Introduction to Modern Physics

A (mainly) historical perspective on

- atomic physics - nuclear physics- particle physics

Theories of Blackbody Radiation

Classical disaster !

Quantum solution

Planck’s “Quantum Theory”

The “oscillators” in the walls can only have certain energies – NOT continuous!

1,

5

kThc

eTI

The Photoelectric Effect

Light = tiny particles!

Wave theory: takes too long to get enough energy to eject electronsParticle theory: energy is concentrated in packets -> efficiently ejects electrons!

The Photoelectric Effect

Energy of molecular oscillator, E = nhfEmission: energy nhf -> (n-1)hf Light emitted in packet of energy E = hf

Einstein’s prediction: hf = KE + W (work function)

c = f

Speed of light3 x 108

meter/secondor

30cm (1 foot) per

nanosecond

Wavelength (meter)

Frequency#vibrations/

second

hf = KE + W (work function)

The Photoelectric Effect

Wave Theory Photon Theory

Increase light intensity ->more electrons with more KE

Increase light intensity -> more photons -> more electronsbut max-KE unchanged !

Frequency of light does not affect electron KE Max-KE = hf - W

If f < f(minimum) , where hf(minimum) = W,Then NO electrons are emitted!

X

X

How many photons from a lightbulb?

100W lightbulb, wavelength = 500nm

Energy/sec = 100 Joules

E = nhf -> n = E/hf = E/hc

n = 100J x 500 x 10-9 = 2.5 x 1020 !! 6.63 x 10-34 J.s x 3 x 108 m/s

So matter contains electrons and light can be emitted in “chunks”… so what does this tell us about atoms??

Possible models of the atom

Which one is correct?

Electric potentialV(r) ~ 1/r

The Rutherford Experiment

Distance of closest approach ~ size of nucleus

At closest point KE -> PE, and PE = charge x potential

KE = PE = 1/40 x 2Ze2/R

R = 2Ze2/ (40 x KE) = 2 x 9 x 109 x 1.6 x 10-19 x Z 1.2 x 10-12 J= 3.8 x 10-16 Z meters = 3.0 x 10-14 m for Z=79 (Gold)

The “correct” model of the atom

…but beware of simple images!

Atomic “signatures”

Rarefied gas

Only discrete lines!

An empirical formula!

2

1211

2 nR

2

1211

2 nR

n = 3,4,…

The Origin of Line Spectra

Newton’s 2nd Law and Uniform Circular Motion

F = ma

Acceleration = v2/rTowards center of circle!

How do we get “discrete energies”?

Linear momentum = mv

Radius r

Angular momentumL = mvr

Bohr’s “quantum” condition – motivated by the Balmer formula

2hnmvrL n ,...3,2,1n

Electron “waves” and the Bohr condition

De Broglie(1923): = h/mv

Only waves with a whole number of wavelengths persist

Quantized orbits!

n = 2r

2hnmvrL n

Same!!

Electrostatic force: Electron/Nucleus

COULOMBS LAW

Combine Coulomb’s Law with the Bohr condition:

maF Newton’s 2nd LawCircular motion r

va2

nn rmv

reZe 2

20

)(4

1

2hnmvrL n

nmrnhv2

1

2

20

22

rZn

mZehnrn

mx

xxxx

mehr

10

1931

1234

20

2

1

10529.0

)10602.1)(1011.9)(14.3()1085.8)(10626.6(

(for Z = 1, hydrogen)

Calculate the total energy for the electron:Total Energy = Kinetic + Potential EnergyElectrostatic potential r

ZerQV

00 41

41

Electrostatic potential energy r

ZeeVU2

041

nn r

ZemvE2

0

2

41

21

Total energy

Substitute

12

2

2220

42 18

EnZ

nhmeZEn

eVJoulesxh

meE 6.13181017.28 2

0

4

1

2

6.13neVEn

So the energy is quantized !… now we can combine this with

hchf

EEhf lu

223

0

42 1'1

8'11

nnchmeZEE

hc n

…and this correctly predicts the line spectrum for hydrogen,…and it gets the Rydberg constant R right!…however, it does not work for more complex atoms…

Experimental results

Quantum Mechanics – or how the atomic world really works (apparently!)

De Broglie(1923): = h/mv

Take the wave description of matter for real: Describe e.g. an electron by a “wavefunction” (x), then this obeys:

)()()(2 2

22

xExxUdxd

mh

Schroedinger’s famous equation

Now imagine we confine an electron in a “box” with infinitely hard/high walls:

Waves must end at the walls so:

and the energy levels for these states are:

Discrete energies!

The probabilities for the electron to be at various places inside the box are:

vs. Classical Mechanics

Uniform probability!

Applying the same quantum mechanical approach to the hydrogen atom:

Probability “cloud”

Bohr radius

The “n = 2” state of hydrogen:

Atomic orbitals

Weird stuff!!

Weird stuff!!

Ghosts!!??

Conclusions- Classical mechanics/electromagnetism does not describe atomic behavior- The Bohr model with a “quantum condition” does better…but only for hydrogen- Quantum mechanics gives a full description and agrees with experiment- …but QM is weird!!