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