Appendix B
Dirac’s Notation and Operator Algebra
In the notation introduced by Dirac [1, 2], a quantum mechanical state is represented
as a ket, for instance |ψ. The set of all the physically meaningful states of a system is
a vector space S, because of the linearity of the time-dependent Schrödinger equation.
In the vector space S we indicate the scalar product of |ψ i and
ψ j
as
ψ i
ψ j
. We
assume the states to be explicitly represented by wavefunctions ψ that depend on the
space (x i , y i and z i ) and spin (s i ) coordinates of N particles, collected in the vector
x ≡ {x 1 , y 1 , z 1 , s 1 . . . x i , y i , z i , s i . . . x n , y n , z n , s n } ≡ {r 1 , s 1 . . . r i , s i . . . r n , s n }
(B.1)
The “bracket” notation is a useful shorthand for the analytical definition of the scalar
product between two wavefunctions, which is
ψ i
ψ j
=
s 1 ...s n
+∞
−∞
ψ
∗
i (x)ψ j (x) dx 1 dy 1 dz 1 . . . dx n dy n dz n
(B.2)
Here we sum over all the integer or half-integer values allowed for the z component
of the spin of each particle, and we integrate over the three Cartesian coordinates,
again for each particle. We see that this definition is consistent with the basic property
of scalar products:
ψ j |ψ i
=
ψ i
ψ j
∗
(B.3)
Now suppose
ψ j
is a linear combination of other states:
ψ j
=
k
c k |φ k
(B.4)
Then
ψ i
ψ j
=
k
c k ψ i |φ k
(B.5)
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