Wave function

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A wave function or wavefunction is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that maps the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states.

For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.

The electron probability density for the first few hydrogen atom electron orbitals shown as cross-sections. These orbitals form an orthonormal basis for the wave function of the electron.

Definition

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}

,

• a complex vector with infinitely many components

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}

,

where each component may be complex function of one or more real variables (a "continuously indexed" complex vector)

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 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 wave function associated with a particle in one dimension is a complex function Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x)\,

defined over the real line. The  positive function Failed to parse (Missing texvc executable; please see math/README to configure.): |\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 Failed to parse (Missing texvc executable; please see math/README to configure.): [a, b]
is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_{ab} = \int_{a}^{b} |\psi(x)|^2\, dx

.

This leads to the normalization condition

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): (-\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 Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x, y, z)\,

defined over three dimensional space, and its complex square is interpreted as a three dimensional probability density function:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_R = \int_R |\psi(x, y, z)|^2 \, dV

The normalization condition is likewise

Failed to parse (Missing texvc executable; please see math/README to configure.): \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, Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(x_1, y_1, z_1, x_2, y_2, z_2) \ , and Failed to parse (Missing texvc executable; please see math/README to configure.): |\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 Failed to parse (Missing texvc executable; please see math/README to configure.): R
and particle two is in region Failed to parse (Missing texvc executable; please see math/README to configure.): S
is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_{R,S} = \int_R \int_S |\psi|^2 \, dV_2 \, dV_1

where Failed to parse (Missing texvc executable; please see math/README to configure.): dV_1 = dx_1 dy_1 dz_1 , and similarly for Failed to parse (Missing texvc executable; please see math/README to configure.): dV_2 .

The normalization condition is then:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int \int |\psi(x, y, z)|^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 Failed to parse (Missing texvc executable; please see math/README to configure.): \psi(p)\,

defined over the real line. The quantity Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi|^2\,
is interpreted as a probability density function in momentum space:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{P}_{ab} = \int_{a}^{b} |\psi(p)|^2\, dp

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}

.

The meaning of the vector's components depends on the basis, but typically Failed to parse (Missing texvc executable; please see math/README to configure.): c_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): c_2

are respectively the coefficients of spin up and spin down in the Failed to parse (Missing texvc executable; please see math/README to configure.): z

direction. In Dirac notation this is:

Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle = c_1 | \uparrow_z \rangle + c_2 | \downarrow_z \rangle

The values Failed to parse (Missing texvc executable; please see math/README to configure.): |c_1|^2 \,

and Failed to parse (Missing texvc executable; please see math/README to configure.): |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

Failed to parse (Missing texvc executable; please see math/README to configure.): |c_1|^2 + |c_2|^2 = 1\,

.

Interpretation

A wave function describes the state of a physical system, Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle\, , by expanding it in terms of other possible states of the same system, Failed to parse (Missing texvc executable; please see math/README to configure.): | \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi

with Failed to parse (Missing texvc executable; please see math/README to configure.): n
components describes how to express the state of the physical system Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle
as the linear combination of finitely many basis elements Failed to parse (Missing texvc executable; please see math/README to configure.): | \phi_i \rangle

, where Failed to parse (Missing texvc executable; please see math/README to configure.): i

runs from Failed to parse (Missing texvc executable; please see math/README to configure.): 1
to Failed to parse (Missing texvc executable; please see math/README to configure.): n

. In particular the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}

,

which is a relation between column vectors, is equivalent to

Failed to parse (Missing texvc executable; please see math/README to configure.): |\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 Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi

is given by the wave function collapse postulate:

If the states Failed to parse (Missing texvc executable; please see math/README to configure.): | \phi_i \rangle

have distinct, definite values, Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi \rangle = \sum_i c_i | \phi_i \rangle

then the probability of measuring Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda_i

is Failed to parse (Missing texvc executable; please see math/README to configure.): |c_i|^2

, and if the measurement yields Failed to parse (Missing texvc executable; please see math/README to configure.): \lambda_i , the system is left in the state Failed to parse (Missing texvc executable; please see math/README to configure.): | \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}

is equivalent to

Failed to parse (Missing texvc executable; please see math/README to configure.): |\psi \rangle = \sum_{i} c_i | \psi_i \rangle

,

where it is understood that the above sum includes all the components of Failed to parse (Missing texvc executable; please see math/README to configure.): \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, Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle , in terms of states with definite position, Failed to parse (Missing texvc executable; please see math/README to configure.): | x \rangle . Thus

Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle = \int_{-\infty}^{\infty} \psi(x) | x \rangle\,dx

.

Note that Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle

is not the same as Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): | x_0 \rangle = \int_{-\infty}^{\infty} \delta(x - x_0) | x \rangle\,dx

and hence the spatial wave function associated with Failed to parse (Missing texvc executable; please see math/README to configure.): | x_0 \rangle

is Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x - x_0)\,
(where Failed to parse (Missing texvc executable; please see math/README to configure.): \delta(x - x_0)\,
is the Dirac delta function).

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 Failed to parse (Missing texvc executable; please see math/README to configure.): H . Some properties of such a space are

1. If Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle

and Failed to parse (Missing texvc executable; please see math/README to configure.): | \phi \rangle
are two allowed states, then

Failed to parse (Missing texvc executable; please see math/README to configure.): a | \psi \rangle + b | \phi \rangle

is also an allowed state, provided Failed to parse (Missing texvc executable; please see math/README to configure.): |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 wave function associated with a particular state may be seen as an expansion of the state in a basis of Failed to parse (Missing texvc executable; please see math/README to configure.): H . For example,

Failed to parse (Missing texvc executable; please see math/README to configure.): \{ |\uparrow_z \rangle, |\downarrow_z \rangle \}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): H . An example of this is the spacial wave function associated with a particle in one dimension which expands the state of the particle in terms of states with definite position.

Every Hilbert space Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \{ | \phi_i \rangle \}\,

, and orthonormal, i.e.

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle \phi_i | \phi_j \rangle = \delta_{ij}.

Then an arbitrary vector Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle

can be expressed as 

Failed to parse (Missing texvc executable; please see math/README to configure.): | \psi \rangle = \sum_i c_i | \phi_i \rangle

where Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \{ | x \rangle \} , the orthonormality condition holds similarly:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle x | x' \rangle = \delta(x - x').

We have the analogous identity

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle x | \int \psi(x') | x' \rangle \,dx' = \int \psi(x') \delta(x - x')\,dx' = \psi(x).

Ontology

Whether the wave function is real, and what it represents, are major questions in the interpretation of quantum mechanics. Many famous physicists have puzzled over this problem, such as Erwin Schrödinger, Albert Einstein and Niels Bohr. Some approaches regard it as merely representing information in the mind of the observer. Others argue that it must be objective.

Notes

See also

• Boson
• Fermion
• Normalisable wave function
• Particle in a box
• Schrödinger equation
• Wave function collapse
• Wave packet

References

• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X. bs:Talasna funkcija

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