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A 'wave function' is a mathematical tool that quantum mechanics uses to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. That is to say, a wave function describes the shape occupied by all of the states of a system overlaid into a single form.
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, the wave function of the electron provides a complete description of how the electron behaves. For this single electron, it can occupy any point between a certain distance from the center of the atom to a large distance away. Because it is most likely to be between these two distances, the wave function can be approximated by a fuzzy sphere, as shown in S-1 in the diagram below.
This sphere 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 configuration of all the electrons, and the wave function describes the probabilities of those configurations. S-2 Is the wave function for two electrons, P2 is the wave function for 3, and so on.

Contents

Definition

Interpretation

One particle in one spatial dimension

One particle in three spatial dimensions

Two distinguishable particles in three spatial dimensions

One particle in one dimensional momentum space

Spin 1/2

Interpretation

Finite vectors

Infinite vectors

Continuously indexed vectors (functions)

Formalism

Ontology

Notes

See also

References

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
:,

★ a complex vector with infinitely many components
:,

★ or 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; the same applies to wave functions. The wave function describing the same physical state takes different forms depending on the basis being used.
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.

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 $psi(x),$ 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 its complex square 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
:$int\; |psi(x,\; y,\; z)|^2,\; dV\; =\; 1$
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 region $S$ is
:$mathbf\{P\}\_\{R,S\}\; =\; int\_R\; int\_S\; |psi|^2\; ,\; dV\_2\; ,\; dV\_1$
where $dV\_1\; =\; dx\_1\; dy\_1\; dz\_1$, and similarly for $dV\_2$.
The normalization condition is then:
:$int\; int\; |psi(x,\; y,\; z)|^2\; ,\; dV\_2\; ,\; dV\_1\; =\; 1$
in which the preceding integral is taken over the full range of all six variables.
Given a wave function of ψ of a systems 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 1/2 particle (ignoring its spatial degrees of freedom) is a column vector
:.
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:
:$|\; psi$
angle = c_1 | uparrow_z
angle + c_2 | downarrow_z
angle
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 wave function 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 $|\; psi$
angle, and the states into which it is being expanded as $|\; phi\_i$
angle. Collectively the latter are referred to as a ''basis'' or ''representation''. In what follows, all wave functions are assumed to be normalized.

Finite vectors

A wave function which is a vector with $n$ components describes how to express the state of the physical system $|\; psi$
angle as the linear combination of finitely many basis elements $|\; phi\_i$
angle, where $i$ runs from $1$ to $n$. In particular the equation
:,
which is a relation between column vectors, is equivalent to
:$|psi$
angle = sum_{i = 1}^n c_i | phi_i
angle,
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 is given by the wave function collapse postulate:
:If the states $|\; phi\_i$
angle 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
::$|psi$
angle = sum_i c_i | phi_i
angle
: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 $|\; phi\_i$
angle.

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
:
is equivalent to
:$|psi$
angle = sum_{i} c_i | psi_i
angle,
where it is understood that the above sum includes all the components of . 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, $|\; psi$
angle, in terms of states with definite position, $|\; x$
angle. Thus
:$|\; psi$
angle = int_{-infty}^{infty} psi(x) | x
angle,dx.
Note that $|\; psi$
angle 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
:$|\; x\_0$
angle = int_{-infty}^{infty} delta(x - x_0) | x
angle,dx
and hence the spatial wave function associated with $|\; x\_0$
angle is $delta(x\; -\; x\_0),$ (where $delta(x\; -\; x\_0),$ 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 $H$. Some properties of such a space are
:1. If $|\; psi$
angle and $|\; phi$
angle are two allowed states, then
:::$a\; |\; psi$
angle + b | phi
angle
: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 wave function associated with a particular state may be seen as an expansion of the state in a basis of $H$. For example,
:$\{\; |uparrow\_z$
angle, |downarrow_z
angle }
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
:$a|uparrow\_z$
angle + b|downarrow_z
angle.
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 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 $H$ 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 $\{\; |\; phi\_i$
angle },, and orthonormal, i.e.
:$langle\; phi\_i\; |\; phi\_j$
angle = delta_{ij}.
Then an arbitrary vector $|\; psi$
angle can be expressed as
:$|\; psi$
angle = sum_i c_i | phi_i
angle
where $c\_i\; =\; langle\; phi\_i\; |\; psi$
angle.
If one chooses a "continuous" basis as, for example, the ''position'' or ''coordinate'' basis consisting of all states of definite position $\{\; |\; x$
angle }, the orthonormality condition holds similarly:
:$langle\; x\; |\; x\text{'}$
angle = delta(x - x').
We have the analogous identity
:$langle\; x\; |\; int\; psi(x\text{'})\; |\; x\text{'}$
angle ,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 Schrödinger. Some approaches regard it as merely representing information in the mind of the observer. Others argue that it must be objective:

"If we are to believe that any one thing in the formalism is 'actually' real for a quantum system, then I think it has to be the wavefunction or state vector that describes quantum reality."

^[1]

Notes

1. Penrose,R. Road To Reality, p508

See also

★ Boson - particles with symmetric wave function under permutation (i.e. switching positions)

★ Fermion - particles with antisymmetric wave function under permutation

★ Normalisable wave function

★ Schrödinger equation

★ Wave function collapse

★ Wave packet

★ Particle in a box

References

★ Introduction to Quantum Mechanics (2nd ed.), Griffiths, David J., , , Prentice Hall, 2004, ISBN 0-13-805326-X