PHYS 30101 Quantum MechanicsPHYS 30101 Quantum Mechanics

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

Lecture 4

Plan of action

1. Basics of QM

2. 1D QMWill 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

Re-cap from lecture 3

1.3 QM tunnelling through a barrier

A eikx

B e-ikxF eikx

V=0

V=V0

Consider a flux of particles, momentum ħk, energy E= ħ2k2/2m approaching a barrier, height V0 (V0 > E), width a.

0 a x

We assume that some flux emerges on the far side…

http://www.sgi.com/fun/java/john/wave-sim.html

Reflection and transmission at a potential barrier:

Quantum mechanical tunnelling

Simple theory of α decayThe 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.

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.

The α-particle makes about 1020 “assaults” on the barrier every second. It can take years before it escapes.

Energy of α particle

Age of Universe

1 microsecond

Half-lives of alpha-emitters

Scanning Tunnelling Scanning Tunnelling MicroscopeMicroscope

VNo applied E-field

With applied

field

Potential energy of electron near the surface of a metal

Image of a surface obtained with by Image of a surface obtained with by scanning tunnelling microscopyscanning tunnelling microscopy

Thermonuclear fusion in Thermonuclear fusion in starsstarsproton proton

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!”

p + p d + e + + νe