19.1 OPERATOR FORMALISM
represent different states, is a ket that represents a continuum of different states
as the complex numbersc 1 andc 2 are varied.
If we need to specify a state more closely – say we know that it correspondsto a plane wave with a wave number whose magnitude isk– then we indicate
this with a label; the corresponding ket vector would be written as|k〉. If we also
knew the direction of the wave then|k〉would be the appropriate form. Clearly,
in general, the more labels we include, the more precisely the corresponding state
is specified.
The Dirac notation for the Hermitian conjugate (dual vector) of the ket vector|ψ〉is written as〈ψ|and is known as abra vector; the wavefunction describing this
state is∫ ψ∗, the complex conjugate ofψ. The inner product of two wavefunctions
ψ∗φdvis then denoted by〈ψ|φ〉or, more generally if a non-unit weight functionρis involved, by
〈ψ|ρ|φ〉, evaluated as∫
ψ∗(r)φ(r)ρ(r)dr. (19.1)Given the (contrived) names for the two sorts of vectors, an inner product like
〈ψ|φ〉becomes a particular type of ‘bra(c)ket’. Despite its somewhat whimsical
construction, this type of quantity has a fundamental role to play in the interpre-
tation of quantum theory, because expectation values, probabilities and transition
rates are all expressed in terms of them. For physical states the inner product of
the corresponding ket with itself, with or without an explicit weight function, is
non-zero, and it is usual to take
〈ψ|ψ〉=1.Although multiplying a ket vector by a constant does not change the statedescribed by the vector, acting upon it with a more general linear operatorA
results (in general) in a ket describing a different state. For example, ifψis a state
that is described in one-dimensionalx-space by the wavefunctionψ(x)=exp(−x^2 )
andAis the differential operator∂/∂x,then
|ψ 1 〉=A|ψ〉≡|Aψ〉is the ket associated with the state whose wavefunction isψ 1 (x)=− 2 xexp(−x^2 ),
clearly a different state. This allows us to attach a meaning to an expression such
as〈φ|A|ψ〉through the equation
〈φ|A|ψ〉=〈φ|ψ 1 〉, (19.2)i.e. it is the inner product of|ψ 1 〉and|φ〉. We have already used this notation in
equation (19.1), but there the effect of the operatorAwas merely multiplication
by a weight function.
If it should happen that the effect of an operator acting upon a particular ket