Angular Momentum

In classical calculations, the quantity that often appears in the result is the square of the angular momentum J². One can often guess at the correct quantum mechanical formula by replacing J² by j (j + 1) h²/4π², where j is the z-component of the angular momentum and h is Planck’s constant. Why is the square of the angular momentum in quantum mechanics proportional to j ( j + 1) instead of just j²?

Answer

We take the space quantization of angular momentum as given, so there will be (2j + 1) components in the z-direction from j to –j, decreasing by an integer each step. Since there is no preferred direction, J² = J²_x + J²_y + J²_z, that is, J² = 3 <J²_z>_avg , where avg represents the average value given by [ j 2 + (j – 1)² + . . . + (–j + 1)² + (–j)²] h²/(4π² [2j + 1]). Using a math table or finding the sum of the series of squared integers directly, one can verify that J² = j (j + 1) h²/4π.