Chapter 1 29-Aug-2015.pptjordan/chem1410-f15/chpt1.pdf · Wave-particle duality Baseball weighs...
Transcript of Chapter 1 29-Aug-2015.pptjordan/chem1410-f15/chpt1.pdf · Wave-particle duality Baseball weighs...
Late 1800’s – Several failures of classical (Newtonian) physicsdiscovered
1905 – 1925 – Development of QM – resolved discrepancies between expt. and classical theory
QM – Essential for understanding many phenomena in Chemistry, Biology, Physics
• photosynthesis + vision (electron excitation)• vibrations/rotations (excitation of
nuclear motion)• magnetic resonance imaging• radioactivity• operation of transistors• lasers (CD + DVD players)• van der Waals interactions
Chapter 1
(from M. Johnson, Yale University)
1. Blackbody Radiationheated objects → light
2
3
2
3
8
8
oscd E dc
kTdc
Planck:(1900)
3
3 /
8 11h kT
hd dc e
originally determined by fitting experiment
= frequency = spectral density (energy volume-1
frequency-1)= average energy/osc
k = Boltzmann const.c = speed of light
Emits ∞ energy at all T
Problems where classical physics is inadequate
Classical theory (Rayleigh-Jeans (1905)
spectral density =
OSCE
Planck later showed that his expression is consistent with the energies of the oscillators making up the blackbody object taking on discrete values
346.626 10h x J s
/ E 1h kT
he
T → 0 0
T → ∞kT Classical
result
Planck's constant:
, 0,1, 2,...E nh n
1 ...xe x Bottom result obtained using Taylor series of ex for small x:
http://en.wikipedia.org/wiki/Planck's_constant
/
0
/
0
nh kT
nn
nh kT
n
nh eE E p n
e
denominator = / 2 / 2/
1 11 ... 1 ...1 1
h kT h kTh kTe e x x
x e
numerator =
/ 2 / 3 /
// / 2 /
2/
0 2 3 ...
1 2 3 ...1
h kT h kT h kT
h kTh kT h kT h kT
h kT
h e e e
h eh e e ee
/
/ /1 1
h kT
h kT h kT
h e hEe e
avg. energy of mode of freq. h
Derivation of Planck’s result
Possible energies: 0, h, 2h, 3h, etc.Probability of having energy nh given by
= Boltzmann factor
/
/
nh kT
nh kT
n
ep ne
normalization(sum over all levels)
/E kTe
2. Photoelectric effect
expected behavior
• light is a wave, so each e-
absorbs small fraction ofthe energy
• e- emitted at all , if intensity (I)great enough• KE of ejected e- > with > intensity
observed
• #e- emitted intensity
• e- emitted if > 0 (critical freq.)• KE > with > , and independent of intensity
metal
light e-
Explained by Einstein in 1905light has energy h (E = h) and acts particle-like, enabling its energy to be focused on one e-
energy of ejected electron = Eel = h - , = work function of metal (~IP) critical frequency (implies critical energy) for production of
photoelectrons
3. Heat capacity of solids (classical = 3R; actual → 0 as T → 0) (R = Nak)
Expt.
Classical result
Einstein’s solution:
Assumed vibrations of lattice are quantized, i.e., energies are nh
Figure from Wikipedia
33
vC NkE NkT
de Broglie (1924): particles have a wavelength: hp
Demonstrated by diffraction of e-, He, H2 from crystalline surfaces
e- with KE = 17 eV has = 3 Å, a typical lattice spacingin a crystal interference (diffraction)
large objects – baseballs, cars, etc., have de Brogliewavelengths too small to be detected
• photoelectric effect light can behave as a particle• diffraction of light light can behave as a wave
4. Wave-particle duality
Baseball weighs ~0.14 kg; 100mph = 44.7m/sλ = 1.05x10-39m
Diffraction experiments
double-slit expt. with e-
the e- goes through both slits!!
In 1977 the expt. was done withthe He atoms Each atomgoes through both slits!!
Comment on single flashes on screen(forces e- back to particle nature)
light incident on a single slit
minima:n = ±1, ±2, ±3, … a: well separated peaks
>> a: can’t see diffraction
sin ,na
5. Spectra of atoms + molecules – discrete lines
Figure from Wikipedia
Rydberg series
spectrum H atom
2 21
1 1HR
n n
n1, n integers, n = n1+1, n1+2, n1+3, …
RH = 109,677.581 cm-1
(1 eV = 8066 cm-1)13.6 eV
IP
Explained by Bohr model (1911)
22
2
v4
e
o
mer r
Coulombforce
centrifugalforce
e-+
Valid solutions: the wave perfectly fits in the “orbit”
2
v2e
hr n np
nhm r n
Bohr based his result on the correspondencePrinciple, rather than the de Broglie relation,which wasn’t proposed until several years later..See plato.Stanford.edu/entries/bohr~correspondence/)
2 20
2
4
e
nrm e
22
2 2 20 0
, 1, 2,38 8
tot kinetic pot
e
E E E
m ee nr h n
gives the definition of Rydberg’s constant
2 22
2 2 30
2 202
v
4
4
e
e
e
e
nm r
m ner m r
nre m
Summary:
• energy + oscillators are quantized
• wave-particle duality
• de Broglie relationship
• these ideas paved the way for QM
NOTE: Frequencies of a guitar string are“quantized” (andguitars are clearlyClassical)
Quantization comes from boundary conditions
Fourier transforms:(frequency time)(position momentum)are conjugate variables
We will come backto these considerations.