(a) what are the possible values of l for n = 4? (enter your answers as a comma-separated list.)
(b) what are the possible values of ml for l = 3? (enter your answers as a comma-separated list.)
(c) what is the smallest possible value of n for which l can be 6?
(d) what is the smallest possible l that can have a z component of 5ℏ?

(a) 0,1,2,3      (b) -3,-2,-1,0,1,2,3              (c) 6               (d) 5

Explanation:

(a) for the principal quantum number 'n', the possible values of I = 0 to n-1. Thus, if the principal quantum number 'n' =4, I = 0,1,2,3.

(b) for a given number of 'I', the possible values of ml = -I to +I. Therefore, if I =3, then ml = -3,-2,-1,0,1,2,3

(c) 'I' which is the orbital angular momentum quantum number usually has values from 0,1,2,⋯,n−1. Therefore, for n greater than or 6, t would be greater than or equal to 5. Thus, the smallest possible value of n for which I can be 6 is 6.

(d) In a 3-dimensional figure,  If the z-component of the orbital angular momentum Lz for which I=5 is measured, The possible outcomes will be:

mħ = -5ħ, -4ħ, -3ħ, -2ħ, -1ħ, 0, 1ħ, 2ħ, 3ħ, 4ħ, 5ħ.

Thus, the smallest possible l that can have a z component of 5ℏ is 5.

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