368 Quantum mechanics
We have introduced a special type of notation for this column vector, “|Ψ〉” with half
of an angle bracket, which is called aDirac ket vector, after its inventor, the English
physicist P. A. M. Dirac (1902–1984). This notation is now standardin quantum
mechanics. The components of|Ψ〉are complex probability amplitudesαkthat are
related to the corresponding probabilities by
Pk=|αk|^2. (9.2.2)The vector|Ψ〉is called thestate vector(conceptually, it bears some similarity to
the phase space vector used to hold the physical state in classicalmechanics). The
dimension of|Ψ〉must be equal to the number of possible states in which the system
might be observed. For example, if the physical system were a coin,then we might
observe the coin in a “heads-up” or a “tails-up” state, and a coin-toss experiment is
needed to realize one of these states. In this example, the dimension of|Ψ〉is 2, and
|Ψ〉could be represented as follows:
|Ψ〉=
(
αH
αT)
, (9.2.3)
whereαHandαTare the (complex) amplitudes for heads-up and tails-up states,
respectively. Since the sum of all the probabilities must be unity
∑kPk= 1, (9.2.4)it follows that ∑
k|αk|^2 = 1. (9.2.5)In the coin-toss example, an unbiased coin would have amplitudesαH=αT= 1/
√
2.
Dirac ket vectors live in a vector space known as theHilbert space, which we will
denote asH. A complementary ordualspace toHcan also be defined in terms of
vectors of the form
〈Ψ|= (α∗ 1 α∗ 2 α∗ 3 ···), (9.2.6)which is known as aDirac bra vector. Hilbert spaces have numerous interesting prop-
erties; however, the most important one for our present purposes is the inner or scalar
product between〈Ψ|and|Ψ〉. This product is defined to be
〈Ψ|Ψ〉=
∑
kα∗kαk=∑
k|αk|^2. (9.2.7)Note that the inner product requires both a bra vector and a ket vector. The terms
“bra” and “ket” are meant to denote two halves of a “bracket” (〈···|···〉), which is
formed when an inner product is constructed. Combining eqn. (9.2.7) with (9.2.5), we
see that|Ψ〉is a unit vector since〈Ψ|Ψ〉= 1.