Week 5

Question 1

Angular momentum equation

Answer 1

L = r x P

Question 2

Magnitude of angular momentum

Answer 2

From crossproduct, rpsin theta

Question 3

What are central forces?

Answer 3

Force acting only in radial direction

Question 4

How do central potentials arise

Answer 4

From central forces (Acting radially only)

Question 5

What are some central potentials?

Answer 5

Gravity, coulomb

Question 6

What happens to L in a central potential?

Answer 6

L (angular momentum) is conserved

Question 8

Prove L conservation in central:

Answer 8

L = r x p + r x p
= (mr x r) + (r x F ) f=central force
= 0 + 0

Question 9

Which operators commute?

Answer 9

Position, momentum, L^2

Question 10

What is formula for rotational energy of single particle?

Answer 10

E rot = L^2 / (2m r^2)

Question 10

What is significant about commutation?

Answer 10

If observables commute, can measure both simulateously and have shared eigenfunction

Question 11

Which observables don’t commute?

Answer 11

Not position and momentum

Question 12

What is the range of the polar angle?

Answer 12

0 to Pi ( polar bears seat pie)

Question 13

What is the angle of the azimuthal angle?

Answer 13

O to Pi (A to Z = whole)

Question 14

What is the linear momentum operator?

Answer 14

P hat sub i = - is bar d/dri

Question 15

What is the linear momentum operator for i?

Answer 15

L hat sub i = - i hbar (r subj * (d/d rsubk) - (r sub k * (d/d rsubj))

Question 16

Why use L^2?

Answer 16

Because L does not commmute

Question 17

What is the operator L ^2 related to?

Answer 17

Rotational energy

Question 18

What commutes with L?

Answer 18

L^2

Question 19

What are the features of the eigenfunctions for L and L squared?

Answer 19

No dependence on r

Question 20

What can you say about Eigen values for common eigenfunctions?

Answer 20

Nothing. They do not have to be the same.

Question 21

What are the boundary conditions for the eigen functions?

Answer 21

As they are based on theta and phi, they are bounded and cannot go to plus or minus infinity

Question 22

What is the eigenvalue associated with the angular momentum operator?

Answer 22

Mh

Question 23

What does the eigenvalue for angular momentum tell us?

Answer 23

Angular momentum is quantised

Question 24

What is the difference between quantum and classical angular momentum?

Answer 24

Quantum is quantised even in free. Space

Question 25

What are the requirements to have solutions to the legendre polynomial?

Answer 25

1. Lambda equals L (L plus 1)H bar^2 2. L is greater or equal to the magnitude of M. 3. L can only be zero, one, two… And only positive.

Question 26

How many way functions in the solutions to Legendre polynomials?

Answer 26

For each combination of L & M. There is a specific wave function.

Question 27

How many possible values of M for each L?

Answer 27

2L +1

Question 28

What is the operator for the magnitude of angular momentum?

Answer 28

L hat = SQRT (L^2)

Question 29

What does the operator L sub Z represent?

Answer 29

The projection of L onto the Z axis

Question 30

What is the value of Lz?

Answer 30

Mh

Question 31

What is the value of Lz and why?

Answer 31

It is always less than the value of L because the Angular momentum can never point exactly a along one axis

Question 32

How can you define LX and LY once you have defined Lz?

Answer 32

They remain undefined

Question 33

On a diagram with L on each axis, what does the circles represent?

Answer 33

The definite values of Lz lead to totally uncertain values of Alex and LY, which are the circles

Question 34

On a diagram with L on each axis, what does the radius of the circle represent?

Answer 34

This is the magnitude of L and equals l(l+1) hbar

Question 35

On a diagram with L on each axis, what does the height of the circles represent?

Answer 35

L

Question 36

What type of polynomials are spherical harmonics?

Answer 36

Homogenous polynomials in XYZ of degree L

Question 37

What requirements are circle harmonics uniquely specified by?

Answer 37

1. Y sub L Sup +/- M is proportional to a polynomial in z^ and X^2 + y^2 all * (x + iy) ^ magnitude of m 2. Must be orthogonal to al previous spherical harmonics

Question 38

What do blue peanuts represent?

Answer 38

Real numbers

Question 39

What do yellow peanuts represent?

Answer 39

Imaginary numbers

Question 40

What can you derive by considering the Hamiltonian operator in 3-D in polar coordinates?

Answer 40

The Hamilton operator in 3D equals Hamilton operator radially plus L^2 operator (theta,phi) divided by(2MR rsquared)

Question 41

Explain you derive by considering the Hamiltonian operator in 3-D in polar coordinates?

Answer 41

The Hamilton operator in 3D equals the radio kinetic and potential energy, plus the angular kinetic energy

Question 42

Which operators have common eigenfunctions?

Answer 42

Hamiltonian, Lz L^2

Question 43

What is the energy eigenfunction equation?

Answer 43

U (r, theta, phi) = R (r) Y (sub L sup m) (theta, phi)

Question 44

What is the radial equation?

Question 45

What is different in the radial equation compared to the standard energy eigenvalue equation?

Answer 45

L(L+1) term looks like an additional potential

Question 46

What is the centrifugal force?

Answer 46

The additional term in the radial equation for potential, which makes up part of the repulsive term, so objects slow as r tends to 0, and then speed up as they are repulsed

Question 48

What is the Eigen value for Lz?

Answer 48

Mh, m = 0, +/-1, +/- 2 …

Question 49

What is the Eigen value for L^2?

Answer 49

L (L+1) hbar^2 L = 0,1,2…

Question 50

What is the name for the eigenfunctions of Lz and L^2?

Answer 50

Spherical harmonics

Question 51

What do spherical harmonics depend on?

Answer 51

No radial dependence

Question 52

What does the radial equation determine?

Answer 52

In 3-D, the radial wavefunctions

Question 53

Which quantum number labels energy

Answer 53

N = principal, quantum number

Question 54

Which quantum number labels angular momentum?

Answer 54

L = orbital, quantum number

Question 55

Which quantum number labels the direction of angular momentum?

Answer 55

M = magnetic quantum number

Question 56

What does the z-component of angular momentum signify?

Answer 56

“Direction”. Of angular momentum

Question 57

What can lift energy degeneracy?

Answer 57

Magnetic field

Question 58

How do you determine the degeneracy of energy?

Answer 58

As each energy has(2L + 1) associated with functions, it is (2L + 1)-fold degenerate