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

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“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]