1. Quantum mechanics unit 2
The Schrödinger equation in 3D
The infinite quantum box in 3D
The Hydrogen atom
Schrödinger equation in spherical polar coordinates
Solution by separation of variables
Angular quantum numbers
Radial equation and principal quantum numbers
Hydrogen-like atoms
Rae – Chapter 3

2. Last time Degeneracy States with the same energy are degenerate
Degeneracy is related to symmetry

3. 3D Harmonic Oscillator 𝑽 𝒓 = 𝟏 𝟐 𝒎 𝝎 𝟐 𝒓 𝟐
Show that the first and second excited states are spherically symmetric
|𝑢 100 | u u ∝ 𝑟 2 exp⁡(− 𝑟 2 𝜆 2 ) 𝑢 𝑢
average
𝑥,𝑦 plane at 𝑧=1

4. Last time Schrödinger equation in spherical polar coordinates
Derivation of Laplacian in spherical polars - Foundation Mathematics, K. F. Riley and M. P. Hobson section , p 473.
Volume element from 𝑑𝑥𝑑𝑦𝑑𝑧 to 𝑟 2 𝑑𝑟 sin 𝜃 𝑑𝜃𝑑𝜙
Units of probability density 𝑢 2 are 1/m3 in 3D
Hydrogenic atom – 𝑣(𝑟) independent of 𝜃, 𝜙 so S.E. is separable and 𝑢(𝑟,𝜃,𝜙)=𝑅 𝑟 Θ 𝜃 Φ(𝜙)
Expect 3 quantum numbers:
principal quantum number, 𝑛=1,2,3,…
angular momentum quantum number, 0≤𝑙<𝑛
magnetic quantum number, −𝑙 ≤ 𝑚 ≤𝑙

5. Separation of Schrödinger equation
Radial equation
− ℏ 2 2 𝑚 𝑒 1 R 𝜕 𝜕𝑟 𝑟 2 𝜕𝑅 𝜕𝑟 + 𝑟 2 𝑣 𝑟 −𝐸 =−𝜆
𝜃, 𝜙 equation
− ℏ 2 2 𝑚 𝑒 1 sin 𝜃 1 𝑌 𝜕 𝜕𝜃 sin 𝜃 𝜕𝑌 𝜕𝜃 − ℏ 2 2 𝑚 𝑒 sin 2 𝜃 1 𝑌 𝜕 2 𝑌 𝜕 𝜙 2 =𝜆
𝑌(𝜃,𝜙) represents the angular dependence of the wavefunction in any spherically symmetric potential

6. Separation of Schrödinger equation
ℏ 2 2 𝑚 𝑒 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕 𝜕𝜃 + 𝜆 2 sin 2 𝜃 Θ=𝜈Θ
Φ equation
ℏ 2 2 𝑚 𝑒 𝜕 2 Φ 𝜕 𝜙 2 =−𝜈Φ
Φ wavefunction
Φ 𝜙 = 1 2𝜋 e im𝜙

7. 𝑌 𝑙𝑚 𝜃,𝜙 = 2𝑙+1 4𝜋 𝑙−|𝑚| ! 𝑙+|𝑚| ! 1 2 −1 𝑚 𝑃 𝑙 𝑚 cos 𝜃 𝑒 𝑖𝑚𝜙
𝑌 00 (𝜃,𝜙)= 1 4𝜋 1/2
𝑌 10 (𝜃,𝜙)= 𝜋 cos 𝜃
𝑌 1−1 (𝜃,𝜙)= 𝜋 sin 𝜃 𝑒 −𝑖𝜙
𝑌 11 (𝜃,𝜙)= − 3 8𝜋 sin 𝜃 𝑒 𝑖𝜙

8. Spherical harmonics and atomic orbitals𝒍 𝒎
𝑌 𝑙𝒎
Orbital
Shape
𝑌 00 =1/ 4𝜋 1/2 𝑠 1
𝑌 10 = 3 4𝜋 1/2 cos 𝜃
𝑝 𝑧 ±1
𝑌 1±1 =∓ 3 8𝜋 sin 𝜃 𝑒 ±𝑖𝜙
𝑝 𝑥 = 3 4𝜋 1/2 sin 𝜃 cos 𝜙
𝑝 𝑦 = 3 4𝜋 1/2 sin 𝜃 sin 𝜙

9. Use of spherical harmonics
Quantum mechanics
Planetary physics
e.g. magnetic fields
Space science – gravity fields
Computer graphics

10. Summary
Spherical harmonics describe the angular dependence of the wavefunction for any spherically symmetric potential
Spherical harmonics are orthonormal
0 𝜋 0 2𝜋 𝑌 𝑙𝑚 ∗ 𝑌 𝑙 ′ 𝑚′ sin 𝜃𝑑𝜃𝑑𝜙= 𝛿 𝑙 𝑙 ′ 𝛿 𝑚𝑚′
Angular momentum quantum number, 0≤ 𝑙< 𝑛
Magnetic quantum number, −𝑙≤ 𝑚≤ 𝑙

11. Separation of Schrödinger equation
Radial equation
− ℏ 2 2 𝑚 𝑒 1 R 𝜕 𝜕𝑟 𝑟 2 𝜕𝑅 𝜕𝑟 + 𝑟 2 𝑣 𝑟 −𝐸 =−𝜆 or
− ℏ 2 2 𝑚 𝑒 1 r 2 𝜕 𝜕𝑟 𝑟 2 𝜕𝑅 𝜕𝑟 + 𝑣 𝑟 + ℏ 2 2 𝑚 𝑒 𝑙(𝑙+1) 𝑟 2 𝑅=𝐸𝑅
Additional ‘potential’, ℏ 2 2 𝑚 𝑒 𝑙(𝑙+1) 𝑟 2 , related to angular momentum
L. I. Schiff, Quantum mechanics (McGraw-Hill, 1955).