J. ANGULAR MOMENTUM

34 Pages·2007·0.44 MB·English

Angular momentum algebra J-1
J. ANGULAR MOMENTUM
Sources:
- D.C. Harris och M.D. Bertolucci, Symmetry and Spectroscopy, Oxford University
Press, 1978.
- J. M. Hollas, Modern Spectroscopy, Wiley, Chichester, 1987.
- G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, 1950.
- G. Herzberg, Infrared and Raman Spectra, Van Nostrand, 1945.
- P.W. Atkins, Molecular Quantum Mechanics, Oxford University Press, 1983.
- H.W. Kroto, Molecular Rotation Spectra, Wiley, 1975.
- E.B. Wilson, J.C. Decius och P.C. Cross, Molecular Vibrations, McGraw-Hill, 1955.
(En ny upplaga: Dover, 1980).
- E. U. Condon och G. H. Shortley, Theory of atomic spectra, Cambridge, 1953.
- L.A. Woodward, Introduction to the Theory of Molecular Vibrations and Vibrational
Spectroscopy, Clarendon Press, 1972.
- S. Califano, Vibrational States, Wiley, 1976.
- E.F.H. Brittain, W.O. George och C.H.J. Wells, Academic Press, 1970.
- M.D. Harmony, Introduction to Molecular Energies and Spectra, Holt & Winston,
1972.
J-2 Molecular spectroscopy
J.1. Basic properties
In the classical mechanics the angular momentumis de(cid:28)ned as
L~ = m~r(cid:2)~v =~r(cid:2)p~: (J:1)
It is thus a vector quantity with the lengt L. It can be split to three Cartesian components,
Lx, Ly and Lz, that are all well de(cid:28)ned.
In quantum mechanics a similar de(cid:28)nition can be used for the orbital angular momentum
but, e.g., for the spin such a de(cid:28)nition is not possible. The generalized de(cid:28)nition of
angular momentum in quantum mechanics is based on the commutation relations
of the components in orthogonal orientations, ^j1, ^j2 and ^j3, of the angular momentum
^
operator~j, i.e.,
[^j1;^j2] = i(cid:22)h^j3
[^j2;^j3] = i(cid:22)h^j1 (J:2)
[^j3;^j1] = i(cid:22)h^j2:
The alternativede(cid:28)nition for the componentsof the orbital angular momentumfollows
from the classical de(cid:28)nition in equation (J.1),
(cid:18) (cid:19)
@ @
L^x = (cid:0)i(cid:22)h y (cid:0)z
@z @y
(cid:18) (cid:19)
@ @
^
Ly = (cid:0)i(cid:22)h z (cid:0)x (J:3)
@x @z
(cid:18) (cid:19)
@ @
^
Lz = (cid:0)i(cid:22)h x (cid:0)y :
@y @x
In atomic units, (cid:22)h = 1.
The equations (J.2) are also equivalent with the de(cid:28)nition
^ ^ ^
~j (cid:2)~j = i~j: (J:4)
In other words, any operator that ful(cid:28)ls this relation it is an angular momentumoperator.
The commutationrelations guarantee that all properties of the angular momentumopera-
torscan be obtained mathematically. This branch of quantummechanicsiscalled angular
momentum algebra or Racah algebra.
The commutatorrelations show that the components do not have commoneigenfunctions.
All components cannot be good quantum numbers at the same time.
Angular momentum algebra J-3
The square of the angular momentum is
2
~^j (cid:17)~^j (cid:1)~^j = ^j12 +^j22 +^j32: (J:5)
It is a scalar quantityand is therefore denoted by ^j2. The square commuteswith one of the
components ^j1, ^j2 and ^j3 but not with all three because the components do not commute.
