1. PHYS 30101 Quantum Mechanics
Lecture 4
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
These slides at:

2. 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

3. 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…

4. Reflection and transmission at a potential barrier:
Quantum mechanical tunnelling

5. 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.

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

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

8. Image of a surface obtained with by scanning tunnelling microscopy

9. 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!”