7

Dirac Notation Identities

7.1    Useful Identities

As described by Dirac (1978) and Feynman et al. (1965), the Dirac notation includes various mathematical properties and allows for various abstractions and permutations. Here, a few useful set of identities and properties of the notation are described.

First, the complex conjugate of 〈ϕ|ψ〉 is defined as

$\langle \text{ϕ}|\text{ψ}\rangle ={\langle \text{ψ}|\text{ϕ}\rangle }^{*}$

(7.1)

Also, the probability amplitude

$\langle \text{ϕ}|\text{ψ}\rangle =\langle \text{ϕ}|j\rangle \langle j|\text{ψ}\rangle$

(7.2)

can be expressed in abstract form as

$|\text{ψ}\rangle =|j\rangle \langle j|\text{ψ}\rangle$

(7.3)

An additional form of abstract notation is

$\langle \text{χ}\left|A\right|\text{ϕ}\rangle =\langle \text{χ}|i\rangle \langle i|A|j\rangle \langle j|\text{ϕ}\rangle$

(7.4)

where A is

$A=|i\rangle \langle i|A|j\rangle \langle j|$

(7.5)

Another abstraction is illustrated by

$A|\text{ϕ}\rangle =|i\rangle \langle i\left|A\right|j\rangle \langle j|\text{ϕ}\rangle$

(7.6)

Further, A can be multiplied by B so that

$\langle \text{χ}\left|BA\right|\text{ϕ}\rangle =\langle \text{χ}|i\rangle \langle i|B|j\rangle \langle j|A|k\rangle \langle k|\text{ϕ}\rangle$

(7.7)

To express

$\langle \text{χ}\left|A\right|\text{ϕ}\rangle =\langle \text{χ}|i\rangle \langle i|A|j\rangle \langle j|\text{ϕ}\rangle$

(7.8)

in the abstract form

$\langle \text{χ}|\text{ψ}\rangle =\langle \text{χ}|i\rangle \langle i|\text{ψ}\rangle$

(7.9)

it is necessary that

$\langle i|\text{ψ}\rangle =\langle i|A|j\rangle \langle j|\text{ϕ}\rangle =\langle i\left|A\right|\text{ϕ}\rangle$

(7.10)

which means that

$\langle \text{χ}|\text{ψ}\rangle =\langle \text{χ}|A|\text{ϕ}\rangle$

(7.11)

Further abstracting leads to

$|\text{ψ}\rangle =|A|\text{ϕ}\rangle$

(7.12)

Other examples of abstractions include

$\langle i|\text{ϕ}\rangle =C_i$

(7.13)

$\langle i|\text{χ}\rangle =D_i$

(7.14)

$|\text{ϕ}\rangle ={\displaystyle \sum _i|i\rangle C_i}$

(7.15)

$|\text{χ}\rangle ={\displaystyle \sum _i|i\rangle D_i}$

(7.16)

$\langle \text{χ}|={\displaystyle \sum _i{D}_i^{*}\langle i|}$

(7.17)

which is the abstracted version of

$\langle \text{χ}|\text{ϕ}\rangle ={\displaystyle \sum _{ij}{D}_j^{*}\langle j|i\rangle C_i}$

(7.18)

and since 〈j|i〉 = δ_ij

$\langle \text{χ}|\text{ϕ}\rangle ={\displaystyle \sum _i{D}_i^{*}{C}_i}$

(7.19)

Finally, using the ultimate abstraction suggested by Dirac (Feynman et al., 1965), Equation 7.19 can be expressed as

$|={\displaystyle \sum _i|i\rangle \langle i|}$

(7.20)

7.1.1    Example

The probability amplitude describing interference in a Mach–Zehnder interferometer can be described as (Duarte, 2003)

$\langle x|s\rangle ={\displaystyle \sum _{kj}\langle x|k\rangle \langle k|j\rangle \langle j|s\rangle }$

(7.21)

Defining

$\langle j|s\rangle =C_j$

(7.22)

$\langle k|x\rangle =D_k$

(7.23)

$\langle x|s\rangle ={\displaystyle \sum _{kj}{D}_k^{*}\langle k|j\rangle C_j}$

(7.24)

and using 〈k|j〉 = δ_kj, we get

$\langle x|s\rangle ={\displaystyle \sum _j{D}_j^{*}{C}_j}$

(7.25)

7.2    Linear Operations

Dirac (1978) describes various mathematical properties related to his ket vectors. First, if c₁ and c₂ are complex numbers, ket vectors can be multiplied by these complex numbers and added to produce a new ket vector

