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.
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Definition
The modern usage of the term wavefunction refers to any vector or function which describes the state of a physical system by expanding it in terms of other states of the same system. Typically, a wavefunction is either:
- a complex vector with finitely many components
- <math>\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}</math>,
- a complex vector with infinitely many components
- <math>\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}</math>,
- or a complex function of one or more real variables (a "continuously indexed" complex vector)
- <math>\psi(x_1, \, \ldots \, x_n)</math>.
In all cases, the wavefunction provides a complete description of the associated physical system. However, it is important to note that the wavefunction associated with a system is not uniquely determined by that system, as many different wavefunctions may describe the same physical scenario.
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 <math>\psi(x)\,</math> defined over the real line. The complex square of the wavefunction, <math>|\psi|^2\,</math>, is interpreted as the probability density associated with the particle's position, and hence the probability that a measurement of the particle's position yields a value in the interval <math>[a, b]</math> is
- <math>\int_{a}^{b} |\psi(x)|^2\, dx \quad </math>.
This leads to the normalization condition
- <math> \int_{-\infty}^{\infty} |\psi(x)|^2\, dx = 1 \quad </math>.
since a measurement of the particle's position must produce a real number.
One particle in three spatial dimensions
The three dimensional case is analogous to the one dimensional case; the wavefunction is a complex function <math>\psi(x, y, z)\,</math> defined over three dimensional space, and its complex square is interpreted as a three dimensional probability density function. The probability that a measurement of the particle's position results in a value which is in the volume <math>R</math> is thus
- <math>\int_R |\psi(x)|^2\, dV</math>.
The normalization condition is likewise
- <math> \int |\psi(x)|^2\, dV = 1</math>
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,
- <math>\psi(x_1, y_1, z_1, x_2, y_2, z_2)\,</math>,
and <math>|\psi|^2\,</math> is a joint probability density function associated with the positions of both particles. 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 then
- <math>\int_R \int_S |\psi|^2 \, dV_2 dV_1 </math>
where <math>dV_1 = dx_1 dy_1 dz_1</math> and similarly for <math>dV_2</math>. The normalization condition is thus
- <math>\int |\psi^2| \, dV_2 dV_1 = 1</math>
where the preceding integral is taken over the full range of all six variables.
It is of crucial importance to realize that, in the case of two particle systems, only the system consisting of both particles need have a well defined wavefunction. That is, it may be impossible to write down a probability density function for the position of particle one which does not depend explicitly on the position of particle two. This gives rise to the phenomenon of quantum entanglement.
One particle in one dimensional momentum space
The wavefunction for a one dimensional particle in momentum space is a complex function <math>\psi(p)\,</math> defined over the real line. The quantity <math>|\psi|^2\,</math> is interpreted as a probability density function in momentum space, and hence the probability that a measurement of the particle's momentum yields a value in the interval <math>[a, b]</math> is
- <math>\int_{a}^{b} |\psi(p)|^2\, dp\quad </math>.
This leads to the normalization condition
- <math>\int_{-\infty}^{\infty} |\psi(p)|^2\, dp = 1 </math>
since a measurement of the particle's momentum always results in a real number.
Spin 1/2
The wavefunction for a spin 1/2 particle (ignoring its spacial degrees of freedom) is a column vector
- <math>\vec \psi = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix}</math>.
The meaning of the vector's components depends on the basis, but typically <math>c_1</math> and <math>c_2</math> are respectively the coefficients of spin up and spin down in the <math>z</math> direction. In Dirac notation this is:
- <math>| \psi \rangle = c_1 | \uparrow_z \rangle + c_2 | \downarrow_z \rangle</math>
The values <math>|c_1|^2 \,</math> and <math>|c_2|^2 \,</math> 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
- <math>|c_1|^2 + |c_2|^2 = 1\,</math>.
