Editing Wave function (section)

==== Application to include spin ====
An example of finite dimensional Hilbert space can be constructed using spin eigenkets of <math display="inline">s</math>-spin particles which forms a <math display="inline">2s+1</math> dimensional [[Hilbert space]]. However, the general wavefunction of a particle that fully describes its state, is always from an infinite dimensional [[Hilbert space]] since it involves a tensor product with [[Hilbert space]] relating to the position or momentum of the particle. Nonetheless, the techniques developed for finite dimensional Hilbert space are useful since they can either be treated independently or treated in consideration of linearity of tensor product.

Since the [[spin operator]] for a given <math display="inline">s</math>-spin particles can be represented as a finite <math display="inline">(2s+1)^2    </math> [[Matrix (mathematics)|matrix]] which acts on <math display="inline">2s+1</math> independent spin vector components, it is usually preferable to denote spin components using matrix/column/row notation as applicable.

For example, each {{math|{{ket|''s<sub>z</sub>''}}}} is usually identified as a column vector:<math display="block">|s\rangle \leftrightarrow \begin{bmatrix} 1 \\ 0 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \quad |s-1\rangle \leftrightarrow \begin{bmatrix} 0 \\ 1 \\ \vdots \\ 0 \\ 0 \\ \end{bmatrix} \,, \ldots \,, \quad |-(s-1)\rangle \leftrightarrow \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 1 \\ 0 \\ \end{bmatrix} \,,\quad |-s\rangle \leftrightarrow  \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ 1 \\ \end{bmatrix}</math>

but it is a common abuse of notation, because the kets {{math|{{ket|''s<sub>z</sub>''}}}} are not synonymous or equal to the column vectors. Column vectors simply provide a convenient way to express the spin components.

Corresponding to the notation, the z-component spin operator can be written as:<math display="block">\frac{1}{\hbar}\hat{S}_z = \begin{bmatrix} s & 0 & \cdots & 0 & 0 \\ 0 & s-1 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & -(s-1) & 0 \\ 0 & 0 & \cdots & 0 & -s \end{bmatrix} </math>

since the [[eigenvector]]s of z-component spin operator are the above column vectors, with eigenvalues being the corresponding spin quantum numbers.

Corresponding to the notation, a vector from such a finite dimensional Hilbert space is hence represented as:

<math display="block">|\phi\rangle  =  \begin{bmatrix} \langle s| \phi\rangle \\ \langle s-1| \phi\rangle \\ \vdots \\ \langle -(s-1)| \phi\rangle \\ \langle -s| \phi\rangle \\ \end{bmatrix} =\begin{bmatrix} \varepsilon_s \\  \varepsilon_{s-1}\\ \vdots \\ \varepsilon_{-s+1} \\ \varepsilon_{-s} \\ \end{bmatrix} </math>where <math display="inline"> \{ \varepsilon_i \}       </math> are corresponding complex numbers.

In the following discussion involving spin, the complete wavefunction is considered as tensor product of spin states from finite dimensional Hilbert spaces and the wavefunction which was previously developed. The basis for this Hilbert space are hence considered: <math>  |\mathbf{r}, s_z\rangle   = |\mathbf{r}\rangle |s_z\rangle   </math>.