Angular Momentum

Question 1

What is the quantum equation for angular momentum L?

Answer 1

L(hat) = r(hat) X p(hat)

Question 2

How do we compute the cross product of r X p if L is L(z)?

Answer 2

L(z)(hat) = r(x)(hat)p(y)(hat) - r(y)(hat)p(x)(hat)

Question 3

What do we get as the final equation for L(z)(hat)?

Answer 3

L(z)(hat) = -iћ(x d/dy - y d/dx)

Question 4

What is L(z)(hat) in spherical polars?

Answer 4

L(z)(hat) = -iћd/dФ

Question 5

What do all the commutators of the different components of L equal?

Answer 5

[L(x)(hat), L(y)(hat)] = iћL(z)(hat), [L(y)(hat), L(z)(hat)] = iћL(x)(hat), [L(z)(hat), L(x)(hat)] = iћL(y)(hat)

Question 6

What does the commutator [L(z)(hat), L^2(hat)] equal?

Answer 6

[L(z)(hat), L(x)(hat)^2] + [L(z)(hat), L(y)(hat)^2] + [L(z)(hat), L(z)(hat)^2] = 0

Question 7

What does finding all these commutators mean?

Answer 7

We cannot know Lz and Lx or Ly precisely and simultaneously, but we can know L^2 and Lz together

Question 8

What do we find from the fact that L(hat)^2 and Lz(hat) commute?

Answer 8

They share a common set of eigenfunctions, which will be spherical harmonics.

Question 9

What is the equation for the angular momentum raising operator?

Answer 9

L(+hat) = L(x)(hat) + iL(y)(hat)

Question 10

What is the equation for the angular momentum lowering operator?

Answer 10

L(-hat) = L(x)(hat) - i(L(y)(hat)

Question 11

What do we get if we multiply the angular momentum raising and lowering operators together?

Answer 11

Get L(hat)^2 - L(z)(hat)^2 + ћL(z)(hat) for L(+hat)L(-hat), and minus the h-bar term for the other way round.

Question 12

What does the commutator of the two ladder operators for angular momentum equal?

Answer 12

[L(+hat), L(-hat)] = 2ћL(z)(hat)

Question 13

What is the eigenvalue equation for L(hat)^2 and L(z)(hat)^2?

Answer 13

L(hat)^2Ψ(αβ) = αΨ(αβ), L(z)(hat)^2Ψ(αβ) = βΨ(αβ)

Question 14

What do the L(+-hat) operators do to the angular momentum?

Answer 14

Raise or lower the z-component by ћ, but do not change the total angular momentum

Question 15

What are the upper and lower limits expected for the z-component of angular momentum?

Answer 15

L(z) <= |L|, so β^2 <= α, giving L(+hat)Ψ(αβmax) = 0, L(ihat)Ψ(αβmin) = 0

Question 16

How can we use L(-hat)*L(+hat) to find an equation for α and β?

Answer 16

Use the answer to show that L(hat)^2Ψ(αβmax) - L(z)(hat)^2Ψ(αβmax) - ћL(z)(hat)Ψ(αβmax) = 0, then plug in eigen value equations for L(z)(hat) =β and L(hat)^2 = α

Question 17

What do we find is the un-simplified equation for α?

Answer 17

α = βmax(βmax+ћ) = βmin(βmin-ћ), so βmax = -βmin

Question 18

How do we use the equation for βmax to find the final equation for α?

Answer 18

L(+-hat) raises β by +-ћ, so 2βmax = nћ, βmax = nћ/2

If we let n = 2l, then we find β = integerћ = m(l)ћ, and α = l(l+1)*ћ

Question 19

What happens if we set n=1 for βmax and βmin?

Answer 19

βmax = ћ/2, βmin = -ћ/2 -> this describes spin angular momentum for a fermion.

Question 20

If we recall that Ψαβ(Ф+2π) = Ψαβ(Ф), what can we find out?

Answer 20

Do L(zhat)Ψαβ = -iћ d/dФ Ψαβ = βΨαβ, and find that Ψαβ = Θ(θ)exp(iβФ), but with β = +- 1/2, we can’t fulfill single-values requirement, so spin angular momentum does not behave like an orbital angular momentum eigenfunction

Question 21

What is a spinor?

Answer 21

A mathematical entity which in general do not look the same on rotation by 2π.

Question 22

What are the 3 Hermitian operators on spin states?

Answer 22

[s(xhat), s(yhat)] = iћs(zhat), s(hat)^2 = s(xhat)^2 + s(yhat)^2 +s(zhat)^2, [s(hat), s(zhat)] = 0

Question 23

What is α and β equal to for spin angular momentum?

Answer 23

β = +-m(s)*ћ = +-1/2 ћ, α = s(s+1)ћ^2 = 3/4 ћ^2

Question 24

What do we get Ψl,m(l)(θ, Ф) is equal to for orbital angular momentum?

Answer 24

Ψl,m(l)(θ, Ф) = Nsin^l(θ)exp(ilФ)

Question 25

In Ψl,m(l)(θ, Ф), what happens if we set l=0?

Answer 25

m(l) = 0, so Ψ00 = N = const

Question 26

How do we find the value for N in Ψl,m(l)(θ, Ф)?

Answer 26

• Integral over everywhere of Ψ00* *Ψ00 dτ = 1 -> Integral over everywhere of N^2 dτ = 1
• E = all solid angle so integral over everywhere of dτ = 4π
• N = +-1/(2*sqrt(π))

Question 27

What does Ψ00 represent?

Answer 27

The lowest spherical harmonic.

Question 28

What does Ψ11 equal?

Answer 28

Ψ11 = sqrt(3/8π) sinθexp(iФ)

Question 29

How can we get Ψ10 from Ψ11?

Answer 29

Ψ10 ∝ L(-hat)Ψ11, where L(-hat) = -ћexp(-iФ)(d/dθ -icotθ d/dФ)

Question 30

What does Ψ10 equal?

Answer 30

Ψ10 = sqrt(3/4π) *cosθ