130_notes.dvi

14.3.1 Parity of the Spherical Harmonics

In spherical coordinates, theparity operationis

r → r

θ → π−θ

φ → φ+π.

The radial part of the wavefunction, therefore, is unchanged and the

R(r)→R(r)

parity of the state is determined from the angular part. We know the stateYℓℓin general.
A parity transformation gives.

Yℓℓ(θ,φ)→Yℓℓ(π−θ,φ+π) =eiℓφeiℓπsinℓ(θ) =eiℓπYℓℓ= (−1)ℓYℓℓ

The states are either even or odd parity depending on the quantumnumberℓ.

parity= (−1)ℓ

The angular momentum operators are axial vectors and do not change sign under a parity transfor-
mation. Therefore,L−does not change under parity and all theYℓmwith have the same parity as
Yℓℓ

L−Yℓℓ→(−1)ℓL−Yℓℓ

Yℓm(π−θ,φ+π) = (−1)ℓYℓm(θ,φ)

14.4 Derivations and Computations

14.4.1 Rotational Symmetry Implies Angular Momentum Conservation

In three dimensions, this means that we can change our coordinates by rotating about any one of
the axes and the equations should not change. Lets try and infinitesimal rotation about thezaxis.
Thexandycoordinates will change.

x′=x+dθy

y′=y−dθx

The original Schr ̈odinger Equation is

Hψ(x,y,z) =Eψ(x,y,z)

and the transformed equation is

Hψ(x+dθy,y−dθx,z) =Eψ(x+dθy,y−dθx,z).