1. Chapter 7
Atomic Physics

2. Schrödinger Equation in 3D infinite square well
− ℏ 2 2𝑚 𝜕 2 𝜓 𝜕 𝑥 𝜕 2 𝜓 𝜕 𝑦 𝜕 2 𝜓 𝜕 𝑧 2 +𝑉𝜓=𝐸𝜓
𝜓 𝑥,𝑦,𝑧 = 𝜓 1 𝑥 𝜓 2 𝑦 𝜓 3 𝑧
𝜓 𝑥,𝑦,𝑧 =𝐴𝑠𝑖𝑛 𝑘 1 𝑥 𝑠𝑖𝑛 𝑘 2 𝑦 𝑠𝑖𝑛 𝑘 3 𝑧
𝐸 𝑛 1 𝑛 2 𝑛 3 = ℏ 2 𝜋 2 2𝑚 𝑛 𝐿 𝑛 𝐿 𝑛 𝐿 3 2
𝑘 𝑖 = 𝑛 𝑖 𝜋 𝐿 𝑖

3. Schrödinger Equation in 3D Spherical Coordinate
Electrons in hydrogen-like atom
− ℏ 2 2𝜇 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕𝜓 𝜕𝑟 − ℏ 2 2𝜇 1 𝑟 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛𝜃 𝜕𝜓 𝜕𝜃 𝑠𝑖𝑛 2 𝜃 𝜕 2 𝜓 𝜕 𝜙 2 +𝑉𝜓=𝐸𝜓
𝑉 𝑟 =− 𝑍𝑘 𝑒 2 𝑟

4. Schrödinger Equation in 3D Spherical Coordinate
− ℏ 2 2𝜇 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕𝜓 𝜕𝑟 − ℏ 2 2𝜇 1 𝑟 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛𝜃 𝜕𝜓 𝜕𝜃 𝑠𝑖𝑛 2 𝜃 𝜕 2 𝜓 𝜕 𝜙 2 − 𝑍𝑘 𝑒 2 𝑟 𝜓=𝐸𝜓
𝜓 𝑟,𝜃,𝜙 =𝑅 𝑟 𝑓 𝜃 𝑔 𝜙
1 𝑅 𝑑 𝑑𝑟 𝑟 2 𝑑𝑅 𝑑𝑟 + 2𝜇 𝑟 2 ℏ 2 𝐸−𝑉 =𝑙 𝑙+1
1 𝑔 𝑑 2 𝑔 𝑑 𝜙 2 =− 𝑚 2
𝑔 𝑚 𝜙 = 𝑒 𝑖𝑚𝜙
𝑓 𝑙𝑚 𝜃 = 𝑠𝑖𝑛𝜃 𝑚 2 𝑙 𝑙! 𝑑 𝑑 𝑐𝑜𝑠𝜃 𝑙+ 𝑚 𝑐𝑜𝑠 2 𝜃−1 𝑙
𝑙 𝑙+1 𝑠𝑖𝑛 2 𝜃+ 𝑠𝑖𝑛𝜃 𝑓 𝑑 𝑑𝜃 𝑠𝑖𝑛𝜃 𝑑𝑓 𝑑𝜃 = 𝑚 2

5. Schrödinger Equation in 3D Spherical Coordinate
𝑌 𝑙𝑚 𝜃,𝜙 = 𝑓 𝑙𝑚 𝜃 𝑔 𝜙 = 𝑒 𝑖𝑚𝜙 𝑠𝑖𝑛𝜃 𝑚 2 𝑙 𝑙! 𝑑 𝑑 𝑐𝑜𝑠𝜃 𝑙+ 𝑚 𝑐𝑜𝑠 2 𝜃−1 𝑙
𝜓 𝑟,𝜃,𝜙 = 𝑅 𝑛𝑙 𝑟 𝑌 𝑙𝑚 𝜃,𝜙
See Table 7-1 for some of the 𝑌 𝑙𝑚 𝜃,𝜙 functions

