Atomic, Molecular, and Laser Physics

Question 1

Bohr equation for hydrogen

/\E =

/\ is a delta

Answer 1

hbar w =

Rhc( Z²/n² - Z²/m² )

Question 2

relate wavenumber and wavelength

Answer 2

v = 1/lambda

Question 3

Rhc =

Answer 3

e²/4πe₀ . 1/2a₀

Question 4

bohr radius

a₀ =

Answer 4

hbar² / (e²/4πe₀)m_e

Question 5

fine structure constant

Answer 5

alpha = v_e/c

= e²/4πe₀ . 1/(hbar.c)

~1/137

Question 6

energy of a magnetic dipole in a magnetic field

Answer 6

E =

-mu.B

mu is the magnetic dipole

Question 7

Bohr magneton

Answer 7

mu_B=

e hbar/2m_e

Question 8

TISE

Answer 8

{-hbar²/2m_e \/² + V(r)} phi = E phi

\/ is nabla

Question 9

To solve schrodiger for hydrogen atom (angular part)

Answer 9

• express \/² in spherical polar also using operator l² to contain angular terms
• separation of variables
• start with angular, separate again with constant m²
• use PHI(phi + 2π) = PHI(phi) => integer m
• apply raising operators to Y until l₊Y = 0 (m_max = l )
• show Y≈ sin^l(theta)e^ilø
• apply lowering ops to find the rest

Question 10

Y_0,0 =

Answer 10

SQRT (1/4π)

Question 11

Y_1,0 =

Answer 11

SQRT ( 3/4π ) cos(theta)

Question 12

Y_1,1

Y_1,-1

Answer 12

-SQRT( 3/8π ) sin(theta) e^+iø

+SQRT( 3/8π ) sin(theta) e^-iø

Question 13

how do we solve shrodinger eqn for a many electron atom

Answer 13

• ignore electron spin
• include a SUM of mutual repulsions
• invent a centrally symmetric potential V(r)
• modify H = H₀ + H₁

such that H₀ centrally symmetric

• eqns are now separable

H₀ >> H₁

_perturbation

Question 14

central field approximation

Answer 14

ignore the risidual el. static interaction from multi atom Hamiltonian.

H₁ = 0

Question 15

Pauli Exclusion Principle. How does it lead to configurations

Answer 15

2 identical fermions cannot occupy the same quantum state simultaneously.

It limits the number of electrons you can put in each state. Otherwise all electrons would be in n=1 l=0

Question 16

What is the quantum defect

Answer 16

a correction applied to the equations of a hydrogenic atom to take into account the fact that inner electrons do not entirely screen their associated charge in the nucleus.

Question 17

electron configuration notation

Answer 17

nl^x

n = shell = 1, 2, 3

l = orbital = s, p, d, = 0, 1, 2…

x = number of electrons, this has maximum 2*(2l + 1)

factor of 2 for the spin

Question 18

why does a higher l correspond to higher (nearer 0) energy state

Answer 18

lower l electrons penetrate closer to the nucleus. Hence electrons with high l are further from the nucleus.

This means that outer electrons are more effectively screened <=> the effective charge experienced is lower.

Question 19

trick for finding energies of first excited state of helium

Answer 19

1s¹2s¹

first electron experiences no screening

2nd experiences perfect screening

1st has the form of hydrogen with z=2

2nd with z=1

Question 20

in helium does the singlet or triplet state have higher energy and why?

Answer 20

singlet

Question 21

can the triplet state in helium

1s¹2s^{1 3}S

decay into the singlet state

1s¹2s^{1 1}S

Answer 21

no. both states are metastable

Question 22

tell me about the sequence of perturbations:

H_atom = H₀ + H₁ + H_SO +[H_HFS]

Answer 22

H₀ is from the central field, it gives the configuration (nl)

H₁ is the residual due to electron repulsion + screening, ot gives the term (^2S+1L) [singlet-triplet splitting]

• together these two make the gross tructure

H_SO is the fine structure due to spin orbit interaction, it gives the level (^2S+1L_J) [triplet state is split into 3 levels]

Question 23

Briefly outline the cause of the fine structure

Answer 23

The combination of the spin and orbital angular momenta.

The spin of the electron gives rise to a magnetic dipole moment, this sits in the magnetic field of the orbiting electrons. This leads to a shift in energy.

