1. Quantum mechanics I Fall 2012
Physics 451
Quantum mechanics I
Fall 2012
Nov 9, 2012
Karine Chesnel

2. HW #18 today Nov 9 by 7pm Homework next week:
Phys 451
Announcements
HW #18 today Nov 9 by 7pm
Homework next week:
HW #19 Tuesday Nov 13 by 7pm
HW #20 Thursday Nov 15 by 7pm

3. The hydrogen atom Phys 451 How to find the stationary states?
Step1: determine the principal quantum number n
Step 2: set the azimuthal quantum number l (0, 1, …n-1)
Step 3: Calculate the coefficients cj in terms of c0
(from the recursion formula, at a given l and n)
Step 4: Build the radial function Rnl(r) and normalize it (value of c0)
Step 5: Multiply by the spherical harmonics (tables)
and obtain 2l +1 functions Ynlm for given (n,l)
(Step 6): Eventually, include the time factor:

4. Phys 451
The hydrogen atom
Representation of
Bohr radius

5. The hydrogen atom Expectation values Pb 4.13 Most probable values
Quantum mechanics
The hydrogen atom
Expectation values
Pb 4.13
Most probable values
Pb 4.14

6. Expectation values for potential
Quantum mechanics
The hydrogen atom
Expectation values for potential
Pb 4.15

7. Phys 451
The angular momentum
Pb 4.19

8. Phys 451
The hydrogen atom
Anisotropy
along Z axis
Representation of

9. The angular momentum Phys 451 Ladder operator
If eigenvector of L2, then eigenvector of L2, same eigenvalue
If eigenvector of Lz with eigen value m
then eigenvector of Lz, new eigenvalue

10. The angular momentum Pb 4.18 Phys 451 Ladder operator Eigenstates Top
Value
=+l
Bottom
Value = -l
Ladder operator
Eigenstates
Pb 4.18

11. Quiz 25 Phys 451 When measuring the vertical component
of the angular momentum (Lz )
of the state , what will we get?
A. 0 B. D. E.

12. in spherical coordinates
Phys 451
The angular momentum
in spherical coordinates x y z r

13. In spherical coordinates
Phys 451
The angular momentum
In spherical coordinates x y z r
Pb 4.21, 4.22

14. The angular momentum eigenvectors Phys 451 y z r x and
were the two angular equations for the spherical harmonics!
Spherical harmonics
are the
eigenfunctions

15. and Schrödinger equation
Phys 451
The angular momentum
and Schrödinger equation x y z r
3 quantum numbers (n,l,m)
Principal quantum number n: integer
Azimutal and magnetic quantum numbers (l,m)
can also be half-integers.