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Singlet And Triplet Operators

rename
Updated 2009-04-27 09:55

Singlet and Triplet Operators

 

Please understand the contents of Adding Angular Momenta before reading this page.



Ql is a projection operator that satisfies the following:
Ql = ΣmL | l mL > < l mL | where mL = -l to +l
< l' m'L | Ql | l' m'L > equals 1 if l=l' and 0 otherwise
L2 = Σl l(l+1) Ql where l = | j-k | to j+k.

 

Operator
Value
Q0singlet operator
Q1/2doublet operator
Q1triplet operator
Q3/2quadruplet operator
Q2quintet operator
Q5/2sextuplet operator
Q3septuplet operator
Ql(2l+1)-multiplet operator
memorize

 

Operator
Value
Q01 - Q1 for j=1/2, k=1/2
Q11 - Q0 for j=1/2, k=1/2
Q1/21 - Q3/2 for j=1, k=1/2
Q3/21 - Q1/2 for j=1, k=1/2
Q11 - Q2 for j=3/2, k=1/2
Q21 - Q1 for j=3/2, k=1/2
Q3/21 - Q5/2 for j=2, k=1/2
Q5/21 - Q3/2 for j=2, k=1/2
Q21 - Q3 for j=5/2, k=1/2
Q31 - Q2 for j=5/2, k=1/2
memorize

 

Operator
Value
Q01 - Q1 - Q2 for j=1, k=1
Q11 - Q0 - Q2 for j=1, k=1
Q21 - Q0 - Q1 for j=1, k=1
Q1/21 - Q3/2 - Q5/2 for j=3/2, k=1
Q3/21 - Q1/2 - Q5/2 for j=3/2, k=1
Q5/21 - Q1/2 - Q3/2 for j=3/2, k=1
Q11 - Q2 - Q3 for j=2, k=1
Q21 - Q1 - Q3 for j=2, k=1
Q31 - Q1 - Q2 for j=2, k=1
Q01 - Q1 - Q2 - Q3 for j=3/2, k=3/2
Q11 - Q0 - Q2 - Q3 for j=3/2, k=3/2
Q21 - Q0 - Q1 - Q3 for j=3/2, k=3/2
Q31 - Q0 - Q1 - Q2 for j=3/2, k=3/2
memorize

 

In the following, M = L2 = l(l+1) = 2 J • K + j(j+1) + k(k+1) holds.

 

Operator
Value
Q0-(4M-8)/8 for j=1/2, k=1/2
Q1 (4M)/8 for j=1/2, k=1/2
Q1/2-(4M-15)/12 for j=1, k=1/2
Q3/2 (4M-3)/12 for j=1, k=1/2
Q1-(4M-24)/16 for j=3/2, k=1/2
Q2 (4M-8)/16 for j=3/2, k=1/2
Q3/2-(4M-35)/20 for j=2, k=1/2
Q5/2 (4M-15)/20 for j=2, k=1/2
Q2 -(4M-48)/24 for j=5/2, k=1/2
Q3  (4M-24)/24 for j=5/2, k=1/2
memorize

 

Operator
Value
Q0 (M-2)(M-6)/12 for j=1, k=1
Q1-(M)(M-6)/8 for j=1, k=1
Q2 (M)(M-2)/24 for j=1, k=1
Q1/2 (4M-15)(4M-35)/384 for j=3/2, k=1
Q3/2-(4M-3)(4M-35)/240 for j=3/2, k=1
Q5/2 (4M-3)(4M-15)/640 for j=3/2, k=1
Q1 (M-6)(M-12)/40 for j=2, k=1
Q2-(M-2)(M-12)/24 for j=2, k=1
Q3 (M-2)(M-6)/60 for j=2, k=1
Q0-(M-2)(M-6)(M-12)/144 for j=3/2, k=3/2
Q1  (M)(M-6)(M-12)/80 for j=3/2, k=3/2
Q2-(M)(M-2)(M-12)/144 for j=3/2, k=3/2
Q3 (M)(M-2)(M-6)/720 for j=3/2, k=3/2
memorize

 

Value of
M = L2 = l(l+1)
j, k combination
giving it
2 J • K + 6/4j=1/2, k=1/2
2 J • K + 11/4j=1,    k=1/2
2 J • K + 18/4j=3/2, k=1/2
2 J • K + 27/4j=2,    k=1/2
2 J • K + 38/4j=5/2, k=1/2
2 J • K + 16/4j=1,    k=1
2 J • K + 23/4j=3/2, k=1
2 J • K + 32/4j=2,    k=1
2 J • K + 30/4j=3/2, k=3/2
memorize

 

Mnemonic:

In the preceding table, starting with M for j=1/2, k=1/2,
for each 1/2 you add to j or k, you can obtain the next value of M by adding
successive odd integers divided by 4 (that is, 5/4, 7/4, 9/4, 11/4, 13/4, etc.):
Raising j or k from 1/2 to 1 adds 5/4 to M,
Raising j or k from 1 to 3/2 adds 7/4 to M,
Raising j or k from 3/2 to 2 adds 9/4 to M,
Raising j or k from 2 to 5/2 adds 11/4 to M, etc.


