angular momentum operators
TRANSCRIPT
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7.1 Angular momentum
Slides: Video 7.1.1 Angular
momentum operatorsText reference: Quantum Mechanics
for Scientists and EngineersChapter 9 introduction and
Section 9.1 (first part)
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Angular momentum
Angular momentum operators
Quantum mechanics for scientists and engineers David M
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Angular momentum operators - preview
We will have operators corresponding toangular momentum about differentorthogonal axes
, , and
though they will not commute withone anotherin contrast to the linear momentum operators
for the different coordinate directions, , and
which do commute
x L ˆ y L
z L
ˆ x p ˆ y p ˆ z p
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Angular momentum operators - preview
We will, however, find another useful angularmomentum operator,
which does commute separately with each of, , and
The eigenfunctions for , , and are simpleThose for
the spherical harmonics, are more complicatebut can be understood relatively simplyand form the angular shapes of the
hydrogen atom orbitals
2ˆ L
x L ˆ y L
z L
x L ˆ y L z L2ˆ L
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y
r
p
origin
posiob
momentum
Classical angular momentum
The classical angularmomentumof a small object
of (vector) linear
momentum pcentered at a point given
by the vectordisplacement r relativeto some originis L r p
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Vector cross product
As usual
where i, j , and k are unit vectors in x, y, and z
and A x is the component of A in the x directiand similarly for the y and z directions
and the components of B
sin
( ) ( ) (
x y z
x y z
y z z y x z z x x y
AB A A A B B B
A B A B A B A B A B
i j k
C A B c
i j k
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Vector cross product
In
C is perpendicular to the plane of A and B
just as the z axis is perpendicular to the planecontaining the x and y axes in right-handed axe is the angle between the vectors A and B
c is a unit vector in the direction of the vector C
sin
( ) ( ) (
x y z
x y z
y z z y x z z x x y
AB A A A B B B
A B A B A B A B A B
i j k
C A B c
i j k
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Vector cross product
Note that, in
the ordering of the multiplications in the second line is
chosen to work also for operators instead of numbersfor one or other vectorthe sequence of multiplications in each term is alwa
in the sequence of the rows from top to bottom
sin
( ) ( ) (
x y z
x y z
y z z y x z z x x y
AB A A A B B B
A B A B A B A B A B
i j k
C A B c
i j k
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Angular momentum operators
With classical angular momentum
we can explicitly write out the various components
Now we can propose a quantum mechanical angularmomentum operator
based on substituting the position and momentumoperators
and similarly write out component operators
x z y L yp zp y x z L zp xp
z y x L xp yp
L r p
ˆ ˆ ˆ i L r p r
L
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Angular momentum operators
Analogously, we obtain three operators
which are each Hermitianand so, correspondingly, is the operator itself
ˆ ˆ ˆ ˆˆ x z y L yp zp i y z z y
ˆ ˆ ˆ ˆˆ y x z L zp xp i z x x z
ˆ ˆˆ ˆ ˆ z y x L xp yp i x y y x
L
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Commutation relations
The operators corresponding to individual coordinatedirections obey commutation relations
These individual commutation relations can be writtenin a more compact form
ˆ ˆ ˆ ˆ ˆ, x y y x x y z L L L L L L i L
ˆ ˆ ˆ ˆ ˆ, y z z y y z x L L L L L L i L
ˆ ˆ ˆ ˆ ˆ, z x x z z x y L L L L L L i L
ˆ ˆ ˆi L L L
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