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 state m of 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 state n of the
system. Then
ψ
∗
n ψ m dτ = =n|m
(3.13)
And
ψ
∗
n H ψ m dτ = =n|H |m
(3.14)
Parity
Parity of a function can be positive or negative, and some functions may not have
any parity.
If ψ(−x, −y, −z) = +ψ(x, y, z) then ψ has positive or even parity.
If ψ(−x, −y, −z) = −ψ(x, y, z) then ψ has negative or odd parity.
(3.15)
Example of even parity is cos x. Example of odd parity is sin x.
For a function like e
x , 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. Considering that the operator of the electric dipole moment has odd parity, the expectation 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)
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 state m of 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 state n of the
system. Then
ψ
∗
n ψ m dτ = =n|m
(3.13)
And
ψ
∗
n H ψ m dτ = =n|H |m
(3.14)
Parity
Parity of a function can be positive or negative, and some functions may not have
any parity.
If ψ(−x, −y, −z) = +ψ(x, y, z) then ψ has positive or even parity.
If ψ(−x, −y, −z) = −ψ(x, y, z) then ψ has negative or odd parity.
(3.15)
Example of even parity is cos x. Example of odd parity is sin x.
For a function like e
x , 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. Considering that the operator of the electric dipole moment has odd parity, the expectation 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)