PHYS 30101 Quantum Mechanics Lecture 4 Dr Jon Billowes Nuclear Physics Group (Schuster Building, room 4.10) j.billowes@manchester.ac.uk These slides at: www.man.ac.uk/dalton/phys30101

Plan of action Basics of QM 1D QM Will be covered in the following order: 1.1 Some light revision and reminders. Infinite well 1.2 TISE applied to finite wells 1.3 TISE applied to barriers – tunnelling phenomena 1.4 Postulates of QM (i) What Ψ represents (ii) Hermitian operators for dynamical variables (iii) Operators for position, momentum, ang. Mom. (iv) Result of measurement 1.5 Commutators, compatibility, uncertainty principle 1.6 Time-dependence of Ψ

A eikx F eikx B e-ikx Re-cap from lecture 3 1.3 QM tunnelling through a barrier V=V0 A eikx F eikx B e-ikx V=0 x a Consider a flux of particles, momentum ħk, energy E= ħ2k2/2m approaching a barrier, height V0 (V0 > E), width a. We assume that some flux emerges on the far side…

Reflection and transmission at a potential barrier: Quantum mechanical tunnelling http://www.sgi.com/fun/java/john/wave-sim.html

Simple theory of α decay The α particle is preformed in the nucleus and bouncing around within the walls formed by the Coulomb barrier. Classically, it is impossible for the particle to escape but in reality it can tunnel through the energy-forbidden region to escape with final kinetic energy equal to the Q value. The chance of tunnelling through depends strongly on the width and height of the barrier, so the higher the Q value is, the greater the chance of escape. Q value The α-particle makes about 1020 “assaults” on the barrier every second. It can take years before it escapes.

Half-lives of alpha-emitters Age of Universe 1 microsecond Energy of α particle

Scanning Tunnelling Microscope No applied E-field V With applied field Potential energy of electron near the surface of a metal

Image of a surface obtained with by scanning tunnelling microscopy

Thermonuclear fusion in stars proton proton p + p d + e + + νe The reverse of α-decay. It happens at surprisingly low temperatures – the average thermal energy of protons is well below the (Coulomb) barrier and fusion takes place by barrier penetration – a slow process, so nuclear fuel lasts for astronomically long times. Sir Arthur Eddington (BSc in Physics, 1st Class, Owens College, Manchester 1902) To doubters that stars were hot enough for fusion he would say “Not hot enough? Go and find a hotter place!”