yawiki.org entry for Wavefunction

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A wavefunction is the 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. 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 whose square gives the probability distribution that the system will be 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 electrons 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.

    Wavefunction
        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
        See also

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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

$vec psi = egin c_1 \ vdots \ c_n end$ ,

a complex vector with infinitely many components

$vec psi = egin c_1 \ vdots \ c_n \ vdots end$ ,

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.

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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.

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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

$mathbf_ = int_^ |psi(x)|^2, dx$ .

This leads to the normalization condition

$int_^ |psi(x)|^2, dx = 1 quad$ .

since the probability of a measurement of the particle's position yielding a value in

(-infty, infty)

is unity.

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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:

$mathbf_R = int_R |psi(x, y, z)|^2 , dV$

The normalization condition is likewise

$int |psi(x, y, z)|^2, dV = 1$

where the preceding integral is taken over all space.

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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 region

S

$mathbf_ = int_R int_S |psi|^2 , dV_2 , dV_1$

where

dV_1 = dx_1 dy_1 dz_1

, and similarly for

dV_2

.

The normalization condition is then:

$int int |psi(x, y, z)|^2 , dV_2 , dV_1 = 1$

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 fuction to a single-particle subsystem. In other words, the particles in the system can be entangled.

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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:

$mathbf_ = int_^ |psi(p)|^2, dp$

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

$int_^ |psi(p)|^2, dp = 1 .$

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Spin 1/2

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

$vec psi = egin c_1 \ c_2 end$ .

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:

$| psi$

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

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

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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 wavefunctions are assumed to be normalized.

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Finite vectors

A wave function which is a vector

vec psi

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

n

. In particular the equation

$vec psi = egin c_1 \ vdots \ c_n end$ ,

which is a relation between column vectors, is equivalent to

$|psi$

angle = sum_^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

vec psi

is given by the wavefunction 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.

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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

$vec psi = egin c_1 \ vdots \ c_n \ vdots end$

is equivalent to

$|psi$

angle = sum_ c_i | psi_i angle,

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).

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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_^ 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_^ delta(x - x_0) | x angle,dx

and hence the spatial wavefunction associated with

| x_0 
angle

delta(x - x_0),

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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 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

$a|uparrow_z$

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 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.

$langle phi_i | phi_j$

angle = delta_.

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

, the orthonormality condition holds similarly:

$langle x | x'$

angle = delta(x - x').

We have the analogous identity

$langle x | int psi(x') | x'$

angle ,dx' = int psi(x') delta(x - x'),dx' = psi(x).

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