1. Reminder: First exam Wednesday Oct. 7th in class
Announcement: No class this Friday Oct. 2nd!!!!!
Try practice problems instead!! Practice problems will be posted to webpage after class
Homework due Wednesday Oct. 7th
Chapter 5: 12,15, 22, 23, 28

2. Simple Harmonic Oscillator
Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.
Consider the Taylor expansion of an arbitrary potential function:
Near a minimum, V1[x-x0] ≈ 0.

3. Simple Harmonic Oscillator
Consider the second-order term of the Taylor expansion of a potential function:
Letting x0 = 0.
Substituting this into Schrödinger’s equation:
We have:
Let and , which yields:

4. The Parabolic Potential Well
The wave function solutions are: where Hn(x) are Hermite polynomials of order n.

5. The Parabolic Potential Wellyn
|yn |2
The Parabolic Potential Well
Classically, the probability of finding the mass is greatest at the ends of motion (because its speed there is the slowest) and smallest at the center.
Contrary to the classical one, the largest probability for the lowest energy states is for the particle to be at (or near) the center.
Classical result

6. Correspondence Principle for the Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase, however, the solution approaches the classical result. This confirms the Correspondence Principle for the quantum-mechanical simple harmonic oscillator.
Classical result

7. The Parabolic Potential Well
The energy levels are given by:
The zero point energy is called the Heisenberg limit:

8. CHAPTER 6 Structure of the Atom
The Atomic Models of Thomson and Rutherford
Rutherford Scattering
The Classic Atomic Model
The Bohr Model of the Hydrogen Atom
Successes & Failures of the Bohr Model
Characteristic X-Ray Spectra and Atomic Number
Atomic Excitation by Electrons
Niels Bohr ( )
Bohr picture: web.gc.cuny.edu/ ashp/nml/copenhagen/
The opposite of a correct statement is a false statement. But the opposite of a profound truth may well be another profound truth.
An expert is a person who has made all the mistakes that can be made in a very narrow field.
Never express yourself more clearly than you are able to think.
Prediction is very difficult, especially about the future.
- Niels Bohr

9. Structure of the Atom
Evidence in 1900 indicated that the atom was not a fundamental unit:
There seemed to be too many kinds of atoms, each belonging to a distinct chemical element (way more than earth, air, water, and fire!).
Atoms and electromagnetic phenomena were intimately related (magnetic materials; insulators vs. conductors; different emission spectra).
Elements combine with some elements but not with others, a characteristic that hinted at an internal atomic structure (valence).
The discoveries of radioactivity, x-rays, and the electron (all seemed to involve atoms breaking apart in some way).

10. Primitive view of an atom
Knowledge of atoms in 1900
Primitive view of an atom
Electrons (discovered in 1897) carried the negative charge.
Electrons were very light, even compared to the atom.
Protons had not yet been discovered, but clearly positive charge had to be present to achieve charge neutrality.

11. Thomson’s Atomic Model
Thomson’s “plum-pudding” model of the atom had the positive charge spread uniformly throughout a sphere the size of the atom, with electrons embedded in the uniform background.
In Thomson’s view, when the atom was heated, the electrons could vibrate about their equilibrium positions, thus producing electromagnetic radiation.
Unfortunately, Thomson couldn’t explain spectra with this model.

12. Experiments of Geiger and Marsden
Rutherford, Geiger, and Marsden conceived a new technique for investigating the structure of matter by scattering a particles from atoms.

13. Experiments of Geiger and Marsden 2
Geiger showed that many a particles were scattered from thin gold-leaf targets at backward angles greater than 90°.

14. Rutherford’s Atomic Model
Experimental observation of many large angle scattering events!
Experimental results were not consistent with Thomson’s atomic model.
Rutherford proposed that an atom has a small positively charged core (nucleus) surrounded by the negative electrons.
Geiger and Marsden confirmed the idea in 1913.
Ernest Rutherford ( )

15. The Classical Atomic Model
Consider an atom as a planetary system. Like gravity, the force on the electron an inverse-square-law force. This is good. Now, by Newton’s 2nd Law:
where v is the tangential velocity of the electron:
So:
Kinetic energy
This is negative, so the system is bound, which is good.
The potential energy is:
The total energy is then:

16. The planetary atom will emit light of a particular frequency.
An electron in an orbit will emit a light wave at the orbital frequency.
Like planets in the solar system, the electron’s orbital frequency will vary according to the electron’s distance from the nucleus.
So this model could, in principle, explain atoms’ discrete spectra. But all frequencies seem possible…

17. The Planetary Model is Doomed!
Because an accelerated electric charge continually radiates energy (electromagnetic radiation), the total energy must continually decrease. So the electron radius must continually decrease!
The electron crashes into the nucleus!
Physics had reached a turning point in 1900 with Planck’s hypothesis of the quantum behavior of radiation, so a radical solution would be considered possible.