The Physics[`.`] command is a computational representation for the scalar product operation between Bras, Kets, and quantum operators set to be so by the Setup command, or returned by the Annihilation, Creation, and Projector commands.

Independent of that, Physics[`.`] is also a handy shortcut for performing simplification of contracted indices in tensor products; for instance if A[mu] is Defined as a tensor, g_[mu, nu] . A[nu] returns A[mu], equivalent to performing Simplify(g_[mu, nu] * A[nu], indices) (see Simplify and the end of the Examples section).

When taking the scalar product between Bras, Kets, or quantum operator, generally speaking, two situations can happen: 1) there is not enough information to perform the operation (for example, $⟨A|B⟩$ for two generic unknown Kets), in which case it is returned unevaluated; or 2) the information available permits performing the operation.

In the following situations, the scalar product is returned unevaluated:

The scalar product between a Bra and a Ket, with or without a quantum operator in between, is returned in terms of the Bracket function.

A product involving only a quantum operator and either a Ket or a Bra is returned as entered, in terms of Physics[`.`].

A product with two or more Kets only (or two or more Bras only) is returned in terms of Physics[*], and represents a state vector of the tensor product of the respective spaces.

In the following situations, the scalar product can be performed:

The Kets belong to the same basis: if the basis, represented by the Ket's first argument, is discrete, then the result involves KroneckerDelta; if the basis is continuous, then it involves the Dirac function. To indicate that a basis is continuous, use the Setup command. Remark: when the discrete or continuous character of the basis has not been set, it is assumed that the basis is discrete.

The label A, entered as the first argument of the Ket, has also been set as a quantum operator by the Setup command; for example, a Ket $\mathrm{Ket}\left(A\,n\,m\right)$ is an eigenvector of $A_1$ with eigenvalue $n$ and of $A_2$ with eigenvalue $m$.

The quantum operator is an annihilation or creation operator, and the Ket (or the Bra) belongs to the basis on which the operator acts.

A bracket rule, for example, for $⟨A|B⟩$ or $⟨A\left|H\right|B⟩$, where $H$ represents a generic quantum operator, has been previously set by Setup, and so determines the result.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}=\mathrm{true}\right)$

$\left[\mathrm{mathematicalnotation}=\mathrm{true}\right]$

$\mathrm{Bra}\left(A\,n\right)·\mathrm{Ket}\left(B\,m\right)$

$⟨A_n|B_m⟩$

$\mathrm{Setup}\left(\mathrm{op}=H\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{op}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumoperators}=\left\{H\right\}\right]$

$\mathrm{Bra}\left(A\,n\right)·H·\mathrm{Ket}\left(B\,m\right)$

$⟨A_n\left|H\right|B_m⟩$

$\mathrm{Bra}\left(A\,n\right)·\mathrm{Ket}\left(A\,m\right)$

${\mathrm{\delta }}_{m,n}$

$\mathrm{Setup}\left(\mathrm{continuousbasis}=C\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{continuousbasis}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumcontinuousbasis}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumcontinuousbasis}=\left\{C\right\}\right]$

$\mathrm{Bra}\left(C\,x\right)·\mathrm{Ket}\left(C\,y\right)$

$\mathrm{\delta }\left(-y+x\right)$

$Q·\mathrm{Ket}\left(C\,y\right)$

$Q\left|C_y\right.⟩$

$\mathrm{Setup}\left(\mathrm{quantumop}=Q\right)$

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{quantumop}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumoperators}\text{'}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{quantumoperators}=\left\{H\,Q\right\}\right]$

$Q·\mathrm{Ket}\left(C\,y\right)$

$Q·\left|C_y\right.⟩$

$Q\left[1\right]·\mathrm{Ket}\left(Q\,m\,n\right)$

$m\left|Q_{m\,n}\right.⟩$

$Q\left[2\right]·\mathrm{Ket}\left(Q\,m\,n\right)$

$n\left|Q_{m\,n}\right.⟩$

$\mathrm{`\%.`}\left(Q_1\,\mathrm{Ket}\left(Q\,m\,n\right)\right)$

$Q_1·\left|Q_{m\,n}\right.⟩$

$\mathrm{value}\left(\right)$

$m\left|Q_{m\,n}\right.⟩$

$\mathrm{\%Bracket}\left(\mathrm{Bra}\left(Q\,m\,n\right)\,\mathrm{Ket}\left(A\,r\,s\right)\right)=A\left(q\right)\mathrm{kd\_}\left[m,r\right]\mathrm{kd\_}\left[n,s\right]$

$⟨Q_{m\,n}|A_{r\,s}⟩=A\left(q\right){\mathrm{\delta }}_{m,r}{\mathrm{\delta }}_{n,s}$

$\mathrm{Setup}\left(\right)$

$\left[\mathrm{bracketrules}=\left\{⟨Q_{m\,n}|A_{r\,s}⟩=A\left(q\right){\mathrm{\delta }}_{m,r}{\mathrm{\delta }}_{n,s}\right\}\right]$

$\mathrm{\%Bracket}\left(\mathrm{Bra}\left(Q\,i\,j\right)\,\mathrm{Ket}\left(A\,k\,l\right)\right)$

