Editing Wave function (section)

==== 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.