1. Quantum mechanics unit 2
The Schrödinger equation in 3D
Infinite quantum box in 3D & 3D harmonic oscillator
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 Time independent Schrödinger equation in 3D
− ℏ 2 2𝑚 𝛻 2 𝑢 𝐫 +𝑉 𝐫 𝑢 𝐫 =𝐸𝑢 𝐫
u 𝐫 must be normalised, u 𝐫 and its spatial derivatives must be finite, continuous and single valued
If 𝑉 𝐫 = 𝑉 1 𝑥 + 𝑉 2 𝑦 + 𝑉 3 𝑧 then 𝑢 𝐫 =𝑋 𝑥 𝑌 𝑦 𝑍 (𝑧) and the 3D S.E. separates into three 1D Schrödinger equations
- obtain 3 different quantum numbers, one for each degree of freedom
Time independent wavefunctions also called stationary states

3. 3D quantum box 𝑉 𝑥,𝑦,𝑧 =0 if −𝑎≤𝑥≤𝑎, −𝑏≤𝑦≤𝑏 and −𝑐≤𝑧≤𝑐
𝐸 𝑛 1 , 𝑛 2 , 𝑛 3 = 𝜋 2 ℏ 2 8𝑚 𝑛 𝑎 𝑛 𝑏 𝑛 𝑐 2
If 𝑎=𝑏 then 𝐸 𝑛 1 , 𝑛 2 , 𝑛 3 = 𝜋 2 ℏ 2 8𝑚 𝑛 𝑛 𝑎 𝑛 𝑐 2
Quantum numbers, 𝑛 1,2,3 =1,2,3…

4. Degeneracy
States are degenerate if energies are equal, eg. 𝐸 1,2,1 = 𝐸 2,1,1
Degree of degeneracy is equal to the number of linearly independent states (wavefunctions) per energy level 𝑢 𝑢
( 𝑢 𝑢 )/2
Degeneracy related to symmetry

5. 3D Harmonic Oscillator 𝑽 𝒓 = 𝟏 𝟐 𝒎 𝝎 𝟐 𝒓 𝟐
Calculate the energy and degeneracies of the two lowest energy levels
𝐸 𝑛 1 𝑛 2 𝑛 3 =ℏ𝜔( 𝑛 1 + 𝑛 2 + 𝑛 )
Ground state 𝐸 000 = 3 2 ℏ𝜔 is undegenerate, or has degeneracy 1
1st excited state 𝐸 100 = 𝐸 010 = 𝐸 001 = 5 2 ℏ𝜔 is 3-fold degenerate
2nd excited state 𝐸 110 = 𝐸 101 = 𝐸 011 = 𝐸 200 = 𝐸 020 = 𝐸 002 = 7 2 ℏ𝜔 has degeneracy 6
- don’t forget 𝑛 1,2,3 =0,1,2,3… for a harmonic oscillator

6. 3D Harmonic Oscillator 𝑽 𝒓 = 𝟏 𝟐 𝒎 𝝎 𝟐 𝒓 𝟐
Show that the lowest three energy levels are spherically symmetric 𝑢 𝑢 𝑢
average

7. Hydrogenic atom Potential (due to nucleus) is spherically symmetric𝑧
Use spherical polar coordinates
𝑥=𝑟 sin 𝜃 cos 𝜙
𝑦=𝑟 sin 𝜃 sin 𝜙
𝑧=𝑟 cos 𝜃 𝑟 𝜃
nucleus 𝑦 𝜙 𝑥

8. Hydrogenic atom 𝑉 𝑟,𝜃,𝜙 =𝑉 𝑟 so, can separate the wavefunction
𝑢 𝑟,𝜃,𝜙 =𝑅 𝑟 𝑌 𝜃,𝜙 =𝑅 𝑟 Θ 𝜃 Φ(𝜙)
Solve separately for 𝑅 𝑟 , Θ 𝜃 , Φ(𝜙)
𝑅, Θ, Φ continuous, finite, single valued, 𝑢 𝑟,𝜃,𝜙 2 d𝜏 = 1
Expect 3 quantum numbers 𝑛,𝑙,𝑚 - as 3 degrees of freedom
Expect 𝑢 𝑟,𝜃,𝜙 →0 as 𝑟→∞ because state is bound
Expect 𝐸 𝑛 = − 𝑚 𝑒 𝑍 2 𝑒 𝜋 𝜖 ℏ 2 𝑛 2 (result from Bohr’s theory)
Expect degenerate excited states

9. Schrödinger equation in spherical polars
−ℏ 2 2 𝑚 𝑒 𝛻 2 𝑢 𝑟,𝜃,𝜙 +𝑉 𝑟 𝑢 𝑟,𝜃,𝜙 =𝐸𝑢 𝑟,𝜃,𝜙 ,
where
𝛻 2 = 1 𝑟 2 𝜕 𝜕𝑟 𝑟 2 𝜕 𝜕𝑟 𝑟 2 sin 𝜃 𝜕 𝜕𝜃 sin 𝜃 𝜕 𝜕𝜃 𝑟 2 sin 2 𝜃 𝜕 2 𝜕 𝜙 2 ,
and
𝑢 𝑟,𝜃,𝜙 =𝑅 𝑟 𝑌 𝜃,𝜙 =𝑅 𝑟 Θ 𝜃 Φ 𝜙 .

10. 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

11. 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