$c_1|\text{ϕ}\rangle +c_2|\text{ψ}\rangle =|\text{θ}\rangle$

(7.26)

Superposition of a state, with itself, yields the original state

$c_1|\text{ϕ}\rangle +c_2|\text{ϕ}\rangle =(c_1+c_2)|\text{ϕ}\rangle$

(7.27)

Additional sum and product conditions are illustrated by

$\langle \text{ϕ}|(|\text{ψ}\rangle +|\text{χ}\rangle )=\langle \text{ϕ}|\text{ψ}\rangle +\langle \text{ϕ}|\text{χ}\rangle$

(7.28)

$\langle \text{ϕ}|(c|\text{ψ}\rangle )=c\langle \text{ϕ}|\text{ψ}\rangle$

(7.29)

and

$(\langle \text{ϕ}|+\langle \text{χ}|)|\text{ψ}\rangle =\langle \text{ϕ}|\text{ψ}\rangle +\langle \text{χ}|\text{ψ}\rangle$

(7.30)

$(c\langle \text{ϕ}|)|\text{ψ}\rangle =c\langle \text{ϕ}|\text{ψ}\rangle$

(7.31)

If α is a linear operator, then

$|\vartheta \rangle =\text{α}|\text{ψ}\rangle$

(7.32)

and

$\text{α}(|\text{ψ}\rangle +|\text{χ}\rangle )=\text{α}|\text{ψ}\rangle +\text{α}|\text{χ}\rangle$

(7.33)

$\text{α}(c|\text{ψ}\rangle )=c\text{α}|\text{ψ}\rangle$

(7.34)

$(\text{α}+\text{β})|\text{ψ}\rangle =\text{α}|\text{ψ}\rangle +\text{β}|\text{ψ}\rangle$

(7.35)

$(\text{α}\text{β})|\text{ψ}\rangle =\text{α}(\text{β}|\text{ψ}\rangle )$

(7.36)

$(\langle \text{ϕ}|\text{α})|\text{ψ}\rangle =\langle \text{ϕ}|(\text{α}|\text{ψ}\rangle )$

(7.37)

Further useful identities introduced by Dirac (1978) are

$|\text{ϕ}\rangle |\text{ψ}\rangle =|\text{ψ}\rangle |\text{ϕ}\rangle$

(7.38)

$|\text{ϕ}\rangle |\text{ψ}\rangle =|\text{ψ}\rangle |\text{ϕ}\rangle =|\text{ϕ}\text{ψ}\rangle$

(7.39)

$|\text{ϕ}\rangle |\text{ψ}\rangle |\text{χ}\rangle \dots =|\text{ϕ}\text{ψ}\text{χ}\dots \rangle$

(7.40)

Also, for more than one particle, Dirac (1978) gives the ket for the assembly as

$|X\rangle =|a_1\rangle |b_2\rangle |c_3 \rangle \dots |g_n\rangle$

(7.41)

7.2.1    Example

The Pryce–Ward probability amplitude, prior to normalization, for entangled photons, with polarizations x and y, traveling in opposite directions 1 and 2, is given by

$|\text{ψ}\rangle =\left(\left|{x\rangle }_1\right|y{\rangle }_2-\left|y{\rangle }_1\right|x{\rangle }_2\right)$

(7.42)

which can also be expressed as

$|\text{ψ}\rangle =(|x,y\rangle -|y,x\rangle )$

(7.43)

Problems

7.1  Write in abstract form the probability amplitude corresponding to a Sagnac interferometer given by (Duarte, 2003)

$\langle x|s\rangle ={\displaystyle \sum _{kj}\langle x|k\rangle \langle k|j\rangle \langle j|s\rangle }$

assuming that 〈k|j〉 = 1 (see Chapter 10).

7.2  The probability amplitude for a multiple-beam interferometer (see Chapter 10) can be expressed as (Duarte, 2003)

$\langle x|s\rangle =\langle x|m\rangle \langle m|l\rangle \langle l|k\rangle \langle k|j\rangle \langle j|s\rangle$

Use the various abstract identities, given in this chapter, to efficiently abstract this probability amplitude.

References

Dirac, P. A. M. (1978). The Principles of Quantum Mechanics, 4th edn. Oxford, London, U.K.

Duarte, F. J. (2003). Tunable Laser Optics, Elsevier-Academic, New York.

Feynman, R. P., Leighton, R. B., and Sands, M. (1965). The Feynman Lectures on Physics, Vol. III, Addison-Wesley, Reading, MA.