Question 24

electron magnetic moment

mu =

and its energy in a magnetic field

Answer 24

-g_s . mu_B/hbar . s

g_s ≈ 2 = Londé factor

E = -mu . B

Question 25

in the fine structure, where does the magnetic field come from? In hydrogen, is it parallel or antiparallel to the angular momentum of the electron?

Answer 25

the charged electrons are _moving_ in the electric field of the nucleus and thus create a magnetic field. NB _B_ = - 1/c² _v_ ^ _E_ (from SR) Parallel

Question 26

Biot Savart Law

Answer 26

_B_ = mu₀/4π x I x int [_r_ ^ d**_s_** / r²] the path integral around the loop current I =

Question 27

when deriving the energy shift in the fine structure of hydrogen, what must you remember?

Answer 27

g_s -\> (g_s - 1) due to Thomas Precession - it is relativistic motion in an accelerated frame because the electron is orbiting the nucleus

Question 28

how to find \< _s_ . _l_ \>/hbar²

Answer 28

_j_ = _l_ + _s_ _j_² - _l_² - _s_² = 2**_l_**._s_ use expectation values of LHS: 2> = hbar² j (j+1) ... \< _s_ . _l_ \>/hbar² = 1/2 [j(j+1) - l(l+1) - s(s+1)]

Question 29

in fine structure, what are the possible values of J and why?

Answer 29

J is the total angular momentum _J_ = _L_ + _S_ so J_max = L + S _Jmin = | L-S |

Question 30

how does the fine structure splitting change for large nuclei

Answer 30

it gets bigger

Question 31

what is L-S coupling used for? (it is also known as Russel-Saunders Coupling)

Answer 31

fine structure of multi electron atoms - easiest case is with 2 valence electrons (eg magnesium)

Question 32

what are _L_ and _S_ and why are they important in LS coupling

Answer 32

_L_ = Σ_l__i _S_ = Σ_s__i _L_² and _S_² both commute with the hamiltonian H₁ (residual electrostatic ham) [L², H₁] = [L_z, H₁] = 0 similarly for S

Question 33

how to find the 'terms' for Si ... 3p¹4p¹

Answer 33

l₁=1, l₂=1 =\> L = 0, 1, 2 s₁=1/2, s₂=1/2 =\> S = 0, 1 so ¹S, ¹P, ¹D, ³S, ³P, ³D

Question 34

what is the condition for LS coupling?

Answer 34

the energies resulting from the residual electrostatic interaction must be larger than the corresponding energies from the spin-orbit interaction: 1> >> SO>

Question 35

Fine structure in LS coupling. what is the 'projection' of \< _s_₁ . _l_₁ \>

Answer 35

\< _s_₁ . _S_ \> _S_ / S(S+1) . \< _l_₁ . _L_ \> _L_​ / L(L+1) ( = some scalar times _S_ . _L_ )

Question 36

E_SO =

Answer 36

beta_LS / hbar² \* \<_S_ . _L_\>

Question 37

what are the differences between LS and jj coupling?

Answer 37

LS: You couple l₁ and l₂ to L (l₁ + l₂ = L), and s₁ and s₂ to S, and then you couple L with S to get LS (L + S = J) JJ: You couple l₁ and s₁ to j₁ (l₁ + s_{1 =} j₁), and l₂ and s₂ to j₂, and then couple j₁ and j₂ to get JJ (j_{1 +} j₂ = J) when E_SO \>≈ E_res use jj coupling

Question 38

what is the ionisation energy of an alkali atom?

Answer 38

-hcR/(n-d_l)² d_l is the quantum defect note that there is no Z as the inner electrons shield (almost) all the nucleus

Question 39

how does the quantum defect change for large l. Does it depend on n?

Answer 39

d_l -\> 0 barely

Question 40

read 3²S_1/2

Answer 40

three - doublet - S - one half

Question 41

What is the dependece on Z in the spin orbit interaction in 1) hydrogen-like atoms 2) alkali atoms

Answer 41

1) Z⁴ 2) Z²

Question 42

in alkali atoms, what kind of splitting is there due to the spin orbit interaction?

