Physics 1September 27, 2005Friday, September 30, 4:00 p.m.special homecoming colloquium by former student Gavin Buffington. Room 104 Physics. Coffee and donuts at 3:45. Physics 1 students want to attend.Education is that which remains when one has forgotten everything learned in school. .A. EinsteinThree areas will be discussed: (1) modeling laser damage to human eyes (because you cant do it experimentally!), (2) doing research in an undergraduate-only department, and (3) comments on a post-UMR career.
Chapter 6Quantum Theory of the Hydrogen Atom6.1 Schrdinger's Equationfor the Hydrogen Atom We have discovered a "new" theoryquantum mechanics as exemplified by Schrdinger's equation.We tested it on three simple model systems in Chapter 5. We ought to test it on something a bit more realistic, like an atom.Whats the simplest atom you can think of?Anyone who is not shocked by quantum theory has not understood it.Neils Bohr
The hydrogen atom is the simplest physical system containing interaction potentials (i.e., not just an isolated particle).Simple: one proton, one electron, and the electrostatic (Coulomb) potential that holds them together. The potential energy in this case is just(the attractive potential between charges of +e and e, separated by a distance r).This is a stationary state potential (no time dependence). We could just plug it in to Schrdingers equation to getbut need to let the symmetry of the problem dictate our mathematical approach.
The spherically symmetric potential tells us to use spherical polar coordinates!http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html
Now we can re-write the 3D Schrdinger equation in three dimensions, and in spherical polar coordinates, as
The preceding equation can be solved for the wave function . It turns out that is the product of three separate functions.On the next slide are the possible hydrogen electron wave functions for n=1, 2, and 3 (n=1 is the ground state). = Rn m m ,with these conditions on the quantum numbers n, , and m: = 0, 1, 2, ..., (n-1) n = 1, 2, 3, ... m = 0, 1, 2, 3, ...,
Lets look at the ground state wave function:This function contains all that is knowable about the ground state electron in hydrogen.Note that for the ground state electron is a function of r only. You will learn in quantum mechanics that * evaluated for some state gives the probability of finding the system in that state.The radial part is the only interesting part.
In spherical polar coordinates, P(r)=R*R r2 dr gives the probability of finding the electron within some small dr in space centered at r.Use the 1s radial wave function and Mathcad to show that the 1s electron in hydrogen is most likely to be found at r=a0. Hints: set a0=1 for simplicity. You could use calculus to find where the radial probability function is maximum. In Mathcad it is easier to plot the probability function and see where the maximum value occurs.
The quantum mechanical equivalent of the average value is the expectation value, given byThe expectation value of the radial part of the electrons wave function is
Use Mathcad to show that the expectation value for the 1s electrons radial coordinate is 1.5a0 (i.e., 1 times as far from the nucleus as its most likely coordinate).Hints: set a0=1 for simplicity. Use the integral function, obtained by entering the & symbol in Mathcad. Mathcad complains if you try to integrate from 0 to . If you integrate from 0 to 50 (meaning 0a0 to 50a0 if you set a0=1), you have gone far enough out in r to approach infinity.Does it make sense that the average electron position is different than its most likely position?Absolutely! You may have to wait until you take Modern Physics to see why.