Lecture #5
 Textbook
Chapter 12 now No! No! No!
  CURMUDGEON GENERAL'S WARNING. These "slides" represent highlights from lecture and are neither complete nor meant to replace lecture. It is advised not to use these as a reliable means to replace missed lecture material. Do so at risk to healthy academic performance in 09-105.
Lecture outline Wave Nature of Matter

Schrodinger's Wave Equation applied to the hydrogen atom

Electron "spin"

ms, spin quantum number

Energies and geometries

Shapes

Heisenberg Uncertainty Principle
If we go to four dimensions (x,y,z,t), a fourth quantum number arises.
The fourth quantum number is what is familiarly referred to as the "spin" of an electron. Spin can be pictured (erroneously, but with little damage and much to be gained for convenience) as being possible in either a clockwise or counterclockwise sense...just two choices...sometimes referred to as "up" and "down" pointed spins.
The 1s wave function (or probability amplitude) as a function of distance from the nucleus.
The square of the 1s wave function or the probability density of the electron in a 1s state
The 2s wave function (or probability amplitude) as a function of distance from the nucleus.
The square of the 2s wave function or the probability density of the electron in a 2s state
The 3s wave function (or probability amplitude) as a function of distance from the nucleus.
The square of the 3s wave function or the probability density of the electron in a 3s state
The square of the 2pz wave function or the probability density of the electron in a 2pz state.
A comparison of the electron density distribution for the 2s and 2p waves. The 2s has a larger average radius than the 2p, but also a greater probability of being found close to the nucleus.
Shape and size of all the hydrogen 2p contours
Shape and size of all the hydrogen 3d contours.
Briefly...what is quantum mechanics?
The Uncertainty Principle is an inherent part of wave-particle duality
The Uncertainty Principle (Section 12.5)
An example calculation showing how location in space and knowledge of momentum are intertwined through wave-particle duality and the Uncertainty Principle
Conclusion of the calculation showing a 16% uncertainty in momentum is unavoidable
A calculation exploring whether or not we can improve our knowledge by using a higher resolution probe of geometry. The uncertainty in momentum is seriously worse than before. (The spread in velocities is about 3 million m/s compared to the average of 1 million m/s)
With wave-particle duality, the description of where a particle is moving -- its pathway through space -- is meaningless. (Absolutely meaningless in the realm where wavelengths are comparable to the size of the system, like atoms and molecules, but unlike baseball) We know only the region of space in which the particle is probably found. We are obligated to talk about probabilities. The concept of orbits, like Bohr's planetary paths, are replaced with wave "orbitals". The word orbitals is used to mean the wave that describes a bound particle.
The probability density measures what fraction of the "electron" is found in a cubical volume element around a particular point (x,y,z) with respect to the nucleus.
We can also write the wave amplitude in spherical coordinates. For a central nucleus, the wave ammplitude then factors into two parts, one of which depends only on the distance r between the nucleus and electron. (You do not need to know the blue-striped equation.)
Another way of describing the density is then by the fraction of the electrons distribution that is in a spherical shell volume element at distance r from the nucleus and covering all angles. (From the previous slide, R is that part of the wave function that depends only on distance from the nucleus, r.)
A graphical comparison of the density distribution (amplitude squared) and the radial density distribution (amplitude squared multiplied by r-squared) for a 1s orbital. The radial density distribution graph is also referred to in the text as an electron density plot. (p. 265)
The radial density distribution for the n = 2 orbitals.
The radial density distributions for the n=3 orbitals.
A close-up of the radial density distribution near the nucleus. The blue "cloud" represents where the inner, 1s electrons would most likely be found.