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 a Scattering Experiment (1911) He found that, once in a while, the a-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 orbits due 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 Photons and 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 series corresponds 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 y Solved for y 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 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)