7.1 Angular momentum

Slides: Video 7.1.1 Angular momentum operators

Text reference: Quantum Mechanics for Scientists and Engineers

Chapter 9 introduction and Section 9.1 (first part)

Angular momentum

Angular momentum operators

Quantum mechanics for scientists and engineers David Miller

Angular momentum operators - preview

We will have operators corresponding to angular momentum about different orthogonal axes

, , and though they will not commute with

one another in contrast to the linear momentum operators

for the different coordinate directions, , andwhich do commute

ˆxL ˆ

yL ˆzL

ˆ xp ˆ yp ˆ zp

Angular momentum operators - preview

We will, however, find another useful angular momentum operator,

which does commute separately with each of , , and

The eigenfunctions for , , and are simple Those for

the spherical harmonics, are more complicated but can be understood relatively simply

and form the angular shapes of the hydrogen atom orbitals

2L̂

ˆxL ˆ

yL ˆzL

ˆxL ˆ

yL ˆzL

2L̂

x

y

r

p q

origin

position of object

momentum

Classical angular momentum

The classical angular momentum of a small object

of (vector) linear momentum pcentered at a point given

by the vector displacement r relative to some originis L r p

Vector cross product

As usual

where i, j, and k are unit vectors in x, y, and z directionsand Ax is the component of A in the x direction

and 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 y x

AB A A AB B B

A B A B A B A B A B A B

i j kC A B c

i j k

Vector cross product

In

C is perpendicular to the plane of A and B just as the z axis is perpendicular to the plane containing the x and y axes in right-handed axes

is the angle between the vectors A and Bc 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 y x

AB A A AB B B

A B A B A B A B A B A B

i j kC A B c

i j k

Vector cross product

Note that, in

the ordering of the multiplications in the second line is chosen to work also for operators instead of numbers for one or other vector

the sequence of multiplications in each term is always 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 y x

AB A A AB B B

A B A B A B A B A B A B

i j kC A B c

i j k

Angular momentum operators

With classical angular momentum

we can explicitly write out the various components

Now we can propose a quantum mechanical angular momentum operator

based on substituting the position and momentum operators

and similarly write out component operators

x z yL yp zp y x zL zp xp z y xL xp yp

L r p

ˆ ˆ ˆ i L r p r

L̂

Angular momentum operators

Analogously, we obtain three operators

which are each Hermitian and so, correspondingly, is the operator itself

ˆ ˆ ˆ ˆˆx z yL yp zp i y zz y

ˆ ˆ ˆˆˆy x zL zp xp i z xx z

ˆ ˆˆ ˆ ˆz y xL xp yp i x yy x

L̂

Commutation relations

The operators corresponding to individual coordinate directions obey commutation relations

These individual commutation relations can be written in a more compact form

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,x y y x x y zL L L L L L i L

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,y z z y y z xL L L L L L i L

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,z x x z z x yL L L L L L i L

ˆ ˆ ˆi L L L

Commutation relations

Unlike operators for position and for linear momentum the different components of this angular momentum operator do not commute with one another

Though a particle can have simultaneously a well-defined position in both the x and y directions

or have simultaneously a well-defined momentum in both the x and y directions

a particle cannot in general simultaneously have a well-defined angular momentum component in more than one direction

7.1 Angular momentum

Slides: Video 7.1.3 Angular momentum eigenfunctions

Text reference: Quantum Mechanics for Scientists and Engineers

Section 9.1 (remainder)

Angular momentum

Angular momentum eigenfunctions

Quantum mechanics for scientists and engineers David Miller

Spherical polar coordinates

The relation between spherical polar and

Cartesian coordinates is

is the polar angle, and is the azimuthal angle

x

y

z

r

(x, y, z)

f

qsin cosx r

sin siny r

cosz r

Spherical polar coordinates

In inverse form

x

y

z

r

(x, y, z)

f

q

2 2 2r x y z

2 21

2 2 2sin

x y

x y z

1tan yx

Angular momentum in spherical polar coordinates

With these definitions of spherical polar coordinates and with standard partial derivative relations of the form

for each of the Cartesian coordinate directionswe can rewrite the angular momentum operator

components in spherical polar coordinates

rx x r x x

Angular momentum in spherical polar coordinates

From

and

we obtain

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,x y y x x y zL L L L L L i L ˆ ˆ ˆ ˆ ˆ ˆ ˆ,y z z y y z xL L L L L L i L

ˆ ˆ ˆ ˆ ˆ ˆ ˆ,z x x z z x yL L L L L L i L

ˆ sin cot cosxL i

ˆ cos cot sinyL i

ˆzL i

Lz eigenfunctions and eigenvalues

Using

we solve for the eigenfunctions and eigenvalues of

The eigen equation is

where is the eigenvalue to be determined The solution of this equation is

ˆzL i

ˆzL

ˆzL m

m

exp im

Lz eigenfunctions and eigenvalues

The requirements that the wavefunction and its derivative are

continuous when we return to where we started i.e., for

mean that m must be an integer positive or negative or zero

Hence we find that the angular momentum around the z axis is quantized

with units of angular momentum of

2