6. Radial Function of Schrödinger Equation
1 𝑅 𝑑 𝑑𝑟 𝑟 2 𝑑𝑅 𝑑𝑟 + 2𝜇 𝑟 2 ℏ 2 𝐸−𝑉 =𝑙 𝑙+1
− ℏ 2 2𝜇 𝑟 2 𝑑 𝑑𝑟 𝑟 2 𝑑𝑅 𝑟 𝑑𝑟 + − 𝑘𝑍 𝑒 2 𝑟 + ℏ 2 𝑙 𝑙+1 2𝜇 𝑟 2 𝑅 𝑟 =𝐸𝑅 𝑟
𝐸 𝑛 =− 𝑘𝑍 𝑒 2 ℏ 2 𝜇 2 𝑛 2 =− 𝑍 2 𝐸 1 𝑛 2

7. Radial Function of Schrödinger Equation
See Table 7-2 for more 𝑅 𝑛𝑙 𝑟 functions
𝑅 10 = 2 𝑎 𝑒 −𝑟/ 𝑎 0
𝑅 20 = 𝑎 − 𝑟 2 𝑎 0 𝑒 −𝑟/ 2𝑎 0
𝑅 21 = 𝑎 𝑟 𝑎 0 𝑒 −𝑟/ 2𝑎 0

8. Summary of the Quantum Numbers
𝜓 𝑟,𝜃,𝜙 = 𝑅 𝑛𝑙 𝑟 𝑌 𝑙𝑚 𝜃,𝜙
Principal quantum number:
𝑛=1,2,3,…
Angular momentum (orbital) quantum number:
𝑙=0,1,2,3,…,(𝑛−1)
Magnetic quantum number:
𝑚=−𝑙,−𝑙+1,…,−2,−1,0,1,2,…,𝑙−1,𝑙

9. Periodic Table and Quantum Numbers

10. Angular Momentum
− ℏ 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛𝜃 𝜕 𝜕𝜃 𝑠𝑖𝑛 2 𝜃 𝜕 2 𝜕 𝜙 2 𝑌 𝑙𝑚 𝜃,𝜙 =𝑙 𝑙+1 ℏ 2 𝑌 𝑙𝑚 𝜃,𝜙
𝐿 2 𝑜𝑝 𝑌 𝑙𝑚 𝜃,𝜙 =𝑙 𝑙+1 ℏ 2 𝑌 𝑙𝑚 𝜃,𝜙 = 𝐿 2 𝑌 𝑙𝑚 𝜃,𝜙
𝐿= 𝑙 𝑙+1 ℏ
𝐿 𝑧 =𝑚ℏ

11. Spectroscopy transitions and selection rules
∆𝑚=0 𝑜𝑟 ±1
∆𝑙=±1

12. Electron Spin
𝑆 =𝑆= 𝑠 𝑠+1 ℏ
𝑚 𝑠 =± 1 2 ℏ
𝑠= 1 2

13. Complete Hydrogen Atom Wave Function
𝜓 100 𝑟,𝜃,𝜙 = 𝐶 100 𝑅 10 𝑟 𝑌 00 𝜃,𝜙
𝜓= 𝐶 1 𝜓 𝑛𝑙 𝑚 𝑙 𝑚 𝑠1 + 𝐶 2 𝜓 𝑛𝑙 𝑚 𝑙 𝑚 𝑠2
𝜓= 𝐶 1 𝜓 100+1/2 + 𝐶 2 𝜓 100−1/2
Pauli Exclusion Principle: Valid for particles with half integer spin (Fermions)
No more than one electron may occupy a given quantum state specified by a particular set of single-particle quantum numbers, n, l, ml, ms

14. Magnetic Moment Define: Magnetic dipole moment 𝜇 =𝐼 𝐴
A loop of current is a magnetic dipole moment!!
Electrons in an atom
Orbital motion:
𝜇 = 𝑞 2 𝑚 𝑒 𝐿
𝜇 =− 𝑒 2 𝑚 𝑒 𝐿

