Calculate the expectation values of the total (root-mean squared) angular moment

Question

Calculate the expectation values of the total (root-mean squared) angular momentum squareroot(L^2), and the projection of the angular momentum onto the z-axis, (L_z), for all the quantum states of a particle on a sphere with l lessthanorequalto s 3. Why is it impossible for the quantum number m_l to be larger than 3 when the orbital angular momentum quantum number is l = 3? In other words, what is the largest value of m_l for which the projection of the angular momentum along the z-axis does not exceed the total (root-mean-squared) angular momentum? Count the number of allowed me values associated with l = 3 to verify that the degeneracy of the corresponding energy level is 2l + 1.

Explanation / Answer

b) ml is magnetic quantum number. Its value depends on orbital angular momentum quantum number l. ml has values from -l to +l including zero. ml has total number of values 2l+1 only. If l = 3.

So ml does not have the value greater than l and value less than -l. Largest value should be +l.

c) For l = 3 possible ml values are

ml = -l to +l

= -3, -2, -1, 0, +1, +2, +3

There are total seven values.

Let us calculate the degenracy.

Degenracy = 2l +1

= 2*3 + 1

= 7

So this values is equal to number of possible values for ml.

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