Clebsch-Gordan Coefficients

This Demonstration illustrates the Clebsch–Gordan coefficients,
〈
j
1
j
2
m
1
m
2
|
j
1
j
2
JM〉
, which give the coupling amplitudes between uncoupled and coupled representations of two angular momenta
j
1
and
j
2
. In the uncoupled representation, the
z
components of each angular momentum,
m
1
and
m
2
, are known; in the coupled representation, the total (resultant) angular momentum
J
and its
z
component
M
are known. The Clebsch–Gordan coefficients are only nonzero when
M
m
1
+
m
2
and
max(M,
j
1
-
j
2
)≤J≤
j
1
+
j
2
; in the Demonstration we show these for
j
1
,
j
2
≤3
. The graphs give a vectorial representation of each
(j,m)
pair, showing the actual value together with all possible
m
values.

Details

In quantum mechanics, angular momentum is quantized in units of
ℏ
. The allowed values are specified by the quantum number
j=1/2,1,3/2,2,…
; for a given
j
, the corresponding total angular momentum has value
j=
j(j+1)
ℏ
. In addition, one Cartesian component—conventionally the
z
component—can also be specified, and can take on values

j
z
=mℏ
where
m=-j,-j+1,…,j
. The other two components
j
x
,
j
y
cannot be specified individually, which is a manifestation of the uncertainty principle.
The Clebsch–Gordan coefficients arise in systems comprising two angular momenta,
j
1
and
j
2
. It is possible to define either states with well-defined individual
z
components
m
1
and
m
2
(the uncoupled representation), or well-defined total angular momentum
J
and its
z
component
M
(the coupled representation). Allowed values in the coupled representation are
J=
j
1
-
j
2
,
j
1
-
j
2
+1,…,
j
1
+
j
2
and
M=-J,-J+1,…,J
. The amplitudes relating the two representations are the Clebsch–Gordan coefficients
〈
j
1
j
2
m
1
m
2
|
j
1
j
2
JM〉
.

External Links

Clebsch–Gordan Coefficient (Wolfram MathWorld)
ClebschGordan (The Wolfram Functions Site)

Permanent Citation

Peter Falloon
​
​"Clebsch-Gordan Coefficients"​
​http://demonstrations.wolfram.com/ClebschGordanCoefficients/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011