Wave Function and Its Significance

Wave function,  in quantum mechanics, variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle’s being there at the time. By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance. The square of the wave function, Ψ2, however, does have physical significance: the probability of finding the particle described by a specific wave function Ψ at a given point and time is proportional to the value of Ψ2.

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The modern usage of the term wave function refers to a complex vector or function, i.e. an element in a complex Hilbert space. Typically, a wave function is either:

  • a complex vector with finitely many components
\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix},
  • a complex vector with infinitely many components
\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix},
  • a complex function of one or more real variables (a continuously indexed complex vector)
\psi(x_1, \, \ldots \, x_n).

In all cases, the wave function provides a complete description of the associated physical system. An element of a vector space can be expressed in different bases; and so the same applies to wave functions. The components of a wave function describing the same physical state take different complex values depending on the basis being used; however the wave function itself is not dependent on the basis chosen; in this respect they are like spatial vectors in ordinary space: choosing a new set of Cartesian axes by rotation of the coordinate frame does not alter the vector itself, only the representation of the vector with respect to the coordinate frame. A basis in quantum mechanics is analogous to the coordinate frame: choosing a new basis does not alter the wavefunction, only its representation, which is expressed as the values of the components above.

Because the probabilities that the system is in each possible state should add up to 1, the norm of the wave function must be 1.

 Spatial interpretation

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

 One particle in one spatial dimension

The spatial wavefunction has no actual value and is impossible to solve.

The spatial wave function associated with a particle in one dimension is a complex function \psi(x)\, defined over the real line. The positive function |\psi|^2\, is interpreted as the probability density associated with the particle’s position. That is, the probability of a measurement of the particle’s position yielding a value in the interval [a,b] is given by

\mathbf{P}_{ab} = \int_{a}^{b} |\psi(x)|^2\, dx .

This leads to the normalization condition

 \int_{-\infty}^{\infty} |\psi(x)|^2\, dx = 1 \quad .

since the probability of a measurement of the particle’s position yielding a value in the range (-\infty, \infty) is unity.

 One particle in three spatial dimensions

The three dimensional case is analogous to the one dimensional case; the wave function is a complex function \psi(x, y, z)\, defined over three dimensional space, and the square of its absolute value is interpreted as a three dimensional probability density function:

\mathbf{P}_R = \int_R |\psi(x, y, z)|^2 \, dV

The normalization condition is likewise

 \int_R |\psi(x, y, z)|^2\, dV = 1

where the preceding integral is taken over all space.

 Two distinguishable particles in three spatial dimensions

In this case, the wave function is a complex function of six spatial variables, \psi(x_1, y_1, z_1, x_2, y_2, z_2) \ , and |\psi|^2\, is the joint probability density associated with the positions of both particles. Thus the probability that a measurement of the positions of both particles indicates particle one is in region R and particle two is in regionS is

\mathbf{P}_{R,S} = \int_R \int_S |\psi|^2 \, dV_2 \, dV_1

where dV1 = dx1dy1dz1, and similarly for dV2.

The normalization condition is then:

\int \int |\psi(x_1, y_1, z_1, x_2, y_2, z_2)|^2 \, dV_2 \, dV_1 = 1

in which the preceding integral is taken over the full range of all six variables.

Given a wave function ψ of a system consisting of two (or more) particles, it is in general not possible to assign a definite wave function to a single-particle subsystem. In other words, the particles in the system can be entangled.

One particle in one dimensional momentum space

The wave function for a one dimensional particle in momentum space is a complex function \psi(p)\, defined over the real line. The quantity |\psi|^2\, is interpreted as a probability density function in momentum space:

\mathbf{P}_{ab} = \int_{a}^{b} |\psi(p)|^2\, dp

As in the position space case, this leads to the normalization condition:

\int_{-\infty}^{\infty} |\psi(p)|^2\, dp = 1 .

 Spin 1/2

The wave function for a spin-½ particle (ignoring its spatial degrees of freedom) is a column vector

\vec \psi = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}.

The meaning of the vector’s components depends on the basis, but typically c1 and c2 are respectively the coefficients of spin up and spin down in the z direction. In Dirac notation this is:

| \psi \rangle = c_1 | \uparrow_z \rangle + c_2 | \downarrow_z \rangle

The values |c_1|^2 \, and |c_2|^2 \, are then respectively interpreted as the probability of obtaining spin up or spin down in the z direction when a measurement of the particle’s spin is performed. This leads to the normalization condition

|c_1|^2 + |c_2|^2 = 1\,.


A wave function describes the state of a physical system, | \psi \rangle\,, by expanding it in terms of other possible states of the same system, | \phi_i \rangle. Collectively the latter are referred to as a basis or representation. In what follows, all wave functions are assumed to be normalized.

 Finite dimensional basis vectors

A wave function which is a vector \vec \psi with n components describes how to express the state of the physical system | \psi \rangle as the linear combination of finitely many basis elements | \phi_i \rangle, where i runs from 1 to n. In particular the equation

\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix},

which is a relation between column vectors, is equivalent to

|\psi \rangle = \sum_{i = 1}^n c_i | \phi_i \rangle,

which is a relation between the states of a physical system. Note that to pass between these expressions one must know the basis in use, and hence, two column vectors with the same components can represent two different states of a system if their associated basis states are different. An example of a wave function which is a finite vector is furnished by the spin state of a spin-1/2 particle, as described above.

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

If the states | \phi_i \rangle have distinct, definite values, λi, of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state

|\psi \rangle = \sum_i c_i | \phi_i \rangle
then the probability of measuring λi is | ci | 2, and if the measurement yields λi, the system is left in the state | \phi_i \rangle.

 Infinite dimensional basis vectors

The case of an infinite vector with a discrete index is treated in the same manner a finite vector, except the sum is extended over all the basis elements. Hence

\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}

is equivalent to

|\psi \rangle = \sum_{i} c_i | \psi_i \rangle,

where it is understood that the above sum includes all the components of \vec \psi. The interpretation of the components is the same as the finite case (apply 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 expands the physical state of the particle, | \psi \rangle, in terms of states with definite position, | x \rangle. Thus

| \psi \rangle = \int_{-\infty}^{\infty} \psi(x) | x \rangle\,dx.

Note that | \psi \rangle is not the same as \psi(x)\,. The former is the actual state of the particle, whereas the latter is simply a wave function describing how to express the former as a superposition of states with definite position. In this case the base states themselves can be expressed as

| x_0 \rangle = \int_{-\infty}^{\infty} \delta(x - x_0) | x \rangle\,dx

and hence the spatial wave function associated with | x_0 \rangle is \delta(x - x_0)\, (where \delta(x - x_0)\, is the Dirac delta function).

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