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Wavefunction

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This article discusses the concept of a wavefunction as it relates to quantum mechanics. The term has a significantly different meaning when used in the context of classical mechanics or classical electromagnetism.

Contents

Definition

The modern usage of the term wavefunction refers to any vector or function. Typically, a wavefunction is either:

[\vec \psi = \begin c_1 \\ \vdots \\ c_n \end],
[\vec \psi = \begin c_1 \\ \vdots \\ c_n \\ \vdots \end],
[\psi(x_1, \, \ldots \, x_n)].
In all cases, the wavefunction provides a complete description of the associated physical system. An element of a vector space can be expressed in different bases; the same applies to wave functions. The wave function describing the same physical state takes different forms depending on the basis being used.

Interpretation

The physical interpretation of the wavefunction 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 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_ = \int_^ |\psi(x)|^2\, dx ].
This leads to the normalization condition

[ \int_^ |\psi(x)|^2\, dx = 1 \quad ].
since the probability of a measurment of the particle's position yielding a value in [(-\infty, \infty)] is unity.

One particle in three spatial dimensions

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

[\mathbf_R = \int_R |\psi(x, y, z)|^2 \, dV]
The normalization condition is likewise

[ \int |\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 wavefunction 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 region [S] is

[\mathbf_ = \int_R \int_S |\psi|^2 \, dV_2 dV_1 ]
where [dV_1 = dx_1 dy_1 dz_1], and similarly for [dV_2].

The normalization condition is then:

[\int |\psi(x, y, z)|^2 \, dV_2 dV_1 = 1]
where the preceding integral is taken over the full range of all six variables.

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

One particle in one dimensional momentum space

The wavefunction 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_ = \int_^ |\psi(p)|^2\, dp ]
As in the position space case, this leads to the normalization condition:

[\int_^ |\psi(p)|^2\, dp = 1 .]

Spin 1/2

The wavefunction for a spin 1/2 particle (ignoring its spatial degrees of freedom) is a column vector

[\vec \psi = \begin c_1 \\ c_2 \end].
The meaning of the vector's components depends on the basis, but typically [c_1] and [c_2] 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\,].

Interpretation

A wavefunction describes the state of a physical system by expanding it in terms of other states of the same system. We shall denote the state of the system under consideration as [| \psi \rangle\,] and the states into which it is being expanded as [| \phi_i \rangle]. Collectively the latter are referred to as a basis or representation. In what follows, all wavefunctions are assumed to be normalized.

Finite vectors

A wavefunction 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 c_1 \\ \vdots \\ c_n \end],
which is a relation between column vectors, is equivalent to

[|\psi \rangle = \sum_^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 wavefunction 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 wavefunction collapse postulate:

If the states [| \phi_i \rangle] have distinct, definite values, [\lambda_i], 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 [\lambda_i] is [|c_i|^2], and if the measurement yields [\lambda_i], the system is left in the state [| \phi_i \rangle].

Infinite 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 c_1 \\ \vdots \\ c_n \\ \vdots \end]
is equivalent to

[|\psi \rangle = \sum_ 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 wavefunction 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_^ \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 wavefunction 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_^ \delta(x - x_0) | x \rangle\,dx]
and hence the spatial wavefunction associated with [| x_0 \rangle] is [\delta(x - x_0)\,].

Formalism

Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a Hilbert space [H]. Some properties of such a space are

1. If [| \psi \rangle] and [| \phi \rangle] are two allowed states, then
::[a | \psi \rangle + b | \phi \rangle]
is also an allowed state, provided [|a|^2+|b|^2=1]. (This condition is due to normalisation.)
2. There is always an orthonormal basis of allowed states of the vector space H.
The wavefunction associated with a particular state may be seen as an expansion of the state in a basis of [H]. For example,

[\]
is a basis for the space associated with the spin of a spin-1/2 particle and consequently the spin state of any such particle can be written uniquely as

[a|\uparrow_z \rangle + b|\downarrow_z \rangle].
Sometimes it is useful to expand the state of a physical system in terms of states which are not allowed, and hence, not in [H]. An example of this is the spacial wavefunction associated with a particle in one dimension which expands the state of the particle in terms of states with definite position.

Every Hilbert space [H] is equipped with an inner product. Physically, the nature of the inner product is contingent upon the kind of basis in use. When the basis is a countable set [\\,], and orthonormal, i.e.

[\langle \phi_i | \phi_j \rangle = \delta_.]
Then an arbitrary vector [| \psi \rangle] can be expressed as

[| \psi \rangle = \sum_i c_i | \phi_i \rangle]
where [c_i = \langle \phi_i | \psi \rangle.]

If one chooses a "continuous" basis as, for example, the position or coordinate basis consisting of all states of definite position [\], the orthonormality condition holds similarly:

[\langle x | x' \rangle = \delta(x - x').]
We have the analogous identity

[\langle x | \int \psi(x') | x' \rangle \,dx' = \int \psi(x') \delta(x - x')\,dx' = \psi(x).]

See also

References

 

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