The “central potenial” problem (example: Hydrogen Atom). This is a potential defined by a radius from a central point, and only by the radius: V(radius)
Or an equation
As in the Bohr model, the values of l correspond to the angular momentum quantum number, but the relation is not quite the same for small l, with a 1 added
Vector model picture
Suppose the force is Coulomb, the V varies like 1/r.
When the equation for r is solved, with solutions in terms of the Laguerre polynomials (powers in r times exponentials), it is found that l?n-1 so n=1,2,3,…, l=0,1,2…n-1, ml=-l,-l+1…0…l and exactly as in the Bohr model, the energy states do not depend on l and:
Spin
Suppose we have a magnetic field along the z axis. Then the energy of the state will depend on the component of ? along B.
The “splitting” of atomic states in a magnetic field is call the Zeeman effect
A “gradient magnet”
Stern used silver for his atoms, detected them with a glass plate, where the beams made silvery spots:
The total angular momentum, J, is the sum of the orbital angular momentum, L, and the spin, S, a vector sum
Many electron atoms, where many can be two, the helium atom. Figuring out the energy levels from observation of the spectral lines, we find it looks like two kinds of hydrogen-like spectra, which were once supposed to be two kinds of helium, para-helium and ortho-helium:
Explanation: has to do with the symmetry of identical particles. Take a spin 1/2 particle. It can be up or down, u or d. Suppose we have two such, 1 and 2. Then there are four combinations possible: u1xu2, u1xd2, u2xd1, d1xd2
We can fix this: take combinations which do have the needed symmetry:
Then it will be true that
When are symmetrical or antisymmetrical wavefunctions found? A remarkable regularity is found. When two identical particles have spin = 1/2, 3/2… the OVERALL wavefunction is ANTISYMMETRICAL, when their spin = 0,1.. the overall wavefunction is SYMMETRICAL.
the two helium spectra correspond to the S=0 (para) for the two electrons, and the S=1 (ortho) states
Electron shells in atoms
Email: willis@nevis1.columbia.edu
Home Page: www.nevis.columbia.edu/~willis
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