Question

Part C

The quantum state of a particle can be specified by giving a complete set of quantum numbers (n,l, ml,ms). How many different quantum states are possible if the principal quantum number is n = 4?

To find the total number of allowed states, first write down the allowed orbital quantum numbers l, and then write down the number of allowed values of ml for each orbital quantum number. Sum these quantities, and then multiply by 2 to account for the two possible orientations of spin.

Express your answer as an integer.

Part E

What is the maximum angular momentum Lmax that an electron with principal quantum number n = 5 can have?

Express your answer in units of ℏ. (You don't need to enter the ℏ, it is in the units field for you.)

Answer 1

Part c)

for n = 4 , we have

l = 0 , 1 , 2 , 3

for each "l", the value of "m" is - l to l

number of allowed states for l = 0

n₁ = 2(2l + 1) = 2 (2(0) + 1) = 2

number of allowed states for l = 1

n₂ = 2(2l + 1) = 2 (2(1) + 1) = 6

number of allowed states for l = 2

n₃ = 2(2l + 1) = 2 (2(2) + 1) = 10

number of allowed states for l = 3

n₄ = 2(2l + 1) = 2 (2(3) + 1) = 14

Total number of allowed states = N = n₁ + n₂ + n₃ + n₄ = 2 + 6 + 10 + 14 = 32

Part E)

for n = 5 , we have

l = 0 , 1 , 2 , 3 ,4

So the maximum value of l is "4"

maximum angular momentum is given as

L_max = sqrt(l (l + 1))ℏ

L_max = sqrt(4 (4 + 1))ℏ

L_max = sqrt(20)ℏ

L_max = 4.5 ℏ