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Orbital Angular Momentum: Formulas, Eigenvalues, and Examples - Prof. Lucien M. Cremaldi, Study notes of Quantum Mechanics

The University of Mississippi (Ole Miss)Quantum Mechanics
Prof. Lucien M. Cremaldi

The concept of orbital angular momentum, providing formulas for l2 and lz, raising and lowering operators, and matrix representation. It also includes examples of calculating probabilities of measuring lx for given initial states.

Typology: Study notes

Pre 2010

Uploaded on 09/24/2009

koofers-user-3di-1
koofers-user-3di-1 🇺🇸

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Chapter 8- Orbital Angular Momentum
Angular Momentum Formula
L2|l,m> = l(l+1)!2|l,m> l=1,2,3,....
LZ|l,m> = ml!|l,m> !l"ml"+l
L2=LX
2+LY
2+LZ
2 LX,LY
#
$%
&=i! LZ (cyclic permutations)
Raising and lowering operators
L+= LX+iLY L!= LX!iLY
L+|l,m> = l(l+1!m(m+1) ! | l,m+1>
L!|l,m> = l(l+1!m(m!1) ! | l,m!1>
Defining LX and LY
LX= 1
2L++ L!
( )
LX= 1
2iL+! L!
( )
L = 1 Case Matrix Representation
L2 =2!2
1 0 0
0 1 0
0 0 1
!
"
#
#
#
$
%
&
&
&
|1,+1>=
1
0
0
!
"
#
#
#
$
%
&
&
&
|1,0 >=
0
1
0
!
"
#
#
#
$
%
&
&
&
|1,'1>=
0
0
1
!
"
#
#
#
$
%
&
&
&
LZ= !
+1 0 0
000
0 0 '1
!
"
#
#
#
$
%
&
&
&
L+|l,m> = 2'm(m+1) ! | l,m+1>
L'|l,m> = 2'm(m'1) ! | l,m'1>
m= +1 0 '1
L+
m'm= <1,m' |
row
"#$ %$ L+|l,m>
column
&
= 2 'm(m+1) !
(
m,m+1 = !
0 2 0
0 0 2
0 0 0
!
"
#
#
#
#
$
%
&
&
&
&
L'
m'm= <1,m' |
row
"#$ %$ L+|l,m>
column
&
= 2 'm(m'1) !
(
m,m'1= !
0 0 0
2 0 0
0 2 0
!
"
#
#
#
$
%
&
&
&
LX= 1
2
L+
m'm+L'
m'm
( )
= !
2
0 1 0
1 0 1
0 1 0
!
"
#
#
#
$
%
&
&
&
LY=1
2i
L+
m'm'L'
m'm
( )
= !
2
0'i0
+i0'i
0+i0
!
"
#
#
#
$
%
&
&
&
LZ
LZ= +1!
L=1
LX
ω
t
2!
LX
LZ= +0!
LZ=!1!
LX and LY values uncert ain in L2 LZ basis .
!LX
( )
2
=LX
2 " !LX
2
# 0
!LY
( )
2
=LY
2 " !LY
2
# 0
pf3
pf4

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Chapter 8- Orbital Angular Momentum

Angular Momentum Formula

L^2 | l , m > = l ( l + 1 )!^2 | l , m > l = 1 , 2 , 3 ,....

LZ | l , m > = ml! | l , m >! l " ml " + l

L^2 = LX^2 + LY^2 + LZ^2 #$ L X , LY %& = i! LZ ( cyclic permutations )

Raising and lowering operators

L +^ = LX + iLY L!^ = LX! iLY

L +^ | l , m > = l ( l + 1! m ( m + 1 )! | l , m + 1 >

L!^ | l , m > = l ( l + 1! m ( m! 1 )! | l , m! 1 >

Defining LX and LY

LX =

( L +^ +^ L !) LX =^

2 i

( L +^!^ L !)

L = 1 Case Matrix Representation L^2 = 2!^2

LZ =!

L +^ | l , m > = 2 ' m ( m + 1 )! | l , m + 1 > L '^ | l , m > = 2 ' m ( m ' 1 )! | l , m ' 1 > m = + 1 0 ' 1 L + m ' m = < 1 , m ' | row

"$# $% L

  • (^) | l , m > column & =^2 '^ m ( m^ +^1 )^ !( m , m + 1 =^!

L ' m ' m = < 1 , m ' | row

"$# $% L

  • (^) | l , m > column & =^2 '^ m ( m^ '^1 )^ !( m , m ' 1 =^!

LX =

( L + m ' m +^ L ' m ' m ) =^

LY =

2 i

( L + m ' m '^ L ' m ' m ) =^

0 ' i 0

  • i 0 ' i 0 + i 0

LZ

LZ = + 1!

L=

LX

LY

ω t

2!

LX LZ = + 0!

LZ =! 1!

LX and LY values uncertain in L^2 LZ basis.

(! LX )

2

= LX^2 "! LX

2

(! LY )

2

= LY^2 "! LY

2

Eigenvalues and Eigenfunctions of Lx LX =! 2

a b c

a b c

a b c

( a + b / 2 = 0 a / 2 ( b + c / 2 = 0 b / 2 ( c = 0 | 1 ,+ 1 > x =

(| 1 ,^1 >^ +^2 |^1 ,^0 >^ +^ |^1 ,(^1 >)

a b c

a + b / 2 = 0 a / 2 + b + c / 2 = 0 b / 2 + c = 0 | 1 ,( 1 > x = 1 2

(| 1 ,^1 >^ (^2 |^1 ,^0 >^ +^ |^1 ,(^1 >)

a b c

b = 0 a + c = 0 b = 0 | 1 , 0 > x =

(| 1 ,^1 >^ +^ |^1 ,(^1 >)

Example #1- If a system is initially in the L=1 m=1 state |1,1> , What is the probability of measuring

LX = +!?

First write | 1 , 1 > in terms of LX eigenfnctions | 1 , 1 > = a | 1 , 1 > X + b | 1 , 0 > X + c | 1 ,! 1 > X a = X < 1 , 1 | 1 , 1 >=

( <^1 ,^1 |^ +^2 <^1 ,^0 |^ +^ <^1 ,!^1 |)i|^1 ,^1 >^ =^

b = X < 1 , 0 | 1 , 1 >= 1 2

(< 1 ,^1 |^ +^ <^1 ,!^1 |) |^1 ,^1 >=^1

a = X < 1 ,! 1 | 1 , 1 >=

(< 1 ,^1 |^ +^2 <^1 ,^0 |^!^ <^1 ,!^1 |) |^1 ,^1 >^ =^

A! 1 , 1

| 1 , 1 > X +

A! 1 , 0

| 1 , 0 > X +

A 1 !,! 1

| 1 ,! 1 > X

A 1 , 1 amplitude to measure m = +" A 1 , 0 amplitude to measure m = 0 " A 1 ,! 1 amplitude to measure m = !" P 1 , 1 = A 1 , 1 2 =

OR

LX = < 1 , 1 | LX | 1 , 1 > =

2

#^ A %^1 ,^1 %^2 $

( +") +^

2

&(^ A '^1 ,^02 ()

( 0 ") +^

2

&^ A '^1 ,!^1 )^2

(^! ")

P 1 , 1 = A 1 , 1

2