Question

How to find the spacing of values as a fraction of magnitude of orbital angular momentum - any help on question c will be great I have done the first 2 questions so if they're wrong let me know

2. A classical electron moves in a circle of radius 10 cm with velocity 10 cm/s.

(a) What is the value of the quantum number l which gives a quantized angular momentum close to the angular momentum of this classical electron?

Got - 86/85.839

(b) How many discrete values are possible for the z component of this orbital angular momentum?

Got - 2l+1 discrete values

(c) How closely spaced are these values as a fraction of the magnitude of the orbital angular momentum?

Answer 1

2]

a] The classic angular momentum is:

L = mvr

equate this to quantum mechanical angular momentum

this gives,

squaring both sides and re-arranging the terms gives a quadratic equation:

the positive solution for this is

b]

Number of discrete values for L_z are: 2l + 1 = 173

c] Angular momentum along z direction is given by:

where m_l can take values from - l to l

so,

this is how closely the values are spaced out as a fraction of orbital angular momentum.