To introduce students to the need for concepts beyond classical mechanics when dealing with phenomena on the atomic and sub-atomic scales. This will be done via discussion of a few key experiments performed at the beginning of the 20th century and the discussion of evidence for quantisation in atoms. The module will discuss why classical approaches do not work for such small systems and will culminate with an introduction to the basics of the quantum mechanical approach and applications in one dimension.
Module learning outcomes
Describe experimental evidence to demonstrate the particle properties of electromagnetic radiation (photoelectric effect and Compton’s experiments) and the wave properties of particles (Davisson and Germer experiment).
Derive the Compton scattering formula and know the photoelectric effect formula and be able to use these to solve problems
State, explain and utilise the energy and momentum forms of the Uncertainty Principle.
Explain what is meant by isotopes, isotones and isobars in nuclei and be able to identify examples of these.
Describe the Rutherford scattering experiment and perform simple calculations for the distance of closest approach between nuclei in a head-on collision.
Explain the origin of atomic emission and absorption spectra.
Discuss evidence for quantisation in atoms (eg line spectra, Franck-Hertz experiment).
Explain the origins of the various series of lines in the hydrogen atom
Describe the Bohr model and the Bohr postulates.
Derive the Rydberg equation using the Bohr model and use it to determine the wavelength of transitions between levels of an atom.
Use the Bohr postulates to calculate the radius and energy of levels in hydrogen –like atoms.
Explain the consequence of the finite nuclear mass on the Bohr model predictions and perform simple calculations using the modified Rydberg constant.
Discuss the problems of the Bohr model in relation to the stability of the atom and the problems of applying the model to multi-electron atoms.
Outline the Schrödinger quantum approach and state two principle postulates of quantum mechanics.
Quote and interpret the time-independent (TISE) Schrodinger equation.
Solve the TISE for simple 1-dimensional physical systems, applying appropriate boundary conditions.
Understand the how energy quantisation arise from the boundary conditions and be able to normalise1-dimensional wave- functions.
Solve the one-dimensional time- independent Schrödinger equation for simple potential steps and barriers and to be able to calculate their transmission and reflection coefficients.
State the significance and importance of operators in representing observable quantities.
Be able to quote operators for position, momentum and energy and describe the quantum mechanical interpretation of a measurement and the significance of eigenvalues as possible results of measurements.
Be able to ustilise the operator for the hydrogen atom to demonstrate that certain solutions are eigenfunctions.
Particle properties of radiation
Light as a particle: Photoelectric effect/ Experiment; Compton scattering/Experiment
Summary and comments on the dual nature of light
Wave properties of matter; de Broglie’s postulate
Davisson – Germer experiment
Wave – particle nature of matter
Uncertainty Principle ; uncertainty in real space and time, and in reciprocal (momentum and energy) space; examples.
Properties and constituents of the atom; Size, mass and constituents
Rutherford scattering and experiment
Constituents of the nucleus ( including concept of isotopes)
Stability of the atom; Quantisation in atoms
Atomic spectra (emission and absorption); the hydrogen atom emission series
The Bohr model, Bohr’s postulates, Bohr’s model
Correction for finite nuclear mass
Failure of classical mechanics to explain spectral lines, atomic bonding, stability of the atom and the ultraviolet catastrophe
Quantum approach - The Schrödinger equation
Postulates of quantum mechanics (main ones only)
The time independent Schrödinger equationExamining the terms of the Schrödinger equationWavefunction of a free particle; Born's probability interpretation of the wavefunction.
Wave-particle duality; Youngs two slit experiment; probability concepts in classical and quantum mechanics
Waves and wavevectors
Time-independent Schrödinger equation (TISE); static potential; stationary states; ‘boundary conditions to be satisfied by physically acceptable solutions of TISE: single-valuedness; normalisability and continuity.
Introduction to operators; observables and their operators; the Hamiltonian operator; position and momentum operators; eigenvalues and eigenfunctions; expectation values; examples.
Particle in an infinite one-dimensional potential well; solutions of TISE, energy eigenvalues and normalised eigenfunctions; orthogonality; orthonormalisation; the Kronecker delta; parity; the finite square well; classically inaccessible regions
Reflection and transmission at steps, barriers, and wells; reflection and transmission coefficients; quantum-mechanical tunnelling; particle flux, probability density and probability current density.
Particle subject to a 1D Coulomb potential; comparison to the spherically symmetric hydrogenic wave functions; hydrogenic energy eigenvalues.