Angular Momentum
المؤلف:
Franklin Potter and Christopher Jargodzki
المصدر:
Mad about Modern Physics
الجزء والصفحة:
p 67
20-10-2016
576
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 = J2x + J2y + J2z, that is, J2 = 3 <J2z>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.
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