What are quantum mechanical three-dimensional wave functions

The Wave function In quantum mechanics, describes the quantum mechanical state of an elementary particle or a system of elementary particles in spatial space and determines its probability of being. According to the Copenhagen interpretation of quantum mechanics, the wave function contains a description of all information of an entity or an entire system.

The wave function is the - mostly complex - solution of a wave equation that can be set up for individual bound particles such as electrons in the orbital model or free particles (e.g. as a wave packet), as well as for entire particle systems. In classical physics, the term has a significantly different meaning.

Quantum as a wave

While the general description of the wave equation can be found under wave (physics), the main focus here is on the properties of the wave function of a quantum particle. The complex nature of the wave function (discussed here) reflects that does not have the real physical meaning, such as the electric field strength a wave of light. In quantum mechanics, wave functions are used for the mathematical description of the quantum mechanical state of a physical system. The wave function a quantum particle can have the shape of a plane wave (free particle):

in which

  • the position vector,
  • A.0 the (complex-valued) amplitude,
  • the wave vector and
  • ω is the angular frequency.

In Schrödinger's quantum mechanics, wave functions result as a solution to the Schrödinger equation or the wave function must always be a solution to the Schrödinger equation. Particles with internal properties are described by wave functions with several components. According to the transformation behavior of the wave functions in Lorentz transformations, the relativistic quantum field theory distinguishes between scalar, tensorial and spinoral wave functions or fields.

Standardization condition and probability of stay

In contrast to classical physics, an exact statement is made about the whereabouts of a particle is generally not possible (Heisenberg's uncertainty principle). Based on the existence of the quantum particle, it must be somewhere (at all times), which is why its wave function is the normalization condition (where ψ * has to satisfy the complex conjugate function of ψ). This leads to the differential probability dP, the particle in place , in the volume element to be found: .

For a normalized wave function the square of the absolute value gives | ψ | 2 = ψψ * thus the probability density for staying at the place for now t at. For particle wave functions in spatial space, the integration of the probability density over a spatial area (an interval in space) results in the probability of finding a particle (e.g. electron) in this spatial area.


A Wave function refers to any vector or function that contains the State of a physical system describes it by representing it as development according to other states of the same system.

Typical wave functions are either:

  • A vector of complex numbers with finitely many components:
  • A vector of complex numbers with a countably infinite number of components (discrete index):
  • or a complex-valued function of one or more continuously changing real variables:

In all cases the wave function provides a complete description of the physical system in question. It is important to note, however, that a wave function assigned to a particular system does not uniquely determine the system; rather, many different wave functions can describe the same physical scenario.

Particle interpretation

The physical interpretation of a wave function depends on the context. 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.

The probability that the particle in the interval [a,b] is therefore to be found


This leads to the normalization condition

because 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 spatial dimensions

The three-dimensional case is analogous to the one-dimensional; The wave function is a complex function defined over the three-dimensional space, and its absolute square is interpreted as the three-dimensional probability density. The probability of a particle being measured in volume R. to find is therefore


The normalization condition is analogous to the one-dimensional case

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,


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

where dV.1 = dx1dy1dz1 and also for dV.2. The normalization condition is therefore


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

It is crucial that in the case of two-particle systems only the system that is made up both Particle exists, must 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. Modern physics calls this phenomenon quantum entanglement or 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 by the set of real numbers. The size is interpreted as a probability density function in momentum space. The probability that a pulse measurement will have a value in the interval [a,b] is consequently


This leads to the normalization condition


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


The meaning of the components of the vector depends on the base used, typically correspond c1 and c2 the coefficient for aligning the spin in z-Direction (spin up) and contrary to the z-Direction (spin down). In Dirac notation this is:

The values and are then interpreted as the probabilities that the spin when measured in z-Direction or against the z- Direction is oriented.

This leads to the normalization condition


Basic interpretation vectors

Wave functions can be understood as elements of a vector space (Hilbert space). A wave function that describes the state of a physical system can be described by the linear combination of other states of the same system. We denote the state of the system under consideration as and the states in which it is developed as . The latter states are said to be one Base of vector space. In the following, all wave functions are assumed to be normalized.

Finite vectors

A vector wave function With n Components, describes how to get the state of the physical system as a linear combination of finitely many basic elements , Which i from 1 to n run, expresses. In particular, the equation is


which is a relation between column vectors, equivalent to


which is a relation between the states of a physical system. It should be noted that when switching between these expressions one has to know the basis used, and consequently two column vectors with the same components represent two different system states if the underlying basis states are different. An example of a finite, vectorial wave function is given by the spin state of a particle with spin 1/2, as described above.

The physical meaning of the components of is given by the postulate of the collapse of the wave function:

When the states unique, finite values ​​λi of some dynamic variables (e.g. pulse, position, etc.), and these variables in a system in the state
to be measured
then the probability is the value λi to measure, given by | ci | 2, and if the measurement λi results, then the system remains in the state .

Infinite vectors

The case of infinite vectors with a discrete index is treated in the same way as a finite vector, with the exception that the sum is expanded over all (infinitely many) basic elements. Hence is

equivalent to


where in the above sum all components of are taken into account. The interpretation of the components is the same as in the finite case (as well as the collapse postulate).

Continuously indexed vectors (functions)

In the case of a continuous index, the sum is replaced by an integral; an example of this is the spatial wave function of a particle in one dimension, which is the physical state of the particle in expressions of states with defined positions developed:


It is not the same as . The former gives the actual state of the particle, while the latter is simply a wave function describing how the state is to be expressed as a superposition of states with defined positions. In this case the base states can be used as

be written, and consequently that is too corresponding spatial wave function δ (xx0). See also: Delta Distribution.


In a given isolated physical system, the permitted states (i.e. the states that the system can adopt without violating the laws of physics) are a subset of a vector space H, the Hilbert room. Specifically, this subset is the set of all vectors with length 1, i.e. the unit sphere around the origin. This follows from the fact that all physically permitted states are normalized. It follows:

  • If and are two allowed states, then is also a permitted state if and only if | α | 2 + | β | 2 = 1 applies (normalization).
  • Because of the normalization, for the vector space H an orthonormal basis from physically permitted states can always be found.

In this context, the wave function of a certain state can be expressed as an evolution of the state on a basis of the vector space H to be viewed as. For example is

a basis of space that describes a particle with spin 1/2, and it follows that the spin state of such a particle is through

is clearly described.

It is usual, H to be provided with an inner product (scalar product), the type of inner product depending on the base used. When there are countably many basic elements which all too H belong, there, then is H provided with the unique inner product which makes this basis orthonormal, e.g. B .:

When that happens, the inner product is of with the development of any vector


The coefficient ci the development of the state | ψ> into the base thus results from projection onto the basis vector .

If the eigenvalues ​​form a continuum, which is the case with the location or coordinate base, for example, no Hilbert space can be built from the corresponding eigenstates, since these eigenstates are not square-integrable. By using Dirac's delta distribution, however, a generalized orthonormalization condition can be formulated for these base states. Such bases are also called improper bases designated. An example of this is the above-mentioned development of the spatial wave function of a particle into states with a certain position x, with the Dirac normalization

so that the analog identity

is satisfied.


The text from the definition section is based on a translation of Article [1] version dated March 12, 2006.

See also

Category: quantum physics