One usually chooses ^j3 to be the commuting component,
[^j2;^j3] = 0: (J:6)
This relation follows directly from the commutation relations above,
[^j2;^j3] = ^j12^j3 +^j22^j3 +^j33 (cid:0)^j3^j12 (cid:0)^j3^j22 (cid:0)^j33
= ^j12^j3 (cid:0)^j1^j3^j1 +^j1^j3^j1 +^j22^j3 (cid:0)^j2^j3^j2 +^j2^j3^j2 (cid:0)^j3^j12 (cid:0)^j3^j22
= ^j1(^j1^j3 (cid:0)^j3^j1)+^j2(^j2^j3 (cid:0)^j3^j2)(cid:0)(^j3^j1 (cid:0)^j1^j3)^j1 (cid:0)(^j3^j2 (cid:0)^j2^j3)^j2 (J:7)
= (cid:0)i^j1^j2 +i^j2^j1 (cid:0)i^j2^j1 +i^j1^j2
= 0:
For the quantum mechanical angular momentum, both the length and the orientation are
quantitized and are described by the quantum numbers j and m, respectively. The
2
length (or norm of the vector) is related to the scalar product j while the orientation is
given by the component along the z axis, jz. A vector with length jjj cannot have a z
component that is larger than the length. Therefore the quantum number m is restricted
to the values +j (cid:20) m (cid:20) (cid:0)j.
J-4 Molecular spectroscopy
J.2. Eigenfunctions
Thetermangularmomentumreferstocircularmotion. Theorbital angularmomentum
is best described in spherical coordinates (r;(cid:18);(cid:30)) rather than in Cartesian coordinates
x;y;z. The transformation from the Cartesian to the spherical coordinates is given by the
equations
1
r = (x2 +y2 +z2)2
(cid:20) (cid:21)
1 1
(cid:18) = arctan (x2 +y2)2 (J:8)
2
(cid:16) (cid:17)
y
(cid:30) = arctan
x
and the inverse transformation by
x = rcos((cid:30))sin((cid:18))
y = rsin((cid:30))sin((cid:18)) (J:9)
z = rcos((cid:18)):
Using these formulae the operators (J.3) and (J.5) can be easily transformed.
The operators are in spherical coordinates
@
^jz = (cid:0)i
@(cid:30)
(cid:20) (cid:18) (cid:19)(cid:21) (J:10)
2
1 @ 1 @ @
^j2 = (cid:0) + sin(cid:18) :
2 2
sin (cid:18) @(cid:30) sin(cid:18) @(cid:18) @(cid:18)
m
The common eigenfunction (cid:9) ((cid:18);(cid:30)) can be solved from these di(cid:27)erential equations. It is
j
a product of an exponential and an associated Legendre polynomial
m m im(cid:30)
(cid:9)j ((cid:18);(cid:30)) = NjmPj ((cid:18))e : (J:11)
m
Thefunction(cid:9) ((cid:18);(cid:30))iscalledspherical harmonicfunction. Afewofthe(cid:28)rstspherical
j
harmonic functions are given in the table below.
Angular momentum algebra J-5
j m Imaginary wavefunction Alternative real wavefunction
q
1
0 0 -
4(cid:25)
q
3
1 0 cos(cid:18) -
4(cid:25)
q q
3 (cid:6)i(cid:30) 3
1 (cid:6)1 sin(cid:18)e sin(cid:18)cos(cid:30)
8(cid:25) 4(cid:25)
q
3
sin(cid:18)sin(cid:30)
4(cid:25)
q
5 2
2 0 (3cos (cid:18)(cid:0)1) -
16(cid:25)
q q
15 (cid:6)i(cid:30) 15
(cid:6)1 sin(cid:18)cos(cid:18)e sin(cid:18)cos(cid:18)cos(cid:30)
8(cid:25) 4(cid:25)
q
15
