Spin and the Exclusion PrincipleModern Ch.7, Physical Systems, 20.Feb.2003 EJZ

Review Hydrogen atom, orbital angular momentum L

Electron spin s

Total angular momentum J = S + L= Spin + orbit

Applications: 21 cm line, Zeeman effect

Good QN and allowed transitions

Pauli exclusion principle

Periodic Table

Lasers

Hydrogen atom : Bohr model

We found rn = n2 r1, En = E1/n2, where the “principle quantum number” n labels the allowed energy levels.

Discrete orbits match observed energy spectrum

Hydrogen atom: Orbits are not discrete

(notice different r scales)

Hydrogen atom: Schrödinger solutions depend on new angular momentum quantum numbers

Quantization of angular momentum direction for l=2

Magnetic field splits l level in (2l+1) values of ml = 0, ±1, ± 2, … ± l

1

12

( 1) 0,1,2,..., 1

cosz l

l l where l n

L m

EE where E Bohr ground state

n l

L

L

Hydrogen atom examples from Giancoli

Hydrogen atom plus L+S coupling:

• Hydrogen atom so far: 3D spherical solution to Schrödinger equation yields 3 new quantum numbers:

l = orbital quantum number

ml = magnetic quantum number = 0, ±1, ±2, …, ±l

ms = spin = ±1/2

• Next step toward refining the H-atom model:

Spin with

Total angular momentum J=L+s

with j=l+s, l+s-1, …, |l-s|

( 1)l l L

1 12 2( 1)s 1

2z ss m

( 1)j j J

Total angular momentum:

• Multi-electron atoms: J = S+L where S = vector sum of spins, L = vector sum of angular momenta

Spectroscopic notation: L=0 1 2 3 S P D F

Allowed transitions (emitting or absorbing a photon of spin 1)

ΔJ = 0, ±1 (not J=0 to J=0) ΔL = 0, ±1

Δmj = 0, ±1 (not 0 to 0 if ΔJ=0) ΔS = 0 Δl = ±1

2 1SJL

Discuss state labels and allowed transitions for sodium

Magnetic moment of electron

Magnetic moment: Bohr magneton models e- as spinning ball (or loop) of charge

We expect but Stern-Gerlach experiment shows that where g = 2.0023…=gyromagnetic ratio(electron is not quite a spinning ball of charge).

arg.

2

2 2 2Be e

ch e evI area where I

time revr eL e

Showthatm m

z B sm

z B sg m

Application of Zeeman effect: 21-cm line

Electron feels magnetic field due to proton magnetic moment (hyperfine splitting).

2 BE B

Pauli Exclusion principle

Identical fermions have antisymmetric wavefunctions, so electrons cannot share the same energy state.

Fill energy levels in up-down pairs:

1s

2s 2p

3s 3p 3d

4s 4p 4d 4f

…

( , ') ( ', )x x x x

LASER = Light Amplification by Stimulated Emission of Radiation

Pump electrons up into metastable excited state.

One transition down stimulates cascade of emissions.

Monochromatic: all photons have same wavelength

Coherent: in phase, therefore intensity ~ N2