321 Quantum Mechanics Unit 2

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

321 Quantum Mechanics Unit 2

Last time• Time independent Schrödinger equation in 3D

• u must be normalised, u and its spatial derivatives must be finite, continuous and single valued

• If then 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

321 Quantum Mechanics Unit 2

3D quantum box

if and if

If then

Quantum numbers,

321 Quantum Mechanics Unit 2

Degeneracy• States are degenerate if energies are equal, eg.

• Degree of degeneracy is equal to the number of linearly independent states (wavefunctions) per energy level

• Degeneracy related to symmetry

|𝑢121|2 |𝑢211|

2 (|𝑢121|¿¿2+|𝑢211|2)/2¿

321 Quantum Mechanics Unit 2

3D Harmonic Oscillator • Calculate the energy and degeneracies of the two lowest

energy levels

Ground state is undegenerate, or has degeneracy 1

1st excited state is 3-fold degenerate

2nd excited state has degeneracy 6

- don’t forget for a harmonic oscillator

321 Quantum Mechanics Unit 2

|𝑢200|2

3D Harmonic Oscillator • Show that the lowest three energy levels are

spherically symmetric

|𝑢110|2 |𝑢020|

2 average

321 Quantum Mechanics Unit 2

Hydrogenic atom• Potential (due to nucleus) is spherically symmetric

Use spherical polar coordinates

nucleus

𝑥

𝑦

𝑧

𝜙

𝜃𝑟

321 Quantum Mechanics Unit 2

Hydrogenic atom• so, can separate the wavefunction

•

• Solve separately for

• continuous, finite, single valued, = 1

• Expect 3 quantum numbers - as 3 degrees of freedom

• Expect as because state is bound

• Expect (result from Bohr’s theory)

• Expect degenerate excited states

321 Quantum Mechanics Unit 2

Schrödinger equation in spherical polars

where

and

321 Quantum Mechanics Unit 2

Separation of Schrödinger equation • Radial equation

• equation

• represents the angular dependence of the wavefunction in any spherically symmetric potential

321 Quantum Mechanics Unit 2