Download Momentum eigenstates

Momentum eigenstates
å
å
å
å
linear momentum eigenstates
orbital angular momentum eigenstates
spin angular momentum eigenstates
addition of angular momenta
l the generic angular momentum operator
We will work mainly with quantum numbers
in the applications
RAF211 - CZJ
1
Linear momentum eigenstates
We will consider motion in one dimension (x):
h d
p x = – i -----2π d x
By solving the eigenvalue equation p x ψ = p x ψ
(where both ψ and p x are unknown), we find
ψ( x) = c ⋅ e
RAF211 - CZJ
 2πi
-------- xp x
 h 
2
Eigenstates of orbital angular momentum
We will look for the simultaneous eigenstates ψ
2
of the operators L and L z
2
2
Lz ψ = Lz ψ , L ψ = L ψ
z
We will use spherical
coordinates: (r,θ,φ)
x
RAF211 - CZJ
θ
r
y
φ
3
Lz eigenstates and eigenvalues
The eigenvalue equation
h
dψ
– i -----= L z gives
2π d φ
ψ ( φ ) = ce
 2πi
-------- φL z
 h 
quantized
h
Condition: ψ ( φ ) = ψ ( φ + 2π ) ⇒ L z = m -----2π
m = 0, ± 1, ± 2, … is the magnetic quantum number
RAF211 - CZJ
4
Eigenstates of L2 and Lz
Spherical harmonics
m ( 2l + 1 ) ( l – m )! 1 ⁄ 2 m
imφ
Y lm ( θ, φ ) = ( – 1 ) -------------------------------------P l ( cos θ )e
4π ( l + m )!
m
where P l ( cos θ ) are the associated Legendre functions
2
h 2
L =  ------ l ( l + 1 ) , l = 0, 1, 2, … the orbital quantum
 2π
number
the magnetic quantum number is restricted by l :
m = – l ,– l + 1 ,…, 0, … ,l – 1 ,l
RAF211 - CZJ
5
Parity eigenstates
The spherical harmonics are also eigenstates
of the parity operator
l
PY lm = ( – 1 ) Y lm
Application: transitions in atoms
RAF211 - CZJ
6
The generic angular momentum operator
h- J and cyclic permutations
J = J x u x + J y u y + J z u z , [ J x, J y ] = i ----2π z
2
h
h- , j = 0, 1, 2, … , m = – j ,– j + 1 ,… ,j
J =  ------ j ( j + 1 ) , L z = m ---- 2π
j
2π
2
By solving the eigenvalue equations for J and J z simultaneously
and assuming n eigenstates, we find that there are two ‘types’ of angular momentum:
a integer j
a half-integer j
2
RAF211 - CZJ
7
The spin angular momentum
Spin does not involve a rotation in space
2
h
S =  ------ s ( s + 1 )
 2π
2
where s is the spin quantum number
h
S z = m s ------ ,
2π
m s = – s ,– s + 1 ,… ,s – 1 ,s
a integer s : bosons
a half-integer s : fermions
RAF211 - CZJ
8
Addition of angular momenta
Let us assume that we want to ‘add’ the quantum numbers l and s to
find the quantum number of the total angular momentum, j , and the
quantum number of the z-component, m j
The rules are:
1. j = l – s , …, l + s
2. for each j , we have 2j + 1 m j ’s: m j = – j ,– j + 1 ,… ,j
Application: L-S coupling and J-J coupling in atoms
RAF211 - CZJ
9