Lecture 6 Quantum

Question 1

What is the vector for spin up?

Answer 1

1,0

Question 2

What is the vector for spin down?

Answer 2

0,1

Question 3

What is spin?

Answer 3

Intrinsic angular momentum unrelated to motion

Question 4

What does angular momentum do to charged particles?

Answer 4

Causes interaction with magnetic field

Question 5

What is the formula for interaction shift in energy when a charged particle interacts with a magnetic field?

Answer 5

Delta E = - Mu . B (mu = magnetic moment)

Question 6

What is the formula for magnetic moment?

Answer 6

Mu =I A (vector loop area))

OR - derived from above identities = - (2w r^2)/2

Question 7

What is the vector loop area formula?

Answer 7

A = A n hat

Question 8

What is the formula for magnitude of the current?

Answer 8

I = - (ew)/ (2PI)

Question 9

What is angular velocity formulas here?

Answer 9

W = r x v / (r^2)

OR = w . N hat

Question 10

How are angular velocity (w) and the unit vector normal to the area surface related?

Answer 10

Both point in same direction

Question 11

What is the formula for the quantum magnetic moment?

Answer 11

Mu = - e L hat (operator) / (2m)

• Can have mu sub z and L hat sub z etc

Question 12

What is the formula for energy shift due to magnetic interaction in terms of angular momentum?

Answer 12

From equating two : - Mu . B = e/ (2m) * L. B

Delta E = eB/ (2m) * Lz hat

Question 13

What does a magnetic field do to the energy of an electron?

Answer 13

Makes it m-dependent

Question 14

By applying the Hamiltonian, what is the energy shift along the Z axis?

Answer 14

eB m hbar / (2m)

Question 15

What is the force acting on atoms in a Bfield proportional to?

Answer 15

Quantum number m

Question 16

What is Bohr Magneton?

Answer 16

A physical constant and the natural unit for magnetic moment of an electron caused by angular momentum

Question 17

What did the Stern-Gerlach experiment show?

Answer 17

Spatial orientation of angular momentum (spin) is quantised i.E. can only exist in discrete states showing atoms have intrinsic quantum properties

Question 18

How many parts will a beam split into when using atoms with angular momentum quantum number l = 0?

Answer 18

(2l + 1) = 1 part

Question 19

What is the Hamiltonian describing the interaction of an electron due to a uniform magnetic fields?

Answer 19

Delta E = e/ (2m) B.Lzhat

Question 20

What role does quantum number play?

Answer 20

Delta E = e/ (2m) B.Lzhat

Question 21

What is the angular momentum eigenvalue?

Answer 21

My

Question 22

Which atoms were used in the Stern-Gerlach experiment?

Answer 22

Silver, with l = 0

Question 23

What is m for the Stern-Gerlach experiment?

Answer 23

M = (2l + 1) -> only 1 mm state, which =0

Question 24

What was surprising in Stern-Gerlach experiment?

Answer 24

Saw 2 beams meaning 2 components of m ,but classical predicted one
- predicted 2 as through to be another type of angular momentum

Question 25

What physical quantity was proved with Stern-Gerlach experiment?

Answer 25

Spin

Question 26

What is m for spin?

Answer 26

Ms = -1/2 or +1/2 as increases in integer steps

Question 27

What are eigenvalues for spin?

Answer 27

+/- h bar /2

Question 29

What is the difference in eigenvalues for spin up and spin down?

Answer 29

Sign

Question 30

What are all the possible values of spin when measured?

Answer 30

+/- h bar /2 -> no other possible values

Question 31

How are spin eigenstates related?

Answer 31

Orthonormality

Question 32

Write orthonormality in Dirac

Answer 32

< Up|Up> = < down | down > = 1 < up | down > = < down | up > = 0

Question 33

Represent any arbitrary spin state in matrices

Answer 33

| X > = C up (1,0) + D down ( 0,1)

Question 34

What is a spinor?

Answer 34

A 2 element vector for any spin state

Question 35

What take positive square root when normalising?

Answer 35

By definition so Wavefunction has well-defined norm

Question 36

Why would you normalise a spinor?

Answer 36

To determine the value of the coefficient

Question 37

How do you normalise a spinor?

Answer 37

Row x complex conjugate of column

Question 38

How to prove orthonormality?

Answer 38

Write conditions then prove A1 X complex conjugate of A2 = 0

Question 39

In SG experiment, what is deflection proportional to?

Answer 39

Spin and B field strength magnitude

Question 40

In SG experiment, what is deflection inversely proportional to?

Answer 40

Particle kinetic energy

Question 41

Eigenvalue of Lx

Answer 41

±ℏl

Question 42

Eigenvalue of L^2

Answer 42

ℏ^2 l ( l+1)

Question 43

What is dagger

Answer 43

- aka Adjoint - (AB) † = B†A †

Question 44

How to check if Hermitian matrix?

Answer 44

- Transpose matrix (diagonal upper right to lower left) - convert complex numbers positive/negative but leave associated real numbers

Question 45

Check if position operator hermitian

Answer 45

Integral -> by parts remembering * for bra <Ψ |p| θ > = < pΨ | θ >

Question 46

Commutator of angular momentum

Answer 46

[Lx, Ly] = i ℏ Lz [Ly, Lz ] = i ℏ Lx [Lz, Lx] = i ℏ Ly