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.