1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Oct 31, 2012
Karine Chesnel

3. Phys 451
Announcements
Homework this week:
HW #16 Friday Nov 2 by 7pm

4. Phys 451
Position- momentum
in 3 dimensions
Pb 4.1

5. Schrödinger equation in 3 dimensions z y x Phys 451 Each stationary
state verifies

6. Schroedinger equation in cartesian coordinates
Phys 451
Schroedinger equation
in cartesian coordinates
Pb 4.2
Infinite cubical well:
V=0 inside a box
Separation of variables

7. Schrödinger equation in spherical coordinates
Phys 451
Schrödinger equation in
spherical coordinates x y z r
The radial equation
The angular equation

8. The angular equation Phys 451 y z r Further separation of variables: x
F equation:
m integer (revolution)
q equation:

9. The angular equation Phys 451 y x z r Physical condition l,m integers
Solution:
Legendre function
Legendre polynomial
Physical condition
l,m integers

10. The angular equation Phys 451 y z r x Legendre function Solution:
are polynoms in cosq
(multiplied by sinq if m is odd)

11. Spherical harmonics l m Phys 451 y z r x Azimuthal quantum number
Magnetic quantum number l m
Simulation:

12. Spherical harmonics Pb 4.3 Quantum mechanics y z rx y z r
Method to build your spherical harmonics:
1. Legendre polynomial
2. Legendre function
3. Plug in cosq
4. Normalization factor
Pb 4.3

13. Which one of the following quantities
Phys 451
Quiz 21
Which one of the following quantities
could not physically correspond to a spherical harmonic? A. B. C. D. E.

14. The radial equation Phys 451 y The radial equation z r x
Change of variables
Centrifugal
term
Form identical to Schrödinger equation !
with an effective potential

15. Phys 451
Normalization z r y x
Radial part or
Angular part

16. Orthonormality Pb 4.3 and 4.5 Phys 451 yz r y x
Spherical harmonics are orthogonal
Angular part
Pb 4.3 and 4.5