Potential energy curve for the α particle
Review: Radioactive decay
KE
+
New nucleus
Nucleus
Alpha particle
(Z-2 protons,
(Z protons,
(2 protons,
& bunch of neutrons) bunch of neutrons) 2 neutrons)
Energy
Small r: Nuclear
force dominates
~30 MeV
‘Large’ r: coulomb force dominates
V(r)
Look at this system as the
distance between the alpha particle
and the nucleus changes.
Strong attractive force
(Nuclear forces)
V (r ) =
1 to 10 MeV
As we bring the α particle closer to the core,
what happens to potential energy?
V(r)
r
Edge of the nucleus (~8x10-15 m),
Nuclear (‘Strong’) force starts acting
strong attraction between nucleons.
Potential energy drops dramatically.
r
kq q
k ( Z − 2)(e)(2e)