Download L - BYU Physics and Astronomy

Hydrogen atom energy levels
deBroglie and Bohr got the spectrum right!
h
1. Wavelength comes from speed:

mv
2. Integer wavelengths fit in orbit of size r
2
3. F = ma:
2
ke
v
m
2
r
r
2 2 e 4 me k 2
E1  
 13.6eV
2
h
13.6eV
En  
n2
n  2 r
rn  n 2 aB
aB  0.0529 nm
Light frequencies emitted/absorbed
are given by initial and final
electron energy levels
hf  E f  Ei
13.6eV
En  
n2
But Bohr got so much wrong…
1. Can’t give energies for anything but H atom.
2. Assumes planar orbits, while atoms are more spherical
3. The electrons are not at a single distance r
4. His ideas about angular momentum were too simple. The quantum
state labels are not just n: they include l,m, and s.
Angular momentum in QM
Classical:
any L, w, KE allowed
L  Iw
1 2
KE  I w
2
L
Quantum:
L, w, KE are quantized
h
L  l (l  1)
2
angular momentum magnitude
l =0,1,2… orbital angular momentum quantum number
l=0: “s” orbital in chemistry
l=1: “p” orbital in chemistry
l=2: “d” orbital in chemistry
Angular momentum in QM
Quantum: the component of L along
any axis measured is also quantized
(call the measurement axis z):
Uncertainty: We can’t know
the direction of L with
precision. We can only know
the z component.
h
Lz  m
2
Lz quantum number
m = -l…l there are 2l+1 values of m
Electron spin
Besides orbital angular momentum, electrons
have spin angular momentum
h
S  s ( s  1)
,
2
1
s  has only one value
2
We can know only the z
component of the spin:
h
1
S z  ms
, ms  
2
2
Clicker
P1. What angle does the upper S cone make vs
the z axis? _______ deg
a) 25-35 b) 35-45 c) 45-55, d) 55-65 e) 65-75
h
S  s ( s  1)
,
2
1
s  has only one value
2
h
1
S z  ms
, ms  
2
2
Quantum states in atoms
To illustrate how bad the idea of the electron “orbiting” the
nucleus is, here is the lowest energy probability:
Because l = 0, there is no angular momentum, so no circular
motion about the nucleus, just motion radially
Quantum states in atoms
Each electron quantum state is labeled by n,l,m,ms
l can be (0 … n-1)
chemistry: n=3, l=1,m=-1, ms=-1/2 would be
3p: _↓_ ___ ___
P2. Including spin, how many total electrons states are there
for n = 3? Count all possible l,m,ms
h
L  l (l  1)
2
h
Lz  m
2
Quantum states in atoms
In H-atom, energy depends only on n
13.6eV
En  
n2
In all other atoms, there is more than one electron, and
electron repulsion makes each electron’s energy depend on all
the other electrons: so state energy depends on n,l,m,ms
Higher kinetic energy is associated with more peaks and
nodes in the wavefunction, radially and angularly
Light and quantum states
Absorption
Light and quantum states
Florescence,
phosphorescence
Florescent minerals
Fluorescent shirts
They are brighter than other cloth because they don’t
scatter much light…they convert much of the visible
spectrum into a single color
Lasers
Lase: Light Amplification by Stimulated Emission
Lasers
Need many atoms in excited
state (“Population inversion”).
Photons bouncing between
mirrors stimulate emission of
other photons that match their
phase, direction.
Laser level diagram
To get a population inversion, need at least electron 3
levels, one of which is long-lived or “metastable” (as
in flourescence)
Diode lasers
Inject energetic electrons into semiconductors: light
emitting diode or laser