15. Magnetic Moment of Electron Orbital
𝜇 𝐿 = 𝑒 2 𝑚 𝑒 𝐿= 𝜇 𝐵 ℏ 𝐿= 𝜇 𝐵 ℏ 𝑙 𝑙+1 ℏ= 𝑙 𝑙+1 𝜇 𝐵
In general:
𝜇 𝐵 = 𝑒ℏ 2 𝑚 𝑒
𝜇= 𝑙 𝑙+1 𝑔 𝜇 𝐵
Bohr magneton:
𝜇 =− 𝑔 𝜇 𝐵 ℏ 𝐿
𝜇 𝑧 =−𝑚𝑔 𝜇 𝐵
𝜇 𝐿𝑧 =− 𝑒ℏ 2 𝑚 𝑒 𝑚=−𝑚 𝜇 𝐵
Gyromagnetic ratio (g-factor):
𝜇 𝐿 = 𝑙 𝑙+1 𝜇 𝐵
𝑔 could be 𝑔 𝐿 or 𝑔 𝑆
𝜇 𝐿𝑧 =− 𝜇 𝐵 𝑚
𝑔 𝐿 =1

16. Magnetic Moment of Electron Spin
𝜇 𝑆 = 𝑔 𝑆 𝜇 𝐵 ℏ 𝑆= 𝑔 𝑆 𝑠 𝑠+1 𝜇 𝐵 = 𝑔 𝑆 𝜇 𝐵
𝜇 𝑆 =− 𝑔 𝑆 𝜇 𝐵 ℏ 𝑆
𝜇 𝑆𝑧 =− 𝑔 𝑆 𝑚 𝑠 𝜇 𝐵 =± 1 2 𝑔 𝑆 𝜇 𝐵
From Stern-Gerlach experiment:
𝑔 𝑆 =

17. Total Angular Momentum
𝐽 = 𝐿 + 𝑆
𝐽 = 𝑗 𝑗+1 ℏ
𝑗=𝑙+𝑠 or
𝑗= 𝑙−𝑠
𝐽 𝑧 = 𝑚 𝑗 ℏ
Total angular momentum of two electrons with different L
𝐽 = 𝐿 𝐿 2
𝑗= 𝑙 1 + 𝑙 2 , 𝑙 1 + 𝑙 2 −1, …, 𝑙 1 − 𝑙 2

18. Magnetic Moment in Magnetic Field
𝜏 = 𝜇 × 𝐵
𝑈 𝑚 =− 𝜇 ∙ 𝐵

19. Stern-Gerlach Experiment
𝐹 =−𝛻 𝑈 𝑚 =−𝛻 − 𝜇 ∙ 𝐵
𝐹 𝑧 = 𝜇 𝑧 𝑑𝐵/𝑑𝑧 =−𝑚𝑔 𝜇 𝐵 𝑑𝐵/𝑑𝑧

20. Stern-Gerlach Experiment
Reality for Silver (l = 0):
Expected:
Electron Spin made it so!
l = 1
l = 0

21. Electron Configurations and Hund’s Rule
Ag: [Kr] 4d10 5s1
Hund’s Rule (finding the ground state electron configuration):
Maximize total electron spin quantum number
Maximize total orbital quantum number
For the outermost subshell, if it is half-filled or less, lowest J has the lowest energy; otherwise, the highest J has the lowest energy.
O: [He] 2s2 2p4
N: [He] 2s2 2p3
C: [He] 2s2 2p2

23. Spectroscopic Notation
𝑛 2𝑠+1 𝑙 𝑗
1 2 𝑆 1/2
(n=1; l=0; s=1/2; j=1/2)
2 2 𝑆 1/2
(n=2; l=0; s=1/2; j=1/2)
2 2 𝑃 1/2
(n=2; l=1; s=1/2; j=1/2)
2 2 𝑃 3/2
(n=2; l=1; s=1/2; j=3/2)

24. Spin-Orbit Coupling
𝑈 𝑚 =− 𝜇 ∙ 𝐵