angular momentum operators

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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)



Angular momentum

Angular momentum operators

Quantum mechanics for scientists and engineers David M



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



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



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



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




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




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




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




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



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

angular momentum operators

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