Dirac Notation Introduction Spring 2014 v3.
1. Braket (Dirac) Notation
Dirac introduced a very beautiful way of expressing the vectors used in quantum mechanics. This
is a short introduction to “braket notation” from the point of view of vector calculus. For those
wanting a clean, logical presentation I know of no better than Dirac’s, The Principles of Quantum
Mechanics sections 6-20. What follows is a brief introduction that focuses on basic definitions and
vector operations.1
Basic idea: A “ket” | ·i is a vector.
The components may be complex, hence the space of kets is a complex vector space. To keep track
of which vector we add a label, replacing · above, with some description. For instance, | +zi might
represent the “spin up in the z direction” vector. Another example are wavefunctions, since (suitable)
functions can be seen to form a vector space, we can write, e.g. | ψi.
Slogan: “Put what you know in the ket.” In quantum mechanics, you identify the vector by the last
measurement on the system. Suppose you have a particle in a box. If you observed the particle on the
left hand side, say 0 < x < L/2, then the ket would be | 0 < x < L/2i.
Basis: If you have N basis vectors | ii, i = 1, 2, . . . , N then any vector | vi is written as
X
| vi = vi | ii.
i
It can also be arranged in a column
v1
v2
.
| vi =
.
.
vN
These expressions are the analog of the usual
~a = ax î + ay ĵ + az k̂
and
ax
~a = ay
az
in the familiar 3D vector space.
Bra: A “bra” h· | is “dual” to a vector which means that, with a ket, the bra gives a complex number,
h· | ·i ∈ C. The bra is an adjoint of the vector,
ha |= (| ai)† .
1The slogans are largely from Chester’s Primer of Quantum Mechanics, a classic and quirky introduction to quantum
mechanics.
1
2
The dagger † is the usual notation for adjoint. The mechanics of the adjoint take kets to bras and the
components to their complex conjugates. For instance, if
i
| neatketi = √ | 3i
2
then
† −i
hneatbra |≡ (| neatketi) = √ h3 |
2
See how the ket switched to a bra and the number became its complex conjugate? The label on the
state does not change. If you are using the column notation for the kets you make a row vector under
the adjoint so
†
v1
v2 = (v1∗ v2∗ v3∗ )
v3
This all agrees nicely with the linear algebra conventions. Note that the adjoint is the “complex
transpose” in that context. This operation is sometimes also called the “Hermitian conjugate”.
Inner Product: The scalar product or inner product is written as h· | ·i. This has many other
interpretations as well. The most important interpretation in quantum mechanics is (fanfare!)
The inner product is the probability amplitude.
So this is the beastie that gives predictions! If you have a state | ψi and what to find out whether the
spin is up in the z direction then you calculate the square modulus of the probability amplitude,
|h+z | ψi|2 = h+z | ψih+z | ψi∗
and that is the probability, (assuming the state | ψi is normalized).
The component, or representation, of a ket vector | ai in the basis | ii is
(| ii)† | ai = hi | ai
So X X
| ai = ai | ii = | iihi | ai
i i
In quantum mechanics, you can write a wavefunction ψ(x) as the wavefunction in the x representation,
i.e. hx | ψi.
We usually work with orthonormal bases so that
hvi | vj i = δij .
You can now write 1 in a new way X
| iihi |= 1!
i
This states that the basis | ii is complete.
Operators: Operators, often written with hats,ˆ(in polite company), take a ket and produce another
ket
Q̂ | ai =| bi.
You can express any operator as a matrix operation by working in a basis like | ii. This is called
“matrix mechanics” (which was discovered by Heisenberg, Born, and Jordan). The operator is entirely
determined by how it acts on every basis vector
X
Q̂ | ii = Qij | ji
j
