of an operator belonging to the eigenvalue is. In wave mechanics this corresponds to the formulation ${\displaystyle |\phi \rangle }$ ${\displaystyle \phi }$ ${\displaystyle \Phi }$

${\displaystyle P=\left\|\int \mathrm {d} ^{3}{\vec {x}}\,\langle \phi |x\rangle \langle x|\psi \rangle \right\|^{2}=\left\|\int \mathrm {d} ^{3}{\vec {x}}\,\phi ^{\dagger }(x)\psi (x)\right\|^{2}}$.

The scalar product of the Hilbert space thus corresponds to an integration over the entire spatial area in the local space. Two wave functions are called orthogonal if the integral vanishes over the entire spatial space of their product. The probability of obtaining the measured value if the system is described by the wave function and and are orthogonal is correspondingly zero. ${\displaystyle \phi }$ ${\displaystyle \psi (x)}$ ${\displaystyle \phi (x)}$ ${\displaystyle \psi (x)}$

The expected value of a measurement in the state is determined by the matrix mechanics ${\displaystyle |\psi \rangle }$

${\displaystyle \langle \Phi \rangle =\langle \psi |\Phi |\psi \rangle }$

described. In wave mechanics, this translates to:

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

Here, the operator in position representation. The following applies to local operators and the double integration is reduced to a simple one: ${\displaystyle \Phi (x,x')}$ ${\displaystyle \Phi ({\vec {x}},{\vec {x}}')=\Phi ({\vec {x}})\delta ^{(3)}({\vec {x}}-{\vec {x}}')}$

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

Particle interpretation

The physical interpretation of a wave function is context dependent. Several examples are given below, followed by an interpretation of the three cases described above.

A particle in one dimension of space

The wave function of a particle in one-dimensional space is a complex function over the set of real numbers . The square of the magnitude of the wave function,, is interpreted as the probability density of the particle position. ${\displaystyle \psi (x)\,}$ ${\displaystyle |\psi |^{2}\,}$

The probability of finding the particle in the interval during a measurement is consequently ${\displaystyle [a,b]}$

${\displaystyle \int _{a}^{b}|\psi (x)|^{2}\,\mathrm {d} x\quad }$.

This leads to the normalization condition

${\displaystyle \int _{-\infty }^{\infty }|\psi (x)|^{2}\,\mathrm {d} x\,{\stackrel {!}{=}}\,1\quad }$

since a measurement of the particle position must result in a real number. That means: the probability of finding a particle in any place is equal to 1.

A particle in three dimensions of space

The three-dimensional case is analogous to the one-dimensional; The wave function is a complex function defined over three-dimensional space, and its square of magnitude is interpreted as a three-dimensional probability density. The probability of finding the particle in the volume during a measurement is therefore ${\displaystyle \psi (x,y,z)\,}$ ${\displaystyle R}$

${\displaystyle \int _{R}|\psi (x,y,z)|^{2}\,\mathrm {d} V}$.

The normalization condition is analogous to the one-dimensional case

${\displaystyle \int |\psi (x,y,z)|^{2}\,\mathrm {d} V=1}$

where the integral extends over the entire space.

Two distinguishable particles in three spatial dimensions

In this case the wave function is a complex function of six space variables,

${\displaystyle \psi (x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})\,}$,

and is the joint probability density function of the positions of both particles. The probability of a position measurement of both particles in the two respective regions R and S is then ${\displaystyle |\psi |^{2}\,}$

${\displaystyle \int _{R}\int _{S}|\psi |^{2}\,\mathrm {d} V_{2}\,\mathrm {d} V_{1}}$

where and also for . The normalization condition is therefore ${\displaystyle \mathrm {d} V_{1}=\mathrm {d} x_{1}\mathrm {d} y_{1}\mathrm {d} z_{1}}$ ${\displaystyle \mathrm {d} V_{2}}$

${\displaystyle \int |\psi |^{2}\,\mathrm {d} V_{2}\,\mathrm {d} V_{1}=1}$,

where the integral presented here covers the entire range of all six variables.

It is of crucial importance that in the case of two-particle systems only the system, which consists of both particles, has to have a well-defined wave function. It follows from this that it may be impossible to define a probability density for particle ONE that does not explicitly depend on the position of particle TWO. The Modern physics calls this phenomenon quantum entanglement and quantum non-locality .

A particle in one-dimensional momentum space

The wave function of a one-dimensional particle in momentum space is a complex function defined on the set of real numbers. The size is interpreted as a probability density function in momentum space. The probability that a pulse measurement will give a value in the interval is consequently ${\displaystyle \psi (p)\,}$ ${\displaystyle |\psi |^{2}\,}$ ${\displaystyle [a,b]}$

${\displaystyle \int _{a}^{b}|\psi (p)|^{2}\,\mathrm {d} p\quad }$.

This leads to the normalization condition

${\displaystyle \int _{-\infty }^{\infty }|\psi (p)|^{2}\,\mathrm {d} p=1}$,

because a measurement of the particle momentum always gives a real number.

Spin 1/2 particle (e.g. electron)

The wave function of a particle with spin 1/2 (without considering its spatial degrees of freedom) is a column vector

${\displaystyle {\vec {\psi }}={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}}$.

The meaning of the components of the vector depends on the basis used, typically correspond to and the coefficients for an alignment of the spin in the direction ( spin up ) and against the direction ( spin down ). In Dirac notation this is: ${\displaystyle c_{1}}$ ${\displaystyle c_{2}}$ ${\displaystyle z}$ ${\displaystyle z}$

${\displaystyle |\psi \rangle =c_{1}|{\mathord {\uparrow }}_{z}\rangle +c_{2}|{\mathord {\downarrow }}_{z}\rangle }$

The values and are then interpreted as the probabilities that the spin is oriented in a measurement in the direction or against the direction. ${\displaystyle |c_{1}|^{2}\,}$ ${\displaystyle |c_{2}|^{2}\,}$ ${\displaystyle z}$ ${\displaystyle z}$

This leads to the normalization condition

${\displaystyle |c_{1}|^{2}+|c_{2}|^{2}=1\,}$.

See also

• Boson - particles with a symmetrical wave function under permutation .
• Fermion - particles with an antisymmetric wave function under permutation.
• Exchange interaction