sin(cid:18)cos(cid:18)sin(cid:30)
4(cid:25)
q q
15 2 (cid:6)2i(cid:30) 15 2
(cid:6)2 sin (cid:18)e sin (cid:18)cos2(cid:30)
32(cid:25) 16(cid:25)
q
15 2
sin (cid:18)sin2(cid:30)
16(cid:25)
q
63 5 3
3 0 ( cos (cid:18)(cid:0)cos(cid:18)) -
16(cid:25) 3
q q
21 2 (cid:6)i(cid:30) 21 2
(cid:6)1 sin(cid:18)(5cos (cid:18)(cid:0)1)e sin(cid:18)(5cos (cid:18)(cid:0)1)cos(cid:30)
64(cid:25) 32(cid:25)
q
21 2
sin(cid:18)(5cos (cid:18)(cid:0)1)sin(cid:30)
32(cid:25)
q q
105 2 (cid:6)2i(cid:30) 105 2
(cid:6)2 sin (cid:18)cos(cid:18)e sin (cid:18)cos(cid:18)cos2(cid:30)
32(cid:25) 16(cid:25)
q
105 2
sin (cid:18)cos(cid:18)sin2(cid:30)
16(cid:25)
q q
35 3 (cid:6)3i(cid:30) 35 3
(cid:6)3 sin (cid:18)e sin (cid:18)cos3(cid:30)
64(cid:25) 32(cid:25)
q
35 3
sin (cid:18)sin3(cid:30)
32(cid:25)
J-6 Molecular spectroscopy
J.3. Eigenvalues
Two new operators are needed in order to calculate the eigenvalues of the eigenvalues of ^j3
and ^j2. The step-up operator ^j+ and the step-down operator ^j(cid:0) are de(cid:28)ned through
the components ^j1 and ^j2,
^j+ = ^j1 +i^j2
(J:12)
^j(cid:0) = ^j1 (cid:0)i^j2:
A number of mathematical relations can be derived for the operators, such as
^j+^j(cid:0) = ^j2 +^j3 (cid:0)^j32
^j(cid:0)^j+ = ^j2 (cid:0)^j3 (cid:0)^j32
(J:13)
[^j(cid:0);^j3] = ^j(cid:0)
[^j+;^j3] = (cid:0)^j+:
The relations are easily proved, e.g.,
^j+^j(cid:0) = (^j1 +i^j2)(^j1 (cid:0)i^j2)
= ^j12 (cid:0)i^j1^j2 +i^j2^j1 +^j22
= ^j12 +^j22 +^j32 (cid:0)^j32 (cid:0)i(^j1^j2 (cid:0)^j2^j1) (J:14)
= ^j12 +^j22 +^j32 (cid:0)^j32 (cid:0)i[^j1;^j2]
= ^j2 +^j3 (cid:0)^j32
and
[^j(cid:0);^j3] = (^j1 (cid:0)i^j2)^j3 (cid:0)^j3(^j1 (cid:0)i^j2)
= ^j1^j3 (cid:0)^j3^j1 (cid:0)i^j2^j3 +i^j3^j2
= [^j1;^j3](cid:0)i[^j2;^j3]
(J:15)
= (cid:0)i^j2 (cid:0)i(cid:1)i^j1
= ^j1 (cid:0)i^j2
= ^j(cid:0):
Let the eigenvalue of the component ^j3 be m and the corresponding eigenfunction (cid:9)m,
i.e.,
^j3(cid:9)m = m(cid:9)m: (J:16)
Operate by the step-down operator ^j(cid:0) on both sides of this eigenvalue equation. This will
result in
^j(cid:0)^j3(cid:9)m = m^j(cid:0)(cid:9)m: (J:17)
Angular momentum algebra J-7
Because
^j(cid:0)^j3 = ^j3^j(cid:0) +^j(cid:0) (J:18)
according to equation (J.13)c one can write
^j3^j(cid:0)(cid:9)m = (m(cid:0)1)^j(cid:0)(cid:9)m: (J:19)
This means that the function ^j(cid:0)(cid:9)m is an eigenfunction of the component ^j3 with the
corresponding eigenvalue m (cid:0)1. The operator ^j(cid:0) transforms a state with the eigenvalue
m to a state with eigenvalue m(cid:0)1. Hence the name (cid:16)step-down operator(cid:17). Observe that
the state ^j(cid:0)(cid:9)m is not normalized,
^j(cid:0)(cid:9)m = C(cid:9)m(cid:0)1; (J:20)
where C is a normalization factor.
