Atomic, Molecular, and Laser Physics Flashcards
Bohr equation for hydrogen
/\E =
/\ is a delta
hbar w =
Rhc( Z2/n2 - Z2/m2 )
relate wavenumber and wavelength
v = 1/lambda
Rhc =
e2/4πe0 . 1/2a0
bohr radius
a0 =
hbar2 / (e2/4πe0)me
fine structure constant
alpha = ve/c
= e2/4πe0 . 1/(hbar.c)
~1/137
energy of a magnetic dipole in a magnetic field
E =
-mu.B
mu is the magnetic dipole
Bohr magneton
muB=
e hbar/2me
TISE
{-hbar2/2me \/2 + V(r)} phi = E phi
\/ is nabla
To solve schrodiger for hydrogen atom (angular part)
- 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 (mmax = l )
- show Y≈ sinl(theta)eilø
- apply lowering ops to find the rest
Y0,0 =
SQRT (1/4π)
Y1,0 =
SQRT ( 3/4π ) cos(theta)
Y1,1
Y1,-1
-SQRT( 3/8π ) sin(theta) e+iø
+SQRT( 3/8π ) sin(theta) e-iø
how do we solve shrodinger eqn for a many electron atom
- 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
central field approximation
ignore the risidual el. static interaction from multi atom Hamiltonian.
H1 = 0
Pauli Exclusion Principle. How does it lead to configurations
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
What is the quantum defect
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.
electron configuration notation
nlx
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
why does a higher l correspond to higher (nearer 0) energy state
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.
trick for finding energies of first excited state of helium
1s12s1
first electron experiences no screening
2nd experiences perfect screening
1st has the form of hydrogen with z=2
2nd with z=1
in helium does the singlet or triplet state have higher energy and why?
singlet
can the triplet state in helium
1s12s1 3S
decay into the singlet state
1s12s1 1S
no. both states are metastable
tell me about the sequence of perturbations:
Hatom = H0 + H1 + HSO +[HHFS]
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
HSO is the fine structure due to spin orbit interaction, it gives the level (2S+1LJ) [triplet state is split into 3 levels]
Briefly outline the cause of the fine structure
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.
electron magnetic moment
mu =
and its energy in a magnetic field
-gs . muB/hbar . s
gs ≈ 2 = Londé factor
E = -mu . B
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?
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
Biot Savart Law
B = mu0/4π x I x int [r ^ ds / r2]
the path integral around the loop
current I =
when deriving the energy shift in the fine structure of hydrogen, what must you remember?
gs -> (gs - 1)
due to Thomas Precession
- it is relativistic motion in an accelerated frame because the electron is orbiting the nucleus
how to find
< s . l >/hbar2
j = l + s
j2 - l2 - s2 = 2l.s
use expectation values of LHS:
<j>2> = hbar2 j (j+1) ...</j>
< s . l >/hbar2 =
1/2 [j(j+1) - l(l+1) - s(s+1)]
in fine structure, what are the possible values of J and why?
J is the total angular momentum
J = L + S
so Jmax = L + S
<span>J</span>min = | L-S |
how does the fine structure splitting change for large nuclei
it gets bigger
what is L-S coupling used for? (it is also known as Russel-Saunders Coupling)
fine structure of multi electron atoms
- easiest case is with 2 valence electrons (eg magnesium)
what are L and S and why are they important in LS coupling
L = Σ_l_i
S = Σ_s_i
L2 and S2 both commute with the hamiltonian H1 (residual electrostatic ham)
[L2, H1] = [Lz, H1] = 0
similarly for S
how to find the ‘terms’ for
Si … 3p14p1
l1=1, l2=1 => L = 0, 1, 2
s1=1/2, s2=1/2 => S = 0, 1
so
1S, 1P, 1D, 3S, 3P, 3D
what is the condition for LS coupling?
the energies resulting from the residual electrostatic interaction must be larger than the corresponding energies from the spin-orbit interaction:
<h>1> >> <h>SO></h></h>
Fine structure in LS coupling.
what is the ‘projection’ of
< s1 . l1 >
< s1 . S > S / S(S+1) . < l1 . L > L / L(L+1)
( = some scalar times S . L )
ESO =
betaLS / hbar2 * <S . L>
what are the differences between LS and jj coupling?
