3.1 Basic Concepts and Formulae 133
Commutators
AB−BA=[A,B] (3.12)
by definition.
Dirac’s Bra and Ket notation
A ket vector, or simply ket, is analogous to the wave function for a state. The symbol
|m >denotes the ket vector that corresponds to the statemof the system. A bra
vector, or bra, is analogous to the complex conjugate of the wave function for a
state. The symbol<n|denotes the bra vector that corresponds to the statenof the
system. Then
∫
ψn∗ψmdτ=〈n|m〉 (3.13)And∫
ψn∗Hψmdτ=〈n|H|m〉 (3.14)Parity
Parity of a function can be positive or negative, and some functions may not have
any parity.
Ifψ(−x,−y,−z)=+ψ(x,y,z) thenψhas positive or even parity.
Ifψ(−x,−y,−z)=−ψ(x,y,z) thenψhas negative or odd parity.(3.15)
Example of even parity is cos x. Example of odd parity is sin x.
For a function likeex, parity cannot be defined. The parity due to orbital angular
momentum is determined by the function (−1)l.
Laporte rule
An integral vanishes between, symmetric limits if the integrand has odd parity. Con-
sidering that the operator of the electric dipole moment has odd parity, the expecta-
tion value of the electric dipole moment has odd parity, the expectation value of the
electric dipole moment as well as the transition probability vanishes unless initial
and final state have different parity.
Even a more restrictive selection rule is
Δl=± 1 (3.16)