Similarly, one can operate on equation (J.16) with ^j+, which will lead to
^j+^j3(cid:9)m = m^j+(cid:9)m: (J:21)
As
^j+^j3 = ^j3^j+ (cid:0)^j+ (J:22)
according to equation (J.13), one can write
^j3^j+(cid:9)m = (m+1)^j+(cid:9)m: (J:23)
Apart from a normalization factor D, one will obtain the state (cid:9)m+1 by operating with
^j+,
^j+(cid:9)m = D(cid:9)m+1: (J:24)
The operator ^j+ will give a ascending series of states, (cid:9)m, (cid:9)m+1, (cid:9)m+2:::. The operator
^j(cid:0) will give descending series (cid:9)m, (cid:9)m(cid:0)1, (cid:9)m(cid:0)2:::. These two series must terminate
because (cid:9)m is a common eigenstate of ^j3 and ^j2 which means that
^j+^j(cid:0)(cid:9)m = (`2 +m(cid:0)m2)(cid:9)m
(J:25)
^j(cid:0)^j+(cid:9)m = (`2 (cid:0)m(cid:0)m2)(cid:9)m
2
where ` is a real number. Further
Z Z
(cid:9)(cid:3)m^j+^j(cid:0)(cid:9)m d(cid:28) = (^j(cid:0)(cid:9)m)(cid:3)(^j(cid:0)(cid:9)m) d(cid:28)
(cid:10) (cid:10)
Z
2 (cid:3) (J:26)
= jCj (cid:9)m(cid:0)1(cid:9)m(cid:0)1 d(cid:28)
(cid:10)
(cid:21) 0:
J-8 Molecular spectroscopy
Equality is valid only if
^j(cid:0)(cid:9)m = C(cid:9)m(cid:0)1 = 0: (J:27)
Correspondingly, the latter equation will give the result
Z Z
(cid:9)(cid:3)m^j(cid:0)^j+(cid:9)m d(cid:28) = (^j+(cid:9)m)(cid:3)(^j+(cid:9)m) d(cid:28)
(cid:10) (cid:10)
Z
2 (cid:3) (J:28)
= jDj (cid:9)m+1(cid:9)m+1 d(cid:28)
(cid:10)
(cid:21) 0:
Equality is valid only if
^j+(cid:9)m = D(cid:9)m+1 = 0: (J:29)
The wavefunction (cid:9)m is normalized which implies that
2 2
` +m(cid:0)m (cid:21) 0;
(J:30)
2 2
` (cid:0)m(cid:0)m (cid:21) 0:
2
For a (cid:28)xed value of ` this is possible only when both the descending and the ascending
series terminate. In that case one will have a minimumvalue mmin and a maximumvalue
mmax of the quantum number m,
(cid:26)
2 2
` +mmin (cid:0)mmin = 0
(J:31)
2 2
` (cid:0)mmax (cid:0)mmax = 0
or 8 q
<mmin = 21 (cid:0) 14 +`2 (cid:9)mmin(cid:0)1 = 0
q (J:32)
: 1 1 2
mmax = (cid:0)2 + 4 +` (cid:9)mmax+1 = 0:
The operators ^j(cid:6) alter the quantum number m by (cid:6)1. Then the di(cid:27)erence mmax - mmin
must be an integer. The number of levels with di(cid:27)erent m is
r
1
2
mmax (cid:0)mmin = 2 +` = heltal = 2j +1; (J:33)
4
where j is the largest allowed value of jmj. As the di(cid:27)erence must be an integer, j can
only adopt the values
1 3
j = 0; ;1; ;2;:::: (J:34)
2 3
By taking the square of each side of the equation one can see that
2 2 2
1+4` = 4j +4j +1 ` = j(j +1): (J:35)
The quantum number m varies between (cid:0)j and j.
The quantity `2 was introduced as the eigenvalue of the operator ^j2. Therefore one can
write the concurrent eigenvalue equations for the common eigenstate j jm >
1 3
^j2 j jm >= j(j +1) j jm >; j = 0; ;1; ;:::
2 2 (J:36)
^j3 j jm >= m j jm >; m = (cid:0)j;(cid:0)j +1;:::;j:
Angular momentum algebra J-9
J.4. The step operators
It is required that all states are normalized. From the equations (J.25), (J.26) and (J.28)
it follows that
jCj2 < m(cid:0)1 j m(cid:0)1 >= jCj2 =< ^j(cid:0)(jm) j ^j(cid:0)(jm) >
2 2
= ` +m(cid:0)m
= j(j +1)(cid:0)m(m(cid:0)1);
(J:37)
jDj2 < m+1 j m+1 >= jDj2 =< ^j+(jm) j ^j+(jm) >
2 2
= ` (cid:0)m(cid:0)m
= j(j +1)(cid:0)m(m+1):
Apart from an arbitrary phase factor, one (cid:28)nds that
(cid:26) p
^j(cid:0) j jm >= j(j +1)(cid:0)m(m(cid:0)1) j j;m(cid:0)1 >
p (J:38)
^j+ j jm >= j(j +1)(cid:0)m(m+1) j j;m+1 > :
This expression was derived from the commutation relations and is therefore valid for all
quantum mechanical angular momenta.
