Lecture 25:Introduction to the Quantum Theory

of Angular Momentum

Phy851 Fall 2009

Goals

1. Understand how to use different coordinatesystems in QM (cartesian, spherical,…)

2. Derive the quantum mechanical propertiesof Angular Momentum• Use an algebraic approach similar to what we

did for the Harmonic Oscillator

3. Use the resulting theory to treat sphericallysymmetric problems in three dimensions• Calculating the Hydrogen atom energy levels

will be our target goal

Motion In 3Dimensions

• For a particle moving in three dimensions,there is a distinct quantum state for everypoint in space.– Thus each position state is now labeled by a

vector

– Vector operators are really three operators

• Coordinate system not unique:– For example, we could use spherical

coordinates

RXv

→

rxr

→

zyx eZeYeXRrrrr

++=

ZYX ,,Scalar Operators Ordinary Vectors

zyx eeerrr,,

),( ΦΘ= reRRrr

ΦΘ,,RScalar Operators

zyxr eeeerrrr

Θ+ΦΘ+ΦΘ=ΦΘ cossinsincossin),(

Vector Operator

),( ΦΘrer

You can never go wrong with CartesianCoordinates

• In all other coordinate systems, the unitvectors are also operators– so must be treated carefully

• Eigenstates:

• For each point in space there is a positioneigenstate

• How we want to label these points is up tous:

rxrrerRerX xx

rrrrrrrr=⋅=⋅=

rrrRrrrr

=

ryrYrr

=

zyxr ,,=r

φθ ,,rr =r

φρ ,, zr =r

Different waysto refer tothe same state

rzrZrr

=

Orthogonality

• In three dimensions, the orthogonalitycondition becomes:

– Cartesian coordinates:

– Spherical coordinates:

– Mixing coordinates:

€

r r

r ′ r = δ 3

r r −

r ′ r ( )

€

xyz ′ x ′ y ′ z = δ x − ′ x ( )δ y − ′ y ( )δ z − ′ z ( )

€

rθφ ′ r ′ θ ′ φ = δ r − ′ r ( )δ θ − ′ θ ( )

rsinθδ φ − ′ φ ( )

r

€

xyz rθφ = δ x − rsinθ cosφ( )δ y − rsinθ sinφ( )δ z − rcosθ( )

Wavefunctions

• Wavefunctions are defined in the usual wayas:

€

d3r∫ ψ(r r ) 2 =1

ψψ xyzzyx =),,(

ψθφφθψ rr =),,(

ψψ rrrr

=)(

€

dx−∞

∞

∫ dy−∞

∞

∫ dz−∞

∞

∫ ψ(x,y,z) 2 =1

€

dr rsinθdθ rdφ0

2π

∫0

π

∫0

∞

∫ ψ(r,θ,φ) 2 =1

Momentum

• The three-dimensional momentum vectoroperator is:

• The three-dimensional Hamiltonian is

• In Cartesian components, this becomes:

zzyyxx PePePePrrrr

++=

€

H =

r P ⋅

r P

2m+ V

r R ( )

( ) ψψ rrVm

Hrrrhr

+∇−= 2

2

2

€

H =Px2

2m+

Py2

2m+

Pz2

2m+ V

r R ( )

( ) ψψ xyzzyxVdz

d

dy

d

dx

d

mHxyz

+

++−= ,,

2 2

2

2

2

2

22h

or

Angular Momentum

• We can decompose the momentum operatoronto spherical components as:

• The unit-vector operators are:

• From the above decomposition, we are led todefine the Angular momentum as:

• This is not the usual definition (L=RxP) but isequivalent.

€

r P =

r e r Θ,Φ( )Pr +

r e θ Θ,Φ( )Pθ +

r e φ Θ,Φ( )Pφ

Pr =r e r Θ,Φ( ) ⋅

r P

Pθ =r e θ Θ,Φ( ) ⋅

r P

Pφ =r e φ Θ,Φ( ) ⋅

r P

zyxr eeeerrrr

Θ+ΦΘ+ΦΘ=ΦΘ cossinsincossin),(

zyx eeeerrrr

Θ+ΦΘ−ΦΘ−=ΦΘ sinsincoscoscos),(θ

yx eeerrr

Φ+Φ−=ΦΘ cossin),(φ

€

r L =

r e θ Θ,Φ( )Pθ +

r e φ Θ,Φ( )Pφ