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Transcript
```Angular Momentum
What was Angular Momentum Again?
• If a particle is confined to going around a sphere:
At any instant the particle
is on a particular circle
r
The particle is some
distance from the origin, r
The particle has angular
momentum, L = r × p
v
The particle or mass m has
some velocity, v and
momentum p
What was Angular Momentum Again?
• So a particle going around in a circle (at any
instant) has angular momentum L:
L=r×p
Determine L’s direction from the “right hand rule”
p
r
What was Angular Momentum Again?
• L like any 3D vector has 3 components:
• Lx : projection of L on a x-axis
• Ly : projection of L on a y-axis
• Lz : projection of L on a z-axis
L=r×p
z
p
x
r
y
What was Angular Momentum Again?
• Picking up L and moving it over to the origin:
L=r×p
L=r×p
z
p
x
r
y
What was Angular Momentum Again?
• Picking up L and moving it over to the origin:
L=r×p
z
Rotate
x
y
What was Angular Momentum Again?
• And re-orienting:
Rotate
x
y
z
L=r×p
What was Angular Momentum Again?
• And re-orienting:
z
L=r×p
y
x
• Now we’re in a viewpoint that will be convenient to
analyse
Angular Momentum Operator
• L is important to us because electrons are
constantly changing direction (turning) when they
are confined to atoms and molecules
• L is a vector operator in quantum mechanics
• Lx : operator for projection of L on a x-axis
• Ly : operator for projection of L on a y-axis
• Lz : operator for projection of L on a z-axis
Angular Momentum Operator
• Just for concreteness L is written in terms of
position and momentum operators as:
with
Angular Momentum Operators
• Ideally we’d like to know L BUT…
One day this will be a lab…
• Lx , Ly and Lz don’t commute!
• By Heisenberg, we can’t measure them
simultaneously, so we can’t know exactly where and
what L is!
Angular Momentum Operators
•
,
•
•
and
does commute with each of
individually
is the length of L squared.
has the simplest mathematical form
• So let’s pick the z-axis as our “reference” axis
Angular Momentum Operators
• So we’ve decided that we will use
substitute for
and
as a
• Because we can simultaneously measure:
• L2 the length of L squared
• Lz the projection of L on the z-axis
BUT we can’t know Lx, Ly and Lz
simultaneously!
z
L
We’ve chosen to know only Lz (and L2)
y
Lz
Ly
Lx
x
Angular Momentum Operators
• So we’ve decided that we will use
substitute for
and
as a
• Because we can simultaneously measure:
• L2 the length of L squared
• Lz the projection of L on the z-axis
z
L
For different L2’s we’ll
have different Lz’s
y
Lz
can be anywhere in a
cone for a given Lz
x
So what are the possible
and
eigenvalues
and what are their eigenfunctions?
Angular Momentum Eigen-System
• Operators that commute have the same
eigenfunctions
•
and
commute so they have the same
eigenfunctions
• Using the commutation relations on the previous
slides along with:
we’d find….
One day this will be a lab too…
Angular Momentum Eigen-System
• Eigenfunctions Y, called: Spherical Harmonics
• l = {0,1,2,3,….} angular momentum quantum number
• ml = {-l, …, 0, …, l} magnetic quantum number
Angular Momentum Vector Diagrams
z
Say l = 2
then m ={-2, -1, 0, 1, 2}
For m =2
For m =2
Angular Momentum Vector Diagrams
z
• Take home messages:
• The magnitude (length) of angular
momentum is quantized:
• Angular momentum can only point
in certain directions:
• Dictated by l and m
Angular Momentum Eigenfunctions
• The explicit form of
and
spherical polar coordinates:
is best expressed in
z
q
x
r
f
For now, our particle is on a
sphere and r is constant
y
• We won’t actually formulate these operators (they are too
messy!), but their wave functions Y, will be in terms of q and f
instead of x, y and z:
• Yl,m(q,f) = Ql,m (q) Fm (f)
Angular Momentum Eigenfunctions
• l = 0, ml = 0
Angular Momentum Eigenfunctions
l = 1 ml ={-1, 0, 1}
Look Familiar?
Angular Momentum Eigenfunctions
l = 2 ml ={-2, -1, 0, 1, 2}
Look Familiar?
Particle on a Sphere
r
q can vary form 0 to p
f can vary form 0 to 2 p
r is constant
Particle on a Sphere
• The Schrodinger equation:
0
Particle on a Sphere
• So for particle on a sphere:
Legendre Polynomials
Spherical harmonics
• Energies are 2l + 1 fold degenerate since:
• For each l, there are {ml} = 2l + 1
eigenfunctions of the same energy
```
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