In the following, N = J • K = (1/2)l(l+1) - j(j+1) - k(k+1) = (1/2)M - j(j+1) - k(k+1) holds.

 

Value of
N = J • K
j, k combination
giving it
(4M-6)/8j=1/2, k=1/2
(4M-11)/8j=1,    k=1/2
(4M-18)/8j=3/2, k=1/2
(4M-27)/8j=2,    k=1/2
(4M-38)/8j=5/2, k=1/2
(4M-16)/8j=1,    k=1
(4M-23)/8j=3/2, k=1
(4M-32)/8j=2,    k=1
(4M-30)/8j=3/2, k=3/2
memorize

 

Mnemonic:

In the preceding table, starting with N for j=1/2, k=1/2,
for each 1/2 you add to j or k, you can obtain the next value of N by subtracting
successive odd integers divided by 8 (that is, 5/8, 7/8, 9/8, 11/8, 13/8, etc.):
Raising j or k from 1/2 to 1 drops N by 5/8,
Raising j or k from 1 to 3/2 drops N by 7/8,
Raising j or k from 3/2 to 2 drops N by 9/8,
Raising j or k from 2 to 5/2 drops N by 11/8, etc.

 

j, k, l
M = l(l+1)
N = J • K
j=1/2, k=1/2, l=0  0  -6/8 =   -3/4
j=1/2, k=1/2, l=1  2   2/8 =    1/4
j=1,    k=1/2, l=1/2  3/4  -8/8 =   -4/4 = -1
j=1,    k=1/2, l=3/215/4   4/8 =    2/4 =   1/2
j=3/2, k=1/2, l=1  2-10/8 =   -5/4
j=3/2, k=1/2, l=2  6   6/8 =    3/4
j=2,    k=1/2, l=3/215/4-12/8 =   -6/4 = -3/2
j=2,    k=1/2, l=5/235/4   8/8 =    4/4 =   1
j=5/2, k=1/2, l=2  6-14/8 =   -7/4
j=5/2, k=1/2, l=312  10/8 =    5/4
memorize

 

j, k, l
M = l(l+1)
N = J • K
j=1,    k=1,    l=0  0-16/8 =   -8/4 = -2
j=1,    k=1,    l=1  2  -8/8 =   -4/4 = -1
j=1,    k=1,    l=2  6   8/8 =    4/4 =   1
j=3/2, k=1,    l=1/2  3/4-20/8 = -10/4 = -5/2
j=3/2, k=1,    l=3/215/4  -8/8 =   -4/4 = -1
j=3/2, k=1,    l=5/235/4  12/8 =    6/4 =   3/2
j=2,    k=1,    l=1  2-24/8 = -12/4 = -3
j=2,    k=1,    l=2  6  -8/8 =   -4/4 = -1
j=2,    k=1,    l=312  16/8 =    8/4 =   2
j=3/2, k=3/2, l=0  0-30/8 = -15/4
j=3/2, k=3/2, l=1  2-22/8 = -11/4
j=3/2, k=3/2, l=2  6  -6/8 =   -3/4
j=3/2, k=3/2, l=312  18/8 =    9/4
memorize

 

Operator
Value
Q0-(4N-1)/4 for j=1/2, k=1/2
Q1 (4N+3)/4 for j=1/2, k=1/2
Q1/2-(4N-2)/6 for j=1, k=1/2
Q3/2 (4N+4)/6 for j=1, k=1/2
Q1 -(4N-3)/8 for j=3/2, k=1/2
Q2 (4N+5)/8 for j=3/2, k=1/2
Q3/2 -(4N-4)/10 for j=2, k=1/2
Q5/2 (4N+6)/10 for j=2, k=1/2
Q2 -(4N-5)/12 for j=5/2, k=1/2
Q3  (4N+7)/12 for j=5/2, k=1/2
memorize

 

Operator
Value
Q0 (N+1)(N-1)/3 for j=1, k=1
Q1-(N+2)(N-1)/2 for j=1, k=1
Q2 (N+2)(N+1)/6 for j=1, k=1
Q1/2 (2N+2)(2N-3)/24 for j=3/2, k=1
Q3/2-(2N+5)(2N-3)/15 for j=3/2, k=1
Q5/2 (2N+5)(2N+2)/40 for j=3/2, k=1
Q1 (N+1)(N-2)/10 for j=2, k=1
Q2-(N+3)(N-2)/6 for j=2, k=1
Q3 (N+3)(N+1)/15 for j=2, k=1
Q0-(4N+11)(4N+3)(4N-9)/1152 for j=3/2, k=3/2
Q1 (4N+15)(4N+3)(4N-9)/640 for j=3/2, k=3/2
Q2-(4N+15)(4N+11)(4N-9)/1152 for j=3/2, k=3/2
Q3  (4N+15)(4N+11)(4N+3)/5760 for j=3/2, k=3/2
memorize

See Also

Explicit Matrices for some of the above operators are given in
Adding Angular Momenta 2 and Adding Angular Momenta 3.