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 + s1 = j1), and l2 and s2 to j2, and then couple j1 and j2 to get JJ (j1 + j2 = J)
when ESO >≈ Eres use jj coupling
what is the ionisation energy of an alkali atom?
-hcR/(n-dl)2
dl is the quantum defect
note that there is no Z as the inner electrons shield (almost) all the nucleus
how does the quantum defect change for large l.
Does it depend on n?
dl -> 0
barely
read
32S1/2
three - doublet - S - one half
What is the dependece on Z in the spin orbit interaction in
1) hydrogen-like atoms
2) alkali atoms
1) Z4
2) Z2
in alkali atoms, what kind of splitting is there due to the spin orbit interaction?
there is a single e- so S=1/2
thus, if
l =/= 1 (so for P, D, F, …)
we get a doublet
Zeeman effect energy shift
EZe = - mu . B
for orbiting electron
mu =
< HZe > =
mu = - muB/hbar l
< HZe > =
muB/hbar B < l . z >
= muB B ml
for spinning electron
mu =
< HZe > =
mu = - gs muB/hbar s
< HZe > =
muB/hbar B < s . z >
= 2 muB B ms
Zeeman effect in LS coupling
< HZe > = ?
mu = - muB/hbar ( L + S )
this is hard to evaluate, but
< mu . j > = constant of motion
< HZe > = - < mu . j > / (hbar2 j(j+1)) j . B
then sub in mu in terms of S and L
EZe = gj muB B Mj
gj = 3/2 + [S(S+1) - L(L+1)]/2J(J+1)
what is the normal Zeeman effect
S=0
=> J = L => gJ = 1
is the Zeeman effect always valid?
it works only if
< HZe > << < HSO > << < H1 >
otherwise we use the Paschen-Back effect
Describe the Paschen-Back effect
the external B-field is stronger than the spin-orbit coupling. L and S precess independently about the B-field axis
-muB/hbar [< L . B > - 2< S . B >]
so
EPB = muB B ( ML + 2 MS )
what quantum numbers are important when there is an applied B field that is:
weak
strong
MJ (Zeeman effect)
ML and MS (Paschen-Back effect)
EHFS
EHFS = - muI . Be
B is parallel to J
muI = gI muN/hbar I
muN = muB me/Mp
when is nuclear spin I non-zero
for half integer I
<=> odd # of protons OR neutrons
I = 0 if # of p and n both even
What is the cause of the hyperfine structure?
the spinning nucleus inside the electronic B-field
hyperfine structure contributions to Be
1) closed shells do not contribute
2) major contribution from s-electron since the wavefunction is non zero at r=0
hyperfine structure s-electron
Be =
-2/3 mu0 gs |phins (r=0)|2 muB/hbar s
outline IJ-coupling
F = I + J
I and J precess about F
< I . J > =
hbar2/s { F(F+1) - I(I+1) - J(J+1) }
hyperfine structure fot l<0
Be =

Born-Oppenheimer approximation
the motion if the electron and nuclei is treated separately because of the great disparity between their masses.
order of magnitude of electronic energy
_/_p ≈ hbar/a
a = atomic radius
Ee ≈ p2/2m
a few eV
order of magnitude of vibrational energy
U(x) = 1/2 mu wv2 x2
mu is the reduced mass
when x is of the order of a, the energy must be of the order Ee
1/2 mu wv2 a2 = hbar2/2ma2
solve for wv
Ev ≈ hbar wv
= SQRT (m/mu) Ee
order of magnitude of rotationl energy
Er = 1/2 I wr2
I = mu a2
the rot frequency is such that:
mu a2 wr ≈ hbar
=>
Er = (m/mu) Ee
condition for optical gain
N2/g2 > N1/g1
necessary (but not sufficient) condition for steady-state population inversion
A21 < g1/g2 1/t1
describe the features of the lineshape function
g(w - w0)
looks like a stretched delta function with FWHM of gamma and centred around w0
How are the lineshapes for absorption and spontaneous and stimulated emission related
they are the same
<=> they have the same frequency dependance