Lecture 15: The Hydrogen Atom

J.J. Thomson’s Plum Pudding Model of the Atom (1897)

He proposed that the electrons are embedded in a positively charged ‘pudding’

Rutherford’s Scattering Experiment (1911)

He found that, once in a while, the -particles were scattered backwards by the target

video clip

Discovery of the Atomic Nucleus

To explain the backscattering, the positive charge must be concentrated in a small region

Rutherford’s Solar System Model of the Atom

The atom consists of electrons orbiting around a small but dense central nucleus

Hydrogen Atom is Unstable?

It is known that accelerating charges emit radiation

Thus, electron should emit radiation, lose energy and eventually fall into the nucleus!

Why doesn’t this happen? Shows that something was wrong with this model of the hydrogen atom

Absorption Spectrum of a Gas

Dark lines will appear in the light spectrum

Absorption spectrum of Sun

Emission spectra of various elements

Balmer’s Formula for Hydrogen

Notice there are four bright lines in the hydrogen emission spectrum

Balmer guessed the following formula for the wavelength of these four lines:

where n = 3, 4, 5 and 6

Bohr’s Model of the Hydrogen Atom(1913)

He proposed that only certain orbits for the electron are allowed

Bohr’s Empirical Explanation

Electrons can only take discrete energies (energy is related to radius of the orbit)

Electrons can jump between different orbitsdue to the absorption or emission of photons

Dark lines in the absorption spectra are due to photons being absorbed

Bright lines in the emission spectra are due to photons being emitted

Absorption / Emission of Photonsand Conservation of Energy

Ef - Ei = hf Ei - Ef = hf

Energy Levels of Hydrogen

Electron jumping to a higher energy level

E = 12.08 eV

Spectrum of Hydrogen

Bohr’s formula:

Hydrogen is therefore a fussy absorber / emitter of light

It only absorbs or emits photons with precisely the right energies dictated by energy conservation

This explains why some nebulae are red or pink in colour

One of the transitions in the Balmer seriescorresponds to the emission of red light

Schrödinger’s Improvement to Bohr’s Model

Showed how to obtain Bohr’s formula using the Schrödinger equation

Electron is described by a wave function

Solved for in the electric potential due to the nucleus of the hydrogen atom

Square Well

Approximate electric (roller coaster) potential by a ‘square well’

System is then identical to the wave equation for a string that is fixed at both ends

Vibrational Modes of a String

fundamental 2nd harmonic 3rd harmonic 4th harmonic

Energy Levels in a Box

Quantum Numbers

Energy levels can only take discrete values

Labelled by a ‘quantum number’ n, which takes values 1, 2, 3, ...

Each level has energy that increases with n

Ground State (n=1)

Lowest or ground-state energy is non-zero

Electron cannot sit still but must be forever ‘jiggling around’

Expected from the Heisenberg uncertainty principle

Vibrational Modes of a Rectangular Membrane

(1,1) mode (1,2) mode (2,1) mode (2,2) mode

Vibrational modes of a circular membrane (drum)

Electron in a Hydrogen Atom

Wave function is like a vibrating string or membrane, but the vibration is in three dimensions

Labelled by three quantum numbers:– n = 1, 2, 3, …– ℓ = 0, 1, …, n-1– m = -ℓ, -ℓ+1, …, ℓ-1, ℓ

For historical reasons, ℓ = 0, 1, 2, 3 is also known as s, p, d, f

1s Orbital

Density of the cloud gives probability of where the electron is located

2s and 2p Orbitals

Another diagram of 2p orbitals

Note that there are three different configurations corresponding to m = -1, 0, 1

3d Orbitals

Now there are five different configurations corresponding to m = -2, -1, 0, 1, 2

4f Orbitals

There are seven different configurations corresponding to m = -3, -2, -1, 0, 1, 2, 3

Summary

Electron does not fly round the nucleus like the Earth around the Sun (Rutherford, Bohr)

Depending on which energy level it is in, the electron can take one of a number of stationary probability cloud configurations (Schrödinger)