1. Chapter 27 Early Quantum Theory
Lecture Notes
Early Quantum Theory 27

2. Chapter Topics Discovery and Properties of the Electron
Planck’s Quantum Hypothesis
Photon Theory of Light
Energy, Mass, and Momentum of a Photon
Photon Interactions
Wave Nature of Matter
The Bohr Model

3. Discovery and Properties of the Electron
In the late 19th century, discharge tubes were made that emitted “cathode rays.”

4. Discovery and Properties of the Electron
It was found that these rays could be deflected by electric or magnetic fields.

5. Discovery and Properties of the Electron
By accelerating the rays through a known potential and then measuring the radius of their path in a known magnetic field, the charge to mass ratio could be measured: E B r
Magnetic force
Velocity selector

6. Discovery and Properties of the Electron
Cathode rays were soon called electrons.
Millikan devised an experiment to measure the charge on the electron by measuring the electric field needed to suspend an oil droplet of known mass between parallel plates.
The mass and charge of each droplet were measured; careful analysis of the data showed that the charge was always an integral multiple of a smallest charge, e.
The currently accepted value of e and m are:

7. electric field of 320 V/m will make the path straight?
Problem 27-01
What is the value of e/m for a particle that moves in a circle of radius 7.0 mm in a 0.86-T magnetic field if a perpendicular
electric field of 320 V/m will make the path straight? E B r

8. Problem 27-03
An oil drop whose mass is determined to be 2.8 x kg is held at rest between two large plates separated by 1.0 cm when the potential difference between them is 340 V. How many excess electrons does this drop have?

9. Planck’s Quantum Hypothesis Stefan-Boltzmann constant
All objects emit radiation whose total intensity is proportional to the fourth power of their temperature. This is called thermal radiation; a blackbody is one that emits thermal radiation only.
Stefan-Boltzmann equation
Power
Absolute temperature
Surface area
Emissivity
Stefan-Boltzmann constant
The spectrum of blackbody radiation has been measured; it is found that the frequency of peak intensity increases linearly with temperature.

10. Planck’s Quantum Hypothesis
This figure shows blackbody radiation curves for three different temperatures. Note that frequency increases to the left.

11. Planck’s Quantum Hypothesis
This spectrum could not be reproduced using 19th-century physics.
A solution was proposed by Max Planck in 1900:
The energy of atomic oscillations within atoms cannot have an arbitrary value; it is related to the frequency:
The constant h is now called Planck’s constant.
Planck’s proposal was that the energy of an oscillation had to be an integral multiple of hf. This is called the quantization of energy.

12. Photon Theory of Light
Einstein suggested that, given the success of Planck’s theory, light must be emitted in small energy packets:
These tiny packets, or particles, are called photons.

13. Photon Theory of Light
The photoelectric effect: If light strikes a metal, electrons are emitted. The effect does not occur if the frequency of the light is too low; the kinetic energy of the electrons increases with frequency.
Photoelectric equation

14. Photon Theory of Light If light is a wave, theory predicts:
a. Number of electrons and their energy should increase with intensity.
b. Frequency would not matter.
If light is particles, theory predicts:
a. Increasing intensity increases number of electrons but not energy.
b. Above a minimum energy required to break atomic bond, kinetic energy will increase linearly with frequency.
c. There is a cutoff frequency below which no electrons will be emitted, regardless of intensity.

15. Photon Theory of Light
The particle theory assumes that an electron absorbs a single photon.
Plotting the kinetic energy vs. frequency:
This shows clear agreement with the photon theory, and not with wave theory.

16. Problem 27-20
In a photoelectric-effect experiment it is observed that no current flows unless the wavelength is less than 570 nm.
(a) What is the work function of this material (in eV)?

17. Problem 27-20
In a photoelectric-effect experiment it is observed that no current flows unless the wavelength is less than 570 nm.
(b) What is the stopping voltage required if light of wavelength 400 nm is used?

18. Problem 27-22
Barium has a work function of 2.48 eV. What is the maximum kinetic energy of electrons if the metal is illuminated by UV light of wavelength 365 nm? What is their speed?
Energy of the photon
Maximum KE of the photoelectrons
Electron’s speed

19. Energy, Mass, and Momentum of a Photon
Clearly, a photon must travel at the speed of light. Looking at the relativistic equation for momentum, it is clear that this can only happen if its rest mass is zero.
We already know that the energy is hf; we can put this in the relativistic energy-momentum relation and find the momentum:

20. Photon Interactions
Photons passing through matter can undergo the following interactions:
A. The photon can produce an electron-positron pair
B. Photoelectric effect: photon is completely absorbed,
electron is ejected
C. Photon may be totally absorbed by electron,
but not have enough energy to eject it;
the electron moves into an excited state
D. The photon can scatter from an atom and lose some energy

21. Photon Interactions
In pair production, energy, electric charge, and momentum must all be conserved.
Energy will be conserved through the mass and kinetic energy of the electron and positron; their opposite charges conserve charge; and the interaction must take place in the electromagnetic field of a nucleus, which can contribute momentum.

22. Problem 27-34
What is the minimum photon energy needed to produce a m+ and m- pair? The mass of each (muon) is 207 times the mass of the electron. What is the wavelength of such a photon?
Photon energy
Wavelength

23. Wave Nature of Matter
Just as light sometimes behaves as a particle, matter sometimes behaves like a wave.
The wavelength of a particle of matter is:
This wavelength is extraordinarily small.
The wave nature of matter becomes more important for very light particles such as the electron.

24. Problem 27-38
What is the wavelength of a neutron ( m = 1.67 x kg )
traveling at 6.5 x 104 m/s.
De Broglie wavelength

25. Problem 27-42
What is the wavelength of an electron of energy 100 eV?
De Broglie wavelength

26. The Bohr Model
Bohr proposed that the possible energy states for atomic electrons were quantized – only certain values were possible. Then the spectrum could be explained as transitions from one level to another.

27. Hydrogen Atom E E1 = 10.2 eV E2 = 12.1 eV E3 = 12.75 eV 0 eV n =  r4
Ground
state
n = 1
-13.6 eV

28. Hydrogen Atom E n =  0 eV r4 -0.54 eV 410 n = 5 434 n = 4 -0.85 eV r3
486
n = 3
-1.5 eV
656 r2
n = 2
-3.4 eV r1
Ground
state
n = 1
-13.6 eV

29. The Bohr Model
The lowest energy level is called the ground state; the others are excited states.

30. END