Chemistry 460
Spring 2015
Dr. Jean M. Standard
January 21, 2015
Dirac Bra-ket Notation
It is often useful to employ notation to simplify the integrals and other quantities that are needed in quantum
mechanics. This notation is called Dirac bra-ket (or bracket) notation and it is widely used in textbooks and in the
literature.
Dirac Notation Involving Integrals
To begin, consider two wavefunctions ψ and φ . The integral over all space of the product of the two
wavefunctions can be expressed in the following way,
€
∞
€
ψ *( x ) φ ( x ) dx .
∫
(1)
−∞
Using Dirac notation, we can rewrite the integral in the following form,
€
∞
ψ *( x ) φ ( x ) dx =
∫
(2)
ψ φ .
−∞
Notice that the complex conjugation is built into the Dirac notation for the wavefunction on the left. The left part of
the Dirac notation is called a "bra"€and the right side is called a "ket" (put them together to get a bracket). Thus, the
bra includes complex conjugation while the ket does not.
If the wavefunctions are indexed, then Dirac notation allows even more simplification,
∞
∫
ψ n* ( x ) ψ m ( x ) dx =
ψn ψm
=
n m .
(3)
−∞
Expectation values also are simplified using Dirac notation. For an operator Aˆ the expectation value may be written
in the following way, €
∞
A
∫
=
€
ψ *( x ) Aˆ ψ ( x ) dx = ψ Aˆ ψ .
(4)
−∞
The form shown above assumes that the wavefunction is properly normalized. The expectation value may be
written in Dirac notation €
in an alternate form,
A
=
ψ Aˆ ψ
=
ψ Aˆ ψ .
(5)
The two forms for the expectation values are equivalent because the operator Aˆ operates on the function to its right
to yield a new function, so the €
bar separating the operator from the ket may be left out.
€
2
Dirac notation may be used for any quantum mechanical integral involving two indexed wavefunctions and an
operator,
∞
∫
ψ *n ( x ) Aˆ ψ m ( x ) dx =
ψ n Aˆ ψ m
n Aˆ m .
=
(6)
−∞
These integrals are sometimes referred to as matrix elements because they are related to the matrix formulation of
quantum mechanics.
€
The effects of complex conjugation are illustrated by the following example. Using regular integral notation, the
complex conjugate of the integral of the product of two wavefunctions ψ and φ can be expressed as
∞
'∞
**
*
) ψ ( x ) φ ( x ) dx , =
ψ ( x ) €φ * ( x )€dx
)(
,+
−∞
−∞
∫
∞
∫
=
∫
φ *( x ) ψ ( x ) dx .
(7)
−∞
In Dirac notation,
€ this becomes
ψ φ
*
(8)
φ ψ .
=
Note that since it is ultimately φ that is the complex conjugate, it must be placed in the bra in Dirac notation.
€
The definition of a Hermitian operator can be expressed in Dirac notation. In regular notation, an operator Aˆ is
Hermitian if
€
∞
∫
∞
f ( x ) Aˆ g( x ) dx =
*
*
g( x ) Aˆ f ( x ) dx .
(
∫
−∞
)
€
(9)
−∞
In Dirac notation, the definition of a Hermitian operator is
€
f
Aˆ g
=
Aˆ f
g .
(10)
Dirac Notation for Wavefunctions
The bra and ket notation may be used separately
to denote wavefunctions. Consider the following expression,
€
f ( x) =
∑ c n ψ n ( x) .
(11)
n
In Dirac notation, the expression becomes
€
f
=
∑ cn
ψn
=
n
∑ cn
n .
(12)
n
Note that in Dirac notation, the coefficients c n are not included in the ket since they are not functions.
€
Finally, the complex conjugate of a wavefunction is represented in terms of a bra. For example,
€
ψ *( x) =
€
ψ .
(13)