It is also possible to derive from these equations the expectation values of the operators ^j1
and ^j2,
1p
< jm(cid:6)1j^j1jjm >= (j (cid:7)m)(j (cid:6)m+1)
2
(J:39)
1 p
< jm(cid:6)1j^j2jjm >= (cid:7) i (j (cid:7)m)(j (cid:6)m+1):
2
For an orbital angular momentumthe eigenvalues of the step-up and step-down operators
can be derived from the de(cid:28)nition L~^ = ~^r(cid:2)^p~. Only integer quantum numbers are allowed
^
for L~.
One can switch to a matrixrepresentation where the state vectors corresponding to a given
quantum number j are 2j +1 dimensional vectors
0 1 0 1
1 0
B0C B1C
B C B C
j j;m = j >= B0C; j j;m = j (cid:0)1 >= B0C; etc: (J:40)
B . C B . C
@ . A @ . A
. .
0 0
and the angular momentum operators are (2j +1)(cid:2)(2j +1) dimensional matrices. The
only non-vanishing elements in the matrices corresponding to the operators ^j(cid:0) and ^j+ are
p p
(j(cid:0))m(cid:0)1;m = j(j +1)(cid:0)m(m(cid:0)1); (j+)m+1;m = j(j +1)(cid:0)m(m+1): (J:41)
J-10 Molecular spectroscopy
1
In the case j = (e.g., the spin of an electron or a proton) one will obtain
2
8 (cid:18) (cid:19) (cid:18) (cid:19)
>> 1 1 1 1 1 0
>j ; >= ; j ;(cid:0) >=
> 2 2 2 2
> 0 1
>
>
> 1 1
>>> (cid:18)+2 (cid:0)2 (cid:19) (cid:18) (cid:19)
>> 1
> + 0 0 0 1
>>>j(cid:0) = 12 ; j+ =
> (cid:0) 1 0 0 0
>>> 2 p q
>>>>m = 21 ) (m(cid:0)1;m) = ((cid:0)21; 21) ) j(j +1)(cid:0)m(m(cid:0)1) = 34 (cid:0) 12((cid:0)21) = 1
> q
> p
>> 1 1 1 3 1 1
>m = (cid:0) ) (m+1;m) = ( ;(cid:0) ) ) j(j +1)(cid:0)m(m+1) = + ( ) = 1
> 2 2 2 4 2 2
>> (cid:18) (cid:19)
>
>> 1 1 0 1
><j1 = 2(j+ +j(cid:0)) = 2
1 0
(cid:18) (cid:19)
> 1 1 0 (cid:0)i
>>>>j2 = (cid:0)2i(j+ (cid:0)j(cid:0)) = 2 i 0
> (cid:20)(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)(cid:18) (cid:19)(cid:21)
>
>> i 0 1 0 (cid:0)i 0 (cid:0)i 0 1
>>>> j3 = (cid:0)i[^j1;^j2] = (cid:0)4 1 0 i 0 (cid:0) i 0 1 0
>
>> (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
>
> i i 0 (cid:0)i 0
>
> = (cid:0) (cid:0)
>
>> 4 0 (cid:0)i 0 i
>> (cid:18) (cid:19)
>
>> 1 1 0
>> =
>> 2 0 (cid:0)1
> (cid:18) (cid:19)
>
>>>:j2 = j21 +j22 +j23 = 43 10 01 :
(J:42)
14
In the case j = 1 (e.g., the nuclear spin of a deuteron or N, or the total spin of a triplet
state) one will obtain similarly
8 0 1 0 1 0 1
> 1 0 0
>
>>>j 11 >= @0A; j 10 >= @1A; j 1;(cid:0)1 >= @0A
>
>
>> 0 0 1
> 0 1 0 p 1
>
>> 0 0 0 0 2 0
> p p
>
>>>j(cid:0) = @ 2 p0 0A; j+ = @0 0 2A
<
0 2 0 0 0 0
0 1 0 1 (J:43)
> 0 1 0 0 (cid:0)i 0
>
>>>>>j1 = p12 @1 0 1A; j2 = p12 @i 0 (cid:0)iA
>
> 0 1 0 0 i 0
>> 0 1 0 1
>
>> 1 0 0 1 0 0
>
>>>j3 = @0 0 0 A; j2 = 2@0 1 0A:
:
0 0 (cid:0)1 0 0 1