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 <math>| \psi \rangle\,</math> and the states into which it is being expanded as <math>| \phi_i \rangle</math>. 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 <math>\vec \psi</math> with <math>n</math> components describes how to express the state of the physical system <math>| \psi \rangle</math> as the linear combination of finitely many basis elements <math>| \phi_i \rangle</math>, where <math>i</math> runs from <math>1</math> to <math>n</math>. In particular the equation
- <math>\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \end{bmatrix}</math>,
which is a relation between column vectors, is equivalent to
- <math>|\psi \rangle = \sum_{i = 1}^n c_i | \phi_i \rangle</math>,
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 <math>\vec \psi</math> is given by the wavefunction collapse postulate:
- If the states <math>| \phi_i \rangle</math> have distinct, definite values, <math>\lambda_i</math>, of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state
- <math>|\psi \rangle = \sum_i c_i | \phi_i \rangle</math>
- then the probability of measuring <math>\lambda_i</math> is <math>|c_i|^2</math>, and if the measurement yields <math>\lambda_i</math>, the system is left in the state <math>| \phi_i \rangle</math>.
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
- <math>\vec \psi = \begin{bmatrix} c_1 \\ \vdots \\ c_n \\ \vdots \end{bmatrix}</math>
is equivalent to
- <math>|\psi \rangle = \sum_{i} c_i | \psi_i \rangle</math>,
where it is understood that the above sum includes all the components of <math>\vec \psi</math>. 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, <math>| \psi \rangle</math>, in terms of states with definite position, <math>| x \rangle</math>. Thus
- <math>| \psi \rangle = \int_{-\infty}^{\infty} \psi(x) | x \rangle\,dx</math>.
Note that <math>| \psi \rangle</math> is not the same as <math>\psi(x)\,</math>. 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
- <math>| x_0 \rangle = \int_{-\infty}^{\infty} \delta(x - x_0) | x \rangle\,dx</math>
and hence the spatial wavefunction associated with <math>| x_0 \rangle</math> is <math>\delta(x - x_0)\,</math>.
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 vector space <math>H</math>, the Hilbert space. That is,
- 1. If <math>| \psi \rangle</math> and <math>| \phi \rangle</math> are two allowed states, then
- <math>a | \psi \rangle + b | \phi \rangle</math>
- is also an allowed state, provided <math>|a|^2+|b|^2=1</math>. (This condition is due to normalisation.)
and,
- 2. Due to normalisation, there is always an orthonormal basis of allowed states of the vector space H.
In this context the wavefunction associated with a particular state may be seen as an expansion of the state in a basis for the vector space <math>H</math>. For example,
- <math>\{ |\uparrow_z \rangle, |\downarrow_z \rangle \}</math>
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
- <math>a|\uparrow_z \rangle + b|\downarrow_z \rangle</math>.
Sometimes it is useful to expand the state of a physical system in terms of states which are not allowed, and hence, not in <math>H</math>. 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. These states are forbidden, however, since they violate the uncertainty principle. Bases such as these as called improper bases.
It is conventional to endow <math>H</math> with an inner product but the nature of the inner product is contingent upon the kind of basis in use. When there are countably many basis elements <math>\{ | \phi_i \rangle \}\,</math> all of which belong to <math>H</math>, <math>H</math> is equipped with the unique inner product that makes this basis orthornormal, i.e.,
- <math>\langle \phi_i | \phi_j \rangle = \delta_{ij}.</math>
When this is done, the inner product of <math>| \phi_i \rangle</math> with the expansion of an arbitrary vector is
- <math>\langle \phi_i | \sum_j c_j | \phi_j \rangle = c_i</math>.
If the basis elements constitute a continuum, as, for example, the position or coordinate basis consisting of all states of definite position <math>\{ | x \rangle \}</math>, it is conventional to choose the Dirac normalization
- <math>\langle x | x' \rangle = \delta(x - x')</math>
so that the analogous identity
- <math>\langle x | \int \psi(x') | x' \rangle \,dx' = \int \psi(x') \delta(x - x')\,dx' = \psi(x)</math>.
holds.
See also
- Wave packet
- Boson - particles with symmetric wavefunction under permutation (i.e. switching positions)
- Fermion - particles with antisymmetric wavefunction under permutation
- Quantum mechanics
- Schrödinger equation
- Normalisable wavefunction
- Von Neumann's catastrophe
References
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