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Chapter 29

Particles and Waves

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There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.

-- William Thomson, Lord Kelvin (Address at the British Association for the Advancement of Science, 1900).

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Quantum Physics• 2nd revolution in physics:

– Starts with Planck ~1900

– Contributions from Einstein, Bohr, Heisenberg, Schrödinger, Born, Dirac, de Broglie …. over 25 years

• Cornerstones:– Wave-particle duality– Uncertainty principle

• Correspondence– Applies for small dimensions

• Planck’s constant: h = 6.6 x 10-34Js• As h -> 0, quantum physics -> classical

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Front row: I. Langmuir, M. Planck, M. Curie, H. A. Lorentz, A. Einstein, P. Langevin, C. E. Guye, C. T. R. Wilson, O. W. Richardson. Second row: P. Debye, M. Knudsen, W. L. Bragg, H. A. Kramers, P. A. M. Dirac, A. H. Compton, L. V. de Broglie, M. Born, N. Bohr. Standing: A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, T. De Donder, E. Schroedinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R. H. Fowler, L. Brillouin.

Solvay conference, 1927

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Front row: I. Langmuir, M. Planck, M. Curie, H. A. Lorentz, A. Einstein, P. Langevin, C. E. Guye, C. T. R. Wilson, O. W. Richardson. Second row: P. Debye, M. Knudsen, W. L. Bragg, H. A. Kramers, P. A. M. Dirac, A. H. Compton, L. V. de Broglie, M. Born, N. Bohr. Standing: A. Piccard, E. Henriot, P. Ehrenfest, E. Herzen, T. De Donder, E. Schroedinger, E. Verschaffelt, W. Pauli, W. Heisenberg, R. H. Fowler, L. Brillouin.

Planck Curie Lorentz

Bragg

Einstein

Dirac

Com

pton

de B

rogl

ie

Born Bohr

Schr

ödin

ger

Hei

senb

erg

Pauli

Solvay conference, 1927

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Part I: Particle nature of light

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1. Blackbody radiation

• All objects radiate and absorb electromagnetic radiation

• At equilibrium, rate of absorption = rate of emission• Best absorber is best emitter

– Perfect absorber is perfect emitter

Cavity is model of a perfect blackbody

a) Blackbody

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Plot of intensity vs wavelength– Depends only on temperature

b) Emmitance spectrum: the problem

Experimental spectrum:

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Classical prediction:

The UV catastrophe

Theory

Based on idea that all oscillations equally probable, more oscillations at lower wavelength

Violates common sense and experiment

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Absorption and emission occur in discrete quanta only

c) Energy quantization: the solution

Energy of quanta proportional to frequency

For small wavelength (high freq), quanta are large. If kT < quantum, radiation not possible.

€

E = nhf ; n = 0,1,2,3...

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• Planck found a “fudge factor” by “happy guesswork” to make the experiment fit. He developed a quantization theory to predict the value h.– “lucky artifact of more fundamental reality yet to be discovered”

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E = nhf = hc /λ ; n = 0,1,2,3...

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h = 6.626 ×10−34 Js

d) Planck’s constant, h

• Nobel prize, 1918

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2. Photoelectric effect

a) The effect

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Experiment Expectation Observation

Increase intensity - Max energy increase

- Current increase

- Time lag decrease

- Max energy constant

- Current increase

- No time lag

Increase Frequency

- Max energy constant

- No threshold

b) Expectations and observations

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Observed frequency dependence

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Experiment Expectation Observation

Increase intensity - Max energy increase

- Current increase

- Time lag decrease

- Max energy constant

- Current increase

- No time lag

Increase Frequency

- Max energy constant

- No threshold

- Max energy prop to freq

- Threshold frequency characteristic of metal

b) Expectations and observations

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KEmax = hf - W0

Energy of photon(from Planck)

Work required toremove electron

c) Einstein theory

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• Found same value for h as Planck had

• Nobel prize in 1921

• In 1913, Planck recommended Einstein for membership in the Prussian Academy. “Notwithstanding his genius, he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta.”

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a) The effect: Scattering of x-ray by electron changes the wavelength

3. The Compton Effect, 1923

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b) The experiment

CrystalX-ray source ()

Bragg reflectiongives ’

graphite

Detector

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c) Classical prediction

- incident wave excites electron at frequency f

- electron radiates at frequency f

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d) Compton’s explanation

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hf = h ′ f + KE

- Conservation of Momentum:

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r p =

r ′ p +

r p e

- Energy-momentum relation for light:

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p =E

c=

hf

c=

h

λ

- Conservation of energy:

Combining these equations gives:

€

′ − =h

mc(1− cosϑ )

’

Nobel prize, 1928

Definitive evidence for photons

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Part II: The wave nature of particles

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a) The hypothesis

4. The de Broglie wavelength, 1924

The dual nature observed in light is present in matter:

€

=h

p

By analogy, de Broglie proposed that a particle with momentum p is associated with a wave with wavelength:

A photon has energy

€

E = hf = hc /λ

From electromagnetism,

€

E = pc }€

=h

p

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b) Electron diffraction/interference

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b) Interpretation of the particle wave

Waves and particles propogate and interfere like waves, but interact like particles.

The intensity of the wave (represented by a wave function at a point in space represents the probability of observing a particle at that location.

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5. The Heisenberg Uncertainty Principle

The wave nature of particles means that position and momentum (wavelength) cannot simultaneously be determined to arbitrary accuracy. The smaller the slit above, the better the y-position is known, but the greater the spread in y-momentum.

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€

ΔpyΔy ≥h

4π

€

ΔpxΔx ≥h

4π

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ΔEΔt ≥h

4π

The principle applies separately to any component of momentum and position:

and to energy and time: