The Dirac Bra-Ket notation is the standard notation in quantum mechanics, although it can also be used to denote abstract vectors and linear functionals in general.

In quantum mechanics, a Ket state vector is used to represent the quantum state of a physical system. In this implementation, Kets are assumed to be non-degenerate, the correspondence between Kets and quantum states is one-to-one, and a set of Kets can always be said to form a basis of a (sub)space of quantum states.

The first argument in $\mathrm{Ket}\left(A,\mathrm{n1},\mathrm{n2}\dots \right)$ is a label identifying the basis to which the Ket belongs. Recall that a single space of quantum states can be represented by different bases, so the scalar product of Kets of these different bases is, generally speaking, not zero. To represent fermionic state vectors, use an anticommutative label; see Setup for more information.

The second and subsequent N arguments of the Ket function represent "quantum numbers," and identify one Ket from another inside the basis. In this sense, the concept of a quantum number is equivalent to the concept of components of a standard vector in a Euclidean N-dimensional space, if the different components are associated with the different values of the quantum number. However, note that in the case of quantum-state vectors, there can be many quantum numbers (the N-dimensional Euclidean analog of that would be a tensor, not a vector). Kets with many quantum numbers belong to the tensor product of spaces that can be represented with Kets of only one quantum number.

When the label, A, of a Ket has also been set as a quantum operator, the Ket is automatically an eigenvector of this operator with a quantum number as its eigenvalue. To specify which quantum number, use the operator indexed with the position of the quantum number; for example, $A_2·\mathrm{Ket}\left(A\,n\,m\right)$ will result in $m\mathrm{Ket}\left(A,n,m\right)$.

A Bra is also a vector in a space of quantum states and corresponds to a Ket vector in the sense that one is obtained from the other through Hermitian conjugation. Bras are elements of the dual of the space, associated with the corresponding Kets.

The set of Kets formed by attributing all possible values to its quantum numbers forms a basis for the corresponding space of quantum states. To construct projectors onto this basis, use the Projector command. To set the dimension, possibly different, of each of the quantum numbers of the Kets of a basis of quantum states, see Setup.

The space of states spanned by a basis of Kets can be either continuous or discrete. 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.

The Physics[`.`] is the computational representation for the scalar (also called inner) product between any two of a Bra, a Ket, or a quantum operator, the latter set to be so by the Setup command or returned by the Annihilation, Creation, or Projector commands.

For scalar products involving Bras and Kets of the same basis and with quantum numbers, the result is expressed in terms of KroneckerDeltas, when the state vectors belong to a discrete space of states, or in terms of Dirac functions when they belong to a continuous space of states. In this sense, Bras and Kets with quantum numbers are analogous to 3-D Euclidean "unit vectors."

When the scalar product cannot be resolved, the operation is returned unevaluated, expressed by using the Bracket command (note the 'c' between Bra and Ket). For details regarding when a scalar product is returned unevaluated, see Physics[`.`].

To represent the tensor product of Bras and Kets, use the Physics[`*`] operator, as in $\&verbar;A>*<B\&verbar;$ (a quantum operator) or $\&verbar;A>*\&verbar;B>$ (a Ket of the tensor product of spaces).

Two important operations, representing the annihilation and creation of 'particles' in a given state, are performed by using the linear operators returned by the Annihilation and Creation commands. These operators diminish and augment by one, respectively, the "quantum numbers" over which they act; see the respective help pages and the examples below.

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

with(Physics):

Setup(mathematicalnotation = true);

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

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

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

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

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

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

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

BK2 := Dagger(K2);

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

BK1 . K2;

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

Bracket(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);

${\mathrm{\&delta;}}^{\left(2\right)}\left(\left[-r\&plus;n\&comma;-s\&plus;m\right]\right)$

Setup(clear, continuousbasis);

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

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

$\left[\mathrm{quantumcontinuousbasis}=\mathrm{none}\right]$

Setup(discretebasis = psi);

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

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

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

am1 := Annihilation(psi, 1);

$\mathrm{am1}≔a^{-}$

am2 := Annihilation(psi, 2, notation = explicit);

$\mathrm{am2}≔{a^{-}}_{{\mathrm{\psi }}_2}$

am1(K1);

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

am2(K2);

$\sqrt{s}\left|{\mathrm{\psi }}_{r\&comma;s-1}\right.⟩$

ap1 := Creation(psi, 1);

$\mathrm{ap1}≔a^{+}$

ap2 := Creation(psi, 2);

$\mathrm{ap2}≔a^{+}$

ap1 . K1;

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

ap2 . K2;

$\sqrt{s+1}\left|{\mathrm{\psi }}_{r\&comma;s+1}\right.⟩$

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

$⟨R_{x\&comma;y\&comma;z}|P_{p_x\&comma;p_y\&comma;p_z}⟩$

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

$⟨R_{x\&comma;y\&comma;z}|P_{p_x\&comma;p_y\&comma;p_z}⟩=\frac{\sqrt{2}{\&ExponentialE;}^{\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(Bra(R, a, b, c), Ket(P, d, e, f));

$⟨R_{a\&comma;b\&comma;c}|P_{d\&comma;e\&comma;f}⟩$

value((23));

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

Setup(continuousbasis = {P,R});

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

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

$\left[\mathrm{quantumcontinuousbasis}=\left\{P\&comma;R\right\}\right]$

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

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left|P_{p_1\&comma;p_2\&comma;p_3}\right.⟩\left.⟨P_{p_1\&comma;p_2\&comma;p_3}\right|\&DifferentialD;p_1\right)\&DifferentialD;p_2\right)\&DifferentialD;p_3$

Bra(R, x[1], x[2], x[3]) . (26);

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\sqrt{2}{\&ExponentialE;}^{\mathrm{I}\left(p_1{x}_1+p_2{x}_2+p_3{x}_3\right)}\left.⟨P_{p_1\&comma;p_2\&comma;p_3}\right|}{4{\mathrm{\pi }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&DifferentialD;p_1\right)\&DifferentialD;p_2\right)\&DifferentialD;p_3$

(26) . Ket(R, x[1], x[2], x[3]);

${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\sqrt{2}{\&ExponentialE;}^{\mathrm{-I}\left(p_1{x}_1+p_2{x}_2+p_3{x}_3\right)}\left|P_{p_1\&comma;p_2\&comma;p_3}\right.⟩}{4{\mathrm{\pi }}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\&DifferentialD;p_1\right)\&DifferentialD;p_2\right)\&DifferentialD;p_3$

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