7.2 The L squared operator

Slides: Video 7.2.1 Separating the L squared operator

Text reference: Quantum Mechanics for Scientists and Engineers

Section 9.2

The L squared operator

Separating the L squared operator

Quantum mechanics for scientists and engineers David Miller

The L2 operator

In quantum mechanics we also consider another operator associated with angular momentum

the operator This should be thought of as

the “dot” product of with itselfand is defined as

2L̂

L̂

2 2 2 2ˆ ˆ ˆ ˆx y zL L L L

The L2 operator

It is straightforward to show then that

where the operator is given by

which is actually the and part of the Laplacian ( ) operator in spherical polar

coordinates hence the notation

2 2 2,L̂

2,

22

, 2 2

1 1sinsin sin

2

Commutation of L2

commutes with each of , , and It is easy to see from

and the form of

that at least and commute

The operation has no effect on functions or operators depending on alone

2L̂ ˆxL ˆ

yL ˆzL

22

, 2 2

1 1sinsin sin

ˆzL i

2L̂ ˆzL

/

Commutation of L2

Of course, the choice of the z direction is arbitraryWe could equally well have chosen the polar axis

along the x or y directions Then it would similarly be obvious that

commutes with orHow can commute with each of , , and

but , , and do not commute with each other?Answer

we can choose the eigenfunctions of to be the same as those of any one of , , and

2L̂ ˆxL ˆ

yL

ˆxL ˆ

yL ˆzL

ˆxL ˆ

yL ˆzL

2L̂

2L̂ ˆxL ˆ

yL ˆzL

Eigenfunctions of L2

We want eigenfunctions of or, equivalently, and so the equation we hope to solve is of the form

We anticipate the answer by writing the eigenvalue in the form

but it is just an arbitrary number to be determined

The notationalso anticipates the final answer

but it is just an arbitrary function to be determined

2L̂ 2,

2, , 1 ,lm lmY l l Y

1l l

,lmY

Separation of variables

We presume that the final eigenfunctions can be separated in the form

whereonly depends on andonly depends on

Substituting this form in gives

,lmY

2, , 1 ,lm lmY l l Y

2

2 2sin 1sin sin

l l

Separation of variables

Multiplying by and rearranging, gives

The left hand side depends only on whereas the right hand side depends only on

so these must both equal a (“separation”) constantAnticipating the answer

we choose a separation constant of where m is still to be determined

2sin /

22

2

1 sin1 sin sinl l

2m

equation

Taking the left hand side of

we now have an equation

The solutions to an equation like this are of the form , or

22 2

2

1 sin1 sin sinl l m

2

22

dm

d

sin m cos m exp im

equation

For the solutions of

we choose the exponential form so is also a solution of the eigen equation

We expect that and its derivative are continuous so this wavefunction must repeat every 2 of angle

Hence, m must be an integer

2

22

dm

d

exp imˆ

zL

ˆzL m

equation

Taking the right hand side of the separation equation

Multiplying by and rearranging gives

This is the associated Legendre equation whose solutions are the associated Legendre functions

2 2sin1 sin sinl l m

2

2

1 sin 1 0sin sin

d d m l ld d

cosmlP

2/ sin

0,1,2,3,...l m ll

The solutions to this equation

require that

(m integer)

The associated Legendre functions can conveniently be defined using Rodrigues’ formula

equation

2

2

1 sin 1 0sin sin

d d m l ld d

cosmlP

/22 21 1 12 !

l mm lml l l m

dP x x xl dx

Associated Legendre functions

For example

00 1P x

1/21 22 3 1P x x x

01P x x

1/21 21 1P x x

1/21 21

1 12

P x x

0 22

1( ) 3 12

P x x

1/21 22

1 12

P x x x

2 22 3 1P x x

2 22

1 18

P x x

l = 0

l = 1

l = 2

Associated Legendre functions

We see that these functions have the following properties The highest power of the argument x is always xl

The functions for a given l for +m and –m are identical other than for numerical prefactors

Less obviously between –1 and +1

and not including the values at those end points

the functions have zeros

mlP x

l m

Eigenfunctions of L2

Putting this all together, the eigen equation is

with spherical harmonics as the eigenfunctionswhich, after normalization, can be written

where , where m is an integer,and the eigenvalues are

2 2ˆ , 1 ,lm lmL Y l l Y

,lmY

!2 1, 1 cos exp

4 !m m

lm l

l mlY P iml m

0,1,2,3,...l m ll

2 1l l

Eigenfunctions of L2 and Lz

As is easily verifiedthese spherical harmonics

are also eigenfunctions of the operator Explicitly, we have the eigen equation

with eigenvalues of being

It makes no difference to the eigenfunctions if we multiply them by a function of

ˆzL

ˆ , ,z lm lmL Y m Y ˆ

zL m

ˆzL