Elementary Quantum Theory (2/4)Presenter: Charlie Payne
September 22, 2017
2017 Student Seminar
1/19
Outline1) Continuity Equation and Born’s Rule2) Angular Momentum3) TISE for Central Potential4) Angular ODE and Spherical Harmonics5) Radial ODE6) Hydrogen Atom
2/19
Continuity Equation
“density”
“current”
3/19
[source: Thomson]
Continuity Equation (cont’d)
trick!
for the TDSE
B
4/19
Born’s Rule
probability density
the probability of detecting the particle between a and b
5/19
Angular Momentum(first quantization)
6/19
Angular Momentum (cont’d)
The components of ang. mom. are NOT compatible => they are NOT simultaneously diag.
“Levi-Civita symbol”
7/19
Angular Momentum (cont’d)
turns out!
The components of ang. mom. ARE compatible (sim. diag.) with full ang. mom. ^2
B
8/19
Angular Momentum (cont’d)
B
ladder operators(wiki. it)
=>
9/19
Spherically Symmetric Central Potential
(can show)by the Heisenberg EoM
10/19
[source: interweb]
Spherically Symmetric Central Potential (cont’d)
(TISE) We’ll need more quantum numbers...
...therefore we’ll leave them blank for now. #Lazyansatz for SoV
11/19
Laplacian in Spherical Coordinates
Left as an exercise for the masochistic…
(likewise with the SoV math)
12/19
Angular Equation (from SoV)
recognize all these derivatives from an earlier slide!?
We can identify the Y-part of the solution to the TISE as our angular momentum eigenstates! :O
13/19
“Spherical Harmonics”
“Associated Legendre Polynomials”
Condon-Shortley Phase
(can show)
14/19
Rodrigues Formula
15/19
[source: interweb]
Radial Equation (from SoV)
(undoing natural units now... ) “Effective
Potential”
16/19
Hydrogen Atom
17/19
Coulomb Potential
“Associated Laguerre Polynomials”Rodrigues Formula
“Radial Wave Function”
Bohr Model (Reproduced!)(via analysis of radial eq’n / wave-FN)
discretized energy levels of H-orbitals
18/19
“Bohr Radius”
radial probability
B
Thanks! Questions?
19/19[source: interweb]
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