Wavefunction

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 : $ec \psi = \begin{bmatrix} c_1 \ dots \ c_n \end{bmatrix}$ , a complex vector with infinitely many components : $ec \psi = \begin{bmatrix} c_1 \ dots \ c_n \ dots \end{bmatrix}$ , 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 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
	:<math>\mathbf{P} {ab}(x)	\int_{a}^{b} \psi(x)^2\, dx </math>
	:<math> \int {-\infty}^{\infty} \psi(x)^2\, Dx	1 \quad </math>
	:<math>\mathbf{P} R	\int_R \psi(x)^2 \, dV</math>
	:<math> \int \psi(x)^2\, DV	1</math>
	:<math>\mathbf{P} {R,S}	\int_R \int_S \psi^2 \, dV_2 dV_1 </math>
	:<math>\int \psi^2 \, DV 2 DV 1	1</math>
	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	"http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/momentum_space" class="copylinks">Momentum Space :
	:<math>\mathbf{P} {ab}(p)	\int_{a}^{b} \psi(p)^2\, dp </math>
	:<math>\int {-\infty}^{\infty} \psi(p)^2\, Dp	1 </math>
	:<math> \psi Angle	c_1 \uparrow_z angle + c_2 \downarrow_z angle</math>
	:<math>c 1^2 + c 2^2	1\,</math>
	:<math>\psi Angle	\sum_{i = 1}^n c_i \phi_i angle</math>,
	::<math>\psi Angle	\sum_i c_i \phi_i angle</math>
	:<math>\psi Angle	\sum_{i} c_i \psi_i angle</math>,
	:<math> \psi Angle	\int_{-\infty}^{\infty} \psi(x) x angle\,dx</math>
	:<math> X 0 Angle	\int_{-\infty}^{\infty} \delta(x - x_0) x angle\,dx</math>
	And Hence The Spatial Wavefunction Associated With <math> X 0 Angle</math> Is	"http://enwikipediaorg/wiki/Dirac_delta" class="copylinks" target="_blank"><math>\delta(x - x_0)\,</math>
	:is Also An Allowed State, Provided <math>a^2+b^2	1</math> (This condition is due to normalisation)
	It Is Conventional To Endow <math>H</math> With An	"http://wwwinformationdelightinfo/encyclopedia/entry/Vrhbosna/inner_product" class="copylinks">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 angle \}\,</math> all of which belong to <math>H</math>, <math>H</math> is equipped with the unique inner product that makes this basis orthornormal, ie,
	:<math>\langle \phi I \phi J Angle	\delta_{ij}</math>
	:<math>\langle \phi I \sum J C J \phi J Angle	c_i</math>
	:<math>\langle X X' Angle	\delta(x - x')</math>
	:<math>\langle X \int \psi(x') X' Angle \,dx'	\int \psi(x') \delta(x - x')\,dx' = \psi(x)</math>

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Wavefunction