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PHYS 30101 Quantum Mechanics
Lecture 3
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: Wavefunctions for a bound particle in a finite well (1-D)
Two types of solution:
V0
E
V=0
Cosine-like
tan ka = μ/k
Sine-like
cot ka = -μ/k
exp(-μx)
Finite square well: solutions for tan ka = μ/k
thus
where
In graph below, y = ka
Compare with
wavefunctions for an
infinite square well:
n=3
n=2
n=1
Useful formulae
TDSE – time dependent
Schrödinger Equation
Vector operators in spherical polar
coordinates
Angular momentum
operators in spherical polars
TISE – time independent S.E.
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…