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One Particle in One Spacial Dimension

In quantum mechanics, the spacial wavefunction associated with a particle in one dimension is a complex-valued function ${\displaystyle \psi \,}$ defined over a subset of the real line. The complex square of the wavefunction, ${\displaystyle |\psi |^{2}\,}$, 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 ${\displaystyle }$ is

${\displaystyle \operatorname {Pr} (a\leq x\leq b)=\int _{a}^{b}|\psi (x)|^{2}\,dx.\quad }$

This leads to the normalization condition

${\displaystyle \int _{-\infty }^{\infty }|\psi (x)|^{2}\,dV=1.\quad }$

since a measurement of the particle's position must produce a real number.

One Particle in Three Spacial Dimensions

The three dimensional case is analogous to the one dimensional case; the wavefunction is a complex function defined over some region of three dimensional space, and the probability that a measurement of the particle's position results in a value which is in a volume ${\displaystyle R}$ is

${\displaystyle \operatorname {Pr} (}$particle measured to be in R${\displaystyle )=\int _{R}|\psi (x)|^{2}\,dV}$.

The normalization condition is likewise

${\displaystyle \int |\psi (x)|^{2}\,dV=1}$

where the preceeding integral is taken over all space.

Two Distinguishable Particles in Three Spacial Dimensions

In this case the wavefunction is a complex function of six spacial variables,

${\displaystyle \psi (x_{1},y_{1},z_{1},x_{2},y_{2},z_{2})\,}$,

and ${\displaystyle |\psi |^{2}\,}$ 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

${\displaystyle \int _{R}\int _{S}|\psi |^{2}\,dV_{2}dV_{1}}$

where ${\displaystyle dV_{1}=dx_{1}dy_{1}dz_{1}}$ and similarly for ${\displaystyle dV_{2}}$. The normalization condition is then

${\displaystyle \int |\psi ^{2}|\,dV_{2}dV_{1}=1}$

where the preceeding 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 entanglement.

The General Case

In the mathematical formulation of quantum mechanics, the state of any system is represented by an object called a ket, which is an element of an abstract mathematical structure called a Hilbert space. For isolated systems, the dynamics (or time evolution) of the system can be described by a one-parameter group of unitary operators. In a wide class of systems this Hilbert space of kets has one or more realizations as a space of complex-valued functions on some space; in this case we refer to these functions as wavefunctions. However, a priori, there is no preferred representation as a Hilbert space of functions. Moreover, in some of these representations the time evolution of the system has the form of a partial differential equation, namely Schrödinger's equation.

Wavefunction representations

An orthonormal basis {e_i}_i in a Hilbert space H provides a representation of elements of H by finite or countable vectors of abstract Fourier coefficients

${\displaystyle {\widehat {\psi }}_{i}=\langle e_{i}|\psi \rangle }$

where < | > is Dirac's bra-ket notation.

Any separable Hilbert space has an orthonormal basis; these bases are not unique however. Nevertheless, for some physical systems there are certain orthonormal bases which have a natural physical interpretation. This fact justifies commonly used expressions regarding quantum states such as they exist in a superposition of basis states, meaning exactly that each state can be represented as a possibly infinite linear combination

${\displaystyle \psi =\sum {\widehat {\psi }}_{i}e_{i}.}$

In fact, there is a far-reaching generalization of an orthonormal representation, which gives an analogous representation with respect to what we could loosely call a continuously indexed orthonormal basis of a Hilbert space. In this representation, ket vectors are represented by functions on the continuous index set and the inner product of the Hilbert space corresponds to the integral of the product of two wavefunctions. In mathematical terms, such continuous orthonormal bases are referred to as diagonalizations, because mathematically they correspond to representing certain commutative algebras of operators as algebras of multiplication operators. The technical details of how this diagonalization is carried out is beyond the scope of this article, but it generalizes the result of linear algebra that a commutative algebra of operators closed under operator adjoint is diagonalized in some orthonormal basis.

Two common diagonalizations used in quantum mechanics are the configuration (position) space representation (which diagonalizes the position operators) and the momentum space representation (which diagonalizes the momentum operators). These are also called by physicists the 'r-space representation' and the 'k-space representation', respectively. Due to the commutation relationship of the position and momentum operators, for a system of spinless particles in Euclidean space the r-space and k-space wavefunctions are Fourier transform pairs. The precise formulation of this last statement is rather subtle and is called the Stone-von Neumann theorem in the mathematical physics literature.

A more general diagonalization in which ket vectors are represented by Hilbert space valued functions on some space occurs naturally, for example, those which involve half-integer spin or systems in which the number of particles or quanta is variable, for example, most of nonlinear quantum optics or atom optics, and any treated by quantum electrodynamics or other quantized-field theories. This diagonal representation is usually called a direct integral of Hilbert spaces.

If the energy spectrum of a system is (partly) discrete, such as for a particle in an infinite potential well or the bound states of the hydrogen atom, then the position representation is continuous while the momentum representation is partly discrete. Wave mechanics are most often used when the number of particles is relatively small and knowledge of spatial configuration or 'shape' is important.

Because the wavefunction relative to the configuration representation has a (comparatively) simple interpretation as a probability in configuration space, many introductory treatments of quantum mechanics are very much wave mechanical. Wave mechanics also dominated many of the more popular older standard textbooks, such as Messiah's Mecanique Quantique. Hence the term wavefunction is sometimes used as a colloquialism for "state vector". This use, however, is deprecated; not only are there systems which cannot be represented by wavefunctions, but the term wavefunction also leads to the belief that there is wave propagation in some medium.

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

References

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• Quantum mechanics