1000 Solved Problems in Modern Physics

136 3 Quantum Mechanics – II

with a trial functionψthat depends on a number of parameters, and varying these
parmeters until the expectation value of the energy is a minimum. This results in an
upper limit for the ground state energy of the system, which will be close if the form
of the trial function resembles that of the eigen function.

Table 3.3Clebsch–Gordan coefficients (C.G.C)

1

2

×^1

2

J= 1101

m 1 m 2 M=+10 0− 1

+ 1 / 2 + 1 / 21

+ 1 / 2 − 1 / 2

√

1 / 2

√

1 / 2

− 1 / 2 + 1 / 2

√

1 / 2 −

√

1 / 2

− 1 / 2 − 1 / 21

1 ×

1

2

J= 3 /2 3/2 1/2 3/2 1/2 3/2

m 1 m 2 M=+3/2 + 1 / 2 + 1 / 2 − 1 / 2 − 1 / 2 − 3 / 2

+ 1 + 1 / 21

+ 1 − 1 / 2

√

1 / 3

√

2 / 3

0 + 1 / 2

√

2 / 3 −

√

1 / 3

0 − 1 / 2

√

2 / 3

√

1 / 3

− 1 + 1 / 2

√

1 / 3 −

√

2 / 3

− 1 − 1 / 21

1 × 1

J= 221210212

m 1 m 2 M=+ 2 + 1 +100 0 − 1 − 1 − 2

+ 1 + 11

+ 10

√

1

2

√

1

2

0 + 1

√

1

2

−

√

1

2

+ 1 − 1

√

1

6

√

1

2

√

1

3

00

√

2

3

0 −

√

1

3

− 1 + 1

√

1

6

−

√

1

2

√

1

3

0 − 1

√

1

2

√

1

2

− 10

√

1

2

−

√

1

2

− 1 − 11