Announcements
Exam 3:
You will not be tested on how well you know the
formulas. A formula sheet will be provided. Also bring
your own double-sided formula/notes sheet.
Material: Chapters 6&7 (Matter waves & 1D Schrodinger)
Lasers (based on lecture notes)
Wave-functions: deBroglie and ψ(x), Ψ(x,t)
Meaning of Schroedinger eqn.
How to solve Schroedinger eqn.
Heisenberg’s uncertainty principle.
Complex numbers.
Applications:
Square well potential, (infinite and finite)
HW 11 due Mon.
Application of Quantum
Tunneling:
Radioactive decay
George Gamow: 1928
Radioactive decay
(Quantum tunneling – George Gamow)
Nucleus is unstable → ejects alpha particle (2 netrons, 2 protons)
Typically found for large atoms with lots of protons and neutrons.
Polonium-210
84 protons,
126 neutrons
Proton (positive charge)
Neutron (no charge)
Nucleus has lots of protons and lots of
neutrons.
Two forces acting in nucleus:
- Coulomb force .. Protons really close
together, so very big repulsion between
protons due to coulomb force.
- Nuclear force (attraction between nuclear
particles is very strong if very close
together) … called the STRONG Force.
Radioactive decay
Proton (positive charge) In alpha-decay, an alpha-particle is
Neutron (no charge)
emitted from the nucleus.
Polonium-210
84 protons,
126 neutrons
Lead-206
82 protons,
124 neutrons
This raises the ratio of neutrons
to protons … makes for a more
stable atom.
(Neutrons are neutral.. no
coulomb repulsion, but nuclear
force attraction)
How to figure out what's going on?
Starting point: Always look at potential energy curve for particle!
+
KE
New nucleus
Nucleus
Alpha particle
(Z-2 protons,
(Z protons,
(2 protons,
& bunch of neutrons) bunch of neutrons-2) 2 neutrons)
Now look at this system… as the
distance between the alpha particle
and the nucleus changes.
As we bring the α particle closer to the core,
what happens to potential energy?
As bring α closer, what happens to potential energy?
V=0 At a great distance
r
A
V(r)
C
B
V(r)
D. Something else
V(r)
r
r
V(r)
First:
Coulomb repulsion
30 MeV
Takes energy to push α towards
the nucleus, so potential energy
must increase.
V(r)
4 to 9MeV
of KE
Coulomb
&Nuclear
Then:
At edge of the nucleus (~8x10-15 m),
Nuclear (Strong) force starts acting
Strong attraction between nucleons.
Potential energy drops dramatically
Potential energy curve for the α particle
+
KE
New nucleus
Nucleus
Alpha particle
(Z-2 protons,
(Z protons,
(2 protons,
& bunch of neutrons) bunch of neutrons) 2 neutrons)
Strong attractive force
(Nuclear forces)
V(r)
Look at this system… as the
distance between the alpha particle
and the nucleus changes.
As we bring the α particle closer to the core,
what happens to potential energy?
r
Coulomb repulsion:
Nucleus:
(Z-2) protons
V=0 for r à ∞
Energy
very small r (~1fm):
nuclear force dominates
~30 MeV
‘Large’ r: coulomb force dominates
V(r)
1 to 10 MeV
r
Edge of the nucleus (~8x10-15 m),
Nuclear (‘Strong’) force starts acting
strong attraction between nucleons.
Potential energy drops dramatically.
Energy
What’s the kinetic energy of this particle inside the
nucleus?
V(r)
D
C
B
x
A
E: Something else
Energy
What would the kinetic energy of that particle be after it
tunneled out from the nucleus?
V(r)
D
C
B
x
A
E: Something else
Energy
So we found that the particle has less kinetic energy
outside than inside the nucleus. Did it loose total
energy?
V(r)
KEoutside
x
KEinside
A)  Yes.
B)  No.
C)  Impossible to tell. Need to
solve Schröd. equ. first.
Energy
So we found that the particle has less kinetic energy
outside than inside the nucleus. Did it loose total
energy?
V(r)
KEoutside
x
KEinside
A)  Yes.
B)  No.
C)  Impossible to tell. Need to
solve Schröd. equ. first.
Energy
Wave function picture:
V(r)
~100MeV
of KE inside
the nucleus
Exponential decay in the barrier
~1-10MeV of KE
outside
Wave function of the free particle:
‘small’ KE à Large wavelength
Wave function of the particle
inside the potential well: Large
KE à small Wavelength
V(r)
Energy
Observations show Alpha-particles from the same
chemical element exit with a range of energies.
9 MeV KE
4 MeV KE
Different KE in different isotopes
# neutrons influence nuclear potential
Observe α particles from different isotopes (same # protons,
different # neutrons), exit with different amounts of energy.
V(r)
Energy
30 MeV
1. Less distance to tunnel
2. V-E is smaller (àsmaller α)
à Wave function doesn’t
decay as much before reaches
other side … more probable!
9MeV KE
4MeV KE
x
The 9 MeV electron more probable…
Isotopes that emit higher energy alpha
particles, have shorter lifetimes!!!
Application: Scanning Tunneling Microscope
à 'See' individual atoms!
Use tunneling to measure very(!) small changes in distance.
Nobel prize winning idea! (1986)
Limitation: only works on conductive surfaces.
Measure current
between tip and
sample