7. Angular momentum

Question 1

What parts of the solution to the 3D TISE form the spherical harmonics?

Answer 1

The angular parts of ϴ and Φ so they are functions of (ϴ,Φ)

Question 2

What are the key points about the solution of ϴ?

Answer 2

• l should be a non -ve number
• there are polynomials in cos(ϴ)
• complex
• For the Coulomb 1/r potential, tthe max quantum number = n-1

Question 3

What do we call the maximum quantum number and what is it equal to?

Answer 3

The principle quantum number, equal to n-1

Question 4

State the symbols for the principal, orbital and magnetic quantum numbers

Answer 4

n - principal quantum number
l - orbital quantum number
m_l - magnetic quantum number

Question 5

State the values for the principal, orbital and magnetic quantum numbers

Answer 5

n = 1,2,3,4...
l = 0,1,2, ... n-1
m_l = -l, -l +1, ... l-1, l

Question 6

State the magnitude for the angular momentum

Answer 6

|L| = mvr

Question 7

Imagine that the angular momentum is parallel to the z axis. State the directions of its components

Answer 7

It is parallel to the z axis (up)

-> orbit is anti clockwise in the x-y plane (RH grip)

Question 8

State the results of the theoretical situation when the angular momentum is parallel to z

Answer 8

v_z = 0, so Δv_z = 0
Δp_z = 0
Δz = infinity NOT ALLOWED BY HUP

Question 9

How much of the angular momentum are we allowed to know?

Answer 9

The magnitude of |L| and one component - usually denoted L_z

Question 10

State the equations of L_z and |L|

Answer 10

L_z = h_bar m_l
|L| = h_bar sqrt( l (l+1) )

Question 11

What was wrong with the old Bohr model of the atom?

Answer 11

• It described the quantisation of the angular momentum wrong ( |L| = n h_bar ). It describes the maximum angular momentum well for large quantum numbers but not for small ones
• It cannot describe a state with 0 |L|

Question 12

State how the angular momentum is quantised and its quantum number

Answer 12

L is quantised with units of h_bar and has a quantum number of l

Question 13

See page 3 of document for the components of the angular momentum

Answer 13

do it lmao

Question 14

What can be said about the axis of the central potential?

Answer 14

It is isotropic - no preferred axis

Question 15

Give an example of a pertubation for a central potential

Answer 15

A magnetic field

Question 16

What can we model and orbiting electron as?

Answer 16

A current loop

Question 17

What is the magnetic moment of a current loop?

Answer 17

m = pi r^2 I z_hat

Question 18

State the interaciton energy (Zeeman effect) equation

Answer 18

ΔE = 2 | m | B_z