$⟨Q_{i\,j}|A_{k\,l}⟩$

$\mathrm{value}\left(\right)$

$A\left(q\right){\mathrm{\delta }}_{i,k}{\mathrm{\delta }}_{j,l}$

$\mathrm{Bra}\left(Q\,r\,s\right)·\mathrm{Ket}\left(A\,t\,s\right)$

$A\left(q\right){\mathrm{\delta }}_{r,t}$

$\mathrm{with}\left(\mathrm{VectorCalculus}\right)\:$

$\mathrm{with}\left(\mathrm{Physics}:-\mathrm{Vectors}\right)\:$

$\mathrm{Bra}\left(A\,n\right)·\mathrm{Ket}\left(A\,m\right)$

${\mathrm{\delta }}_{m,n}$

$\mathrm{Bra}\left(C\,x\right)·\mathrm{Ket}\left(C\,y\right)$

$\mathrm{\delta }\left(-y+x\right)$

$Q·\mathrm{Ket}\left(C\,y\right)$

$Q·\left|C_y\right.⟩$

$\mathrm{AB}≔\mathrm{A\_}·\mathrm{B\_}$

$\mathrm{AB}≔\overrightarrow{A}·\overrightarrow{B}$

$\mathrm{A\_},\mathrm{B\_}≔\mathrm{\_i}+2\mathrm{\_j}+3\mathrm{\_k},4\mathrm{\_i}+5\mathrm{\_j}+6\mathrm{\_k}$

$\overrightarrow{A},\overrightarrow{B}≔\stackrel{\wedge }{i}+2\stackrel{\wedge }{j}+3\stackrel{\wedge }{k},4\stackrel{\wedge }{i}+5\stackrel{\wedge }{j}+6\stackrel{\wedge }{k}$

$\mathrm{lprint}\left(\mathrm{A\_}\right)$

_i+2*_j+3*_k

$\mathrm{type}\left(\mathrm{A\_}\,\mathrm{algebraic}\right)$

$\mathrm{true}$

$\mathrm{AB}$

$32$

$\mathrm{A\_},\mathrm{B\_}≔⟨1\left|2\right|3⟩,⟨4\left|5\right|6⟩$

$\mathrm{lprint}\left(\mathrm{A\_}\right)$

Vector[row](3,{1 = 1, 2 = 2, 3 = 3},datatype = anything,storage = rectangular,
order = Fortran_order,attributes = [coords = cartesian],shape = [])

$\mathrm{type}\left(\mathrm{A\_}\,\mathrm{algebraic}\right)$

$\mathrm{false}$

$\mathrm{A\_}·\mathrm{B\_}$

$32$

$\mathrm{Define}\left(A\,B\right)$

$\mathrm{Defined\; objects\; with\; tensor\; properties}$

$\left\{A\,B\,{\mathrm{\gamma }}_{\mathrm{\mu }}\,{\mathrm{\sigma }}_{\mathrm{\mu }}\,{\partial }_{\mathrm{\mu }}\,g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{\epsilon }}_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}\right\}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]$

$A_{\mathrm{\mu }}{\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·A\left[\mathrm{\mu }\right]$

$A_{\mathrm{\alpha }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\mu }}{B}_{\mathrm{\nu }}{\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}$

$\mathrm{Simplify}\left(\,\mathrm{indices}\right)$

$A_{\mathrm{\nu }}{B}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·A\left[\mathrm{\mu }\right]\right)\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\alpha }}{B}_{\mathrm{\nu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\nu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]·B\left[\mathrm{\nu }\right]\right)$

$B_{\mathrm{\alpha }}{A}_{\mathrm{\mu }}{g}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\mu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\mu }}$

$\left(\left(\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]A\left[\mathrm{\mu }\right]\right)·\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\nu }\right]\right)B\left[\mathrm{\nu }\right]$

$A_{\mathrm{\nu }}{B}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]\left(A\left[\mathrm{\mu }\right]+B\left[\mathrm{\mu }\right]\right)$

$\left(A_{\mathrm{\mu }}+B_{\mathrm{\mu }}\right){\mathrm{\delta }}_{\mathrm{\alpha }\phantom{\mathrm{\mu }}}^{\phantom{\mathrm{\alpha }}\mathrm{\mu }}$

$\mathrm{g\_}\left[\mathrm{\alpha },\mathrm{\mu }\right]·\left(A\left[\mathrm{\mu }\right]+B\left[\mathrm{\mu }\right]\right)$

$A_{\mathrm{\alpha }}+B_{\mathrm{\alpha }}$

$\mathrm{g\_}\left[\mathrm{sc}\right]$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Default\; differentiation\; variables\; for\; d\_,\; D\_\; and\; dAlembertian\; are:}\left\{X=\left(r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right)\right\}$

$\mathrm{Setting}\mathrm{lowercaselatin\_is}\mathrm{letters\; to\; represent}\mathrm{space}\mathrm{indices}$

$\mathrm{The\; Schwarzschild\; metric\; in\; coordinates}\left[r\,\mathrm{\theta }\,\mathrm{\phi }\,t\right]$

$\mathrm{Parameters:}\left[m\right]$

$\mathrm{Signature:}\left(\mathrm{-\; -\; -\; +}\right)$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$g_{\mathrm{\mu },\mathrm{\nu }}=\left[\right]$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]$

$R_{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}{R}_{\phantom{}\phantom{\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }}$

$\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]·\mathrm{Riemann}\left[\mathrm{\alpha },\mathrm{\beta },\mathrm{\mu },\mathrm{\nu }\right]$

$\frac{48m^2}{r^6}$

Cohen-Tannoudji, C.; Diu, B.; and Laloe, F. Quantum Mechanics. Chapter II. Paris, France: Hermann, 1977.