A Ket state vector represents the quantum state of a physical system. A Bra is also a vector in a space of quantum states and corresponds to a Ket vector in that one is obtained from the other through Hermitian conjugation.

The Bracket(B, K) command (note the 'c' between Bra and Ket) performs the scalar product, also called the inner product, between a Bra B and a Ket K. The Bracket(B, H, K) command, where H represents a linear quantum operator, defined as such by the Setup command, or returned by one of the Annihilation, Creation, or Projector commands, returns the scalar product between the Bra B and the Ket resulting from the action of H on the Ket K. When H is not an operator, Bracket(B, H, K) transforms into H*Bracket(B, K).

The %Bracket command is the inert form of Bracket; that is, it represents the same mathematical operation while displaying the operation unevaluated. To evaluate the operation, use the value command.

A shortcut notation for Bracket consists of the following:

- if an even number of arguments N are passed, and the first and (N/2 + 1)th are names (so they can represent labels for spaces of states), and the rest of the arguments are not quantum operators, then for example, Bracket(B, n1,n2, K, m1,m2) is transformed into Bracket(Bra(B, n1,n2), Ket(K, m1,m2)).

- if an odd number of arguments N are passed, where the (N+1)/2th argument is $H$, then for example, Bracket(B, n1,n2, H, K, m1,m2) is transformed into Bracket(Bra(B, n1,n2), H, Ket(K, m1,m2)).

The scalar product between $B$ and $K$ (or $H$ and $K$) is computed also by using $B·K$ (or $H·K$), where $\mathrm{`.`}$ is the Physics[`.`] scalar (inner) product operator.

The Bracket between a Bra and a Ket is expressed in terms of KroneckerDeltas, when the state vectors belong to a discrete space of states, or by using Dirac functions, when the state vectors belong to a continuous space of states. To make the label (first argument) of a Bra or Ket be associated with a continuous space of states, use the Setup command, as in $\mathrm{Setup}\left(\mathrm{continuousbasis}=A\right)$; otherwise, it is assumed that the Bra or Ket corresponds to a discrete space of states.

To set bracket rules involving Bra and Ket state vectors, use the Setup command; see the examples below.

When the Bracket cannot be resolved, the operation is returned unevaluated; see details in Physics[`.`].

with(Physics):

Setup(mathematicalnotation = true);

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

K1 := Ket(psi, n, m);

$\mathrm{K1}≔\left|{\mathrm{\psi }}_{n\,m}\right.⟩$

K2 := Ket(psi, r, s);

$\mathrm{K2}≔\left|{\mathrm{\psi }}_{r\,s}\right.⟩$

BK1 := Bra(psi, n, m);

$\mathrm{BK1}≔\left.⟨{\mathrm{\psi }}_{n\,m}\right|$

BK2 := Dagger(K2);

$\mathrm{BK2}≔\left.⟨{\mathrm{\psi }}_{r\,s}\right|$

Bracket(BK1, K2);

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

BK1 . K2;

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

Setup(continuousbasis = psi);

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

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

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

Bracket(BK1, K2);

${\mathrm{\δ}}^{\left(2\right)}\left(\left[-r\+n\,-s\+m\right]\right)$

Bracket(Bra(psi, x), Ket(phi, y));

$⟨{\mathrm{\psi }}_x|{\mathrm{\phi }}_y⟩$

Setup(discretebasis = psi, redo);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{discretebasis}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{quantumdiscretebasis}\text{'}$

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

$\left[\mathrm{quantumdiscretebasis}=\left\{\mathrm{\psi }\right\}\right]$

am1 := Annihilation(psi, 1);

$\mathrm{am1}≔a^{-}$

ap2 := Creation(psi, 2);

$\mathrm{ap2}≔a^{+}$

am1 . K1;

$\sqrt{n}\left|{\mathrm{\psi }}_{n-1\,m}\right.⟩$

ap2 . K1;

$\sqrt{m+1}\left|{\mathrm{\psi }}_{n\,m+1}\right.⟩$

%Bracket(Dagger(K1), am1, K2);

$⟨{\mathrm{\psi }}_{n\,m}\left|a^{-}\right|{\mathrm{\psi }}_{r\,s}⟩$

value((16));

$\sqrt{r}{\mathrm{\delta }}_{n,r-1}{\mathrm{\delta }}_{m,s}$

%Bracket(Dagger(K1), ap2, K2);

$⟨{\mathrm{\psi }}_{n\,m}\left|a^{+}\right|{\mathrm{\psi }}_{r\,s}⟩$

value((18));

$\sqrt{s+1}{\mathrm{\delta }}_{n,r}{\mathrm{\delta }}_{m,s+1}$

Bracket(Bra(R, x, y, z), Ket(P, p[x], p[y], p[z]));

$⟨R_{x\,y\,z}|P_{p_x\,p_y\,p_z}⟩$

(20) = 1/(2*Pi)^(3/2)*exp(I*(x*p[x] + y*p[y]+z*p[z]));

$⟨R_{x\,y\,z}|P_{p_x\,p_y\,p_z}⟩=\frac{\sqrt{2}{\ⅇ}^{\mathrm{I}\left(x{p}_x+y{p}_y+z{p}_z\right)}}{4{\mathrm{\pi }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

Setup((21));

%Bracket(R, a,b,c, P, d,e,f);

$⟨R_{a\,b\,c}|P_{d\,e\,f}⟩$

value((23));

$\frac{\sqrt{2}{\ⅇ}^{\mathrm{I}\left(ad+be+fc\right)}}{4{\mathrm{\pi }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$

Setup(continuousbasis = {P,R});

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

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

$\left[\mathrm{quantumcontinuousbasis}=\left\{P\,R\right\}\right]$

Projector(Ket(P, p[1], p[2], p[3]));

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|P_{p_1\,p_2\,p_3}\right.⟩\left.⟨P_{p_1\,p_2\,p_3}\right|\ⅆp_1\right)\ⅆp_2\right)\ⅆp_3$

'Bracket'(R, x[1],x[2],x[3] , (26), R,y[1],y[2],y[3]);

$⟨R_{x_1\,x_2\,x_3}\left|{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|P_{p_1\,p_2\,p_3}\right.⟩\left.⟨P_{p_1\,p_2\,p_3}\right|\ⅆp_1\right)\ⅆp_2\right)\ⅆp_3\right|R_{y_1\,y_2\,y_3}⟩$

(27);

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\ⅇ}^{\mathrm{I}\left(x_1-y_1\right)p_1+\mathrm{I}\left(x_2-y_2\right)p_2+\mathrm{I}\left(x_3-y_3\right)p_3}}{8{\mathrm{\pi }}^3}\ⅆp_1\right)\ⅆp_2\right)\ⅆp_3$

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