Transcript lect4 - Personal Webpages (The University of Manchester)

PHYS 30101 Quantum Mechanics
Lecture 4
Dr Jon Billowes
Nuclear Physics Group (Schuster Building, room 4.10)
[email protected]
These slides at: www.man.ac.uk/dalton/phys30101
Plan of action
1. Basics of QM
Will be covered in the following order:
2. 1D QM
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
V=V0
A eikx
F eikx
B e-ikx
V=0
0
a
x
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
Q value chance of escape.
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
V
No applied
E-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!”