Editing Wave function (section)

=== Finite dimensional Hilbert space ===
While [[Hilbert space]]s originally refer to infinite dimensional [[Complete metric space|complete]] [[inner product space]]s they, by definition, include finite dimensional [[Complete metric space|complete]] [[inner product space]]s as well.{{sfn | Treves | 2006 | p=112-125}}
In physics, they are often referred to as ''finite dimensional Hilbert spaces''.<ref name=":0">{{Cite web |last=B. Griffiths |first=Robert |author-link=Robert B. Griffiths |title=Hilbert Space Quantum Mechanics |url=https://quantum.phys.cmu.edu/QCQI/qitd114.pdf |page=1}}</ref> For every finite dimensional Hilbert space there exist [[orthonormal basis]] kets that [[Span (mathematics)|span]] the entire Hilbert space.

If the {{math|''N''}}-dimensional set <math display="inline">\{ |\phi_i\rangle \}</math> is orthonormal, then the projection operator for the space spanned by these states is given by:

<math display="block">P = \sum_i |\phi_i\rangle\langle \phi_i | = I </math>where the projection is equivalent to identity operator since <math display="inline">\{ |\phi_i\rangle \}</math> spans the entire Hilbert space, thus leaving any vector from Hilbert space unchanged. This is also known as completeness relation of finite dimensional Hilbert space.

The wavefunction is instead given by:

<math display="block">|\psi\rangle = I|\psi\rangle = \sum_i |\phi_i\rangle\langle \phi_i |\psi\rangle  </math>where <math display="inline">\{ \langle \phi_i |\psi\rangle \} </math>, is a set of complex numbers which can be used to construct a wavefunction using the above formula.

==== Probability interpretation of inner product ====
If the set <math display="inline">\{ |\phi_i\rangle \}</math> are eigenkets of a non-[[Degenerate energy levels|degenerate]] [[observable]] with eigenvalues <math display="inline">\lambda_i</math>, by the [[postulates of quantum mechanics]], the probability of measuring the observable to be <math display="inline">\lambda_i</math> is given according to [[Born rule]] as:{{sfn | Landsman | 2009}}

<math display="block">P_\psi(\lambda_i) = |\langle \phi_i|\psi \rangle|^2    </math>

For non-degenerate <math display="inline">\{ |\phi_i\rangle \}</math> of some observable, if eigenvalues <math display="inline">\lambda</math> have subset of eigenvectors labelled as <math display="inline">\{ |\lambda^{(j)}\rangle \}</math>, by the [[Mathematical formulation of quantum mechanics|postulates of quantum mechanics]], the probability of measuring the observable to be <math display="inline">\lambda</math> is given by:

<math display="block">P_\psi(\lambda) =\sum_j |\langle \lambda^{(j)}|\psi \rangle|^2 = |\widehat P_\lambda |\psi \rangle |^2          </math>where <math display="inline">\widehat P_\lambda =\sum_j|\lambda^{(j)}\rangle\langle\lambda^{(j)}|         </math> is a projection operator of states to subspace spanned by <math display="inline">\{ |\lambda^{(j)}\rangle \}</math>. The equality follows due to orthogonal nature of <math display="inline">\{ |\phi_i\rangle \}</math>.

Hence, <math display="inline">\{ \langle \phi_i |\psi\rangle \} </math> which specify state of the quantum mechanical system, have magnitudes whose square gives the probability of measuring the respective <math display="inline">|\phi_i\rangle   </math> state.

==== Physical significance of relative phase ====
While the relative phase has observable effects in experiments, the global phase of the system is experimentally indistinguishable. For example in a particle in superposition of two states, the global phase of the particle cannot be distinguished by finding expectation value of observable or probabilities of observing different states but relative phases can affect the expectation values of observables.

While the overall phase of the system is considered to be arbitrary, the relative phase for each state <math display="inline">|\phi_i\rangle   </math> of a prepared state in superposition can be determined based on physical meaning of the prepared state and its symmetry. For example, the construction of spin states along x direction as a superposition of spin states along z direction, can done by applying appropriate rotation transformation on the spin along z states which provides appropriate phase of the states relative to each other.

==== 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>.