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

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

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 | 2 + u 010 2 + u 001 2 ∝ 𝑟 2 exp⁡(− 𝑟 2 𝜆 2 ) 𝑢 100 2 𝑢 010 2 average 𝑥,𝑦 plane at 𝑧=1

Last time Schrödinger equation in spherical polar coordinates Derivation of Laplacian in spherical polars - Foundation Mathematics, K. F. Riley and M. P. Hobson section 11.8.2, 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

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 1 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

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

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)= 3 4𝜋 1 2 cos 𝜃 𝑌 1−1 (𝜃,𝜙)= 3 8𝜋 1 2 sin 𝜃 𝑒 −𝑖𝜙 𝑌 11 (𝜃,𝜙)= − 3 8𝜋 1 2 sin 𝜃 𝑒 𝑖𝜙

Spherical harmonics and atomic orbitals 𝒍 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 1 2 sin 𝜃 𝑒 ±𝑖𝜙 𝑝 𝑥 = 3 4𝜋 1/2 sin 𝜃 cos 𝜙 𝑝 𝑦 = 3 4𝜋 1/2 sin 𝜃 sin 𝜙

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

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

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).