Answer 42

there is a single e^- so S=1/2 thus, if l =/= 1 (so for P, D, F, ...) we get a doublet

Question 43

Zeeman effect energy shift

Answer 43

E_Ze = - _mu_ . _B_

Question 44

for orbiting electron _mu_ = \< H_Ze \> =

Answer 44

_mu_ = - mu_B/hbar _l_ \< H_Ze \> = mu_B/hbar B \< _l_ . _z_ \> = mu_B B m_l

Question 45

for spinning electron _mu_ = \< H_Ze \> =

Answer 45

_mu_ = - g_s mu_B/hbar _s_ \< H_Ze \> = mu_B/hbar B \< _s_ . _z_ \> = 2 mu_B B m_s

Question 46

Zeeman effect in LS coupling \< H_Ze \> = ?

Answer 46

_mu_ = - mu_B/hbar ( _L_ + _S_ ) this is hard to evaluate, but \< _mu_ . _j_ \> = constant of motion \< H_Ze \> = - \< _mu_ . _j_ \> / (hbar² j(j+1)) _j_ . _B_ then sub in _mu_ in terms of S and L E_Ze = g_j mu_B B M_j g_j = 3/2 + [S(S+1) - L(L+1)]/2J(J+1)

Question 47

what is the normal Zeeman effect

Answer 47

S=0 =\> J = L =\> g_J = 1

Question 48

is the Zeeman effect always valid?

Answer 48

it works only if \< H_Ze \> \<\< \< H_SO \> \<\< \< H₁ \> otherwise we use the Paschen-Back effect

Question 49

Describe the Paschen-Back effect

Answer 49

the external B-field is stronger than the spin-orbit coupling. L and S precess independently about the B-field axis -mu_B/hbar [\< _L_ . _B_ \> - 2\< _S_ . _B_ \>] so E_PB = mu_B B ( M_L + 2 M_S )

Question 50

what quantum numbers are important when there is an applied B field that is: weak strong

Answer 50

M_J (Zeeman effect) M_L and M_S (Paschen-Back effect)

Question 51

E_HFS

Answer 51

E_HFS = - _mu__I . _B__{e B is parallel to J} _mu__I = g_I mu_N/hbar _I_ mu_N = mu_B m_e/M_p

Question 52

when is nuclear spin I non-zero

Answer 52

for half integer I \<=\> odd # of protons OR neutrons I = 0 if # of p and n both even

Question 53

What is the cause of the hyperfine structure?

Answer 53

the spinning nucleus inside the electronic B-field

Question 54

hyperfine structure contributions to B_e

Answer 54

1) closed shells do not contribute 2) major contribution from s-electron since the wavefunction is non zero at r=0

Question 55

hyperfine structure s-electron _B__e =

Answer 55

-2/3 mu₀ g_s |phi_ns (r=0)|² mu_B/hbar _s_

Question 56

outline IJ-coupling

Answer 56

F = I + J I and J precess about F

Question 57

\< I . J \> =

Answer 57

hbar²/s { F(F+1) - I(I+1) - J(J+1) }

Question 58

hyperfine structure fot l\<0 B_e =

Answer 58

Question 59

Born-Oppenheimer approximation

Answer 59

the motion if the electron and nuclei is treated separately because of the great disparity between their masses.

Question 60

order of magnitude of electronic energy

Answer 60

_/\_p ≈ hbar/a a = atomic radius E_e ≈ p²/2m a few eV

Question 61

order of magnitude of vibrational energy

Answer 61

U(x) = 1/2 mu w_v² x² mu is the reduced mass when x is of the order of a, the energy must be of the order E_e 1/2 mu w_v² a² = hbar²/2ma² solve for w_v Ev ≈ hbar w_v = SQRT (m/mu) E_e

Question 62

order of magnitude of rotationl energy

Answer 62

E_r = 1/2 I w_r² I = mu a² the rot frequency is such that: mu a² w_r ≈ hbar =\> E_r = (m/mu) E_e

Question 63

condition for optical gain

Answer 63

N₂/g₂ \> N₁/g₁

Question 64

necessary (but not sufficient) condition for steady-state population inversion

Answer 64

A₂₁ \< g₁/g₂ 1/t₁

Question 65

describe the features of the lineshape function

Answer 65

g(w - w₀) looks like a stretched delta function with FWHM of gamma and centred around w₀

Question 66

How are the lineshapes for absorption and spontaneous and stimulated emission related

Answer 66

they are the same \<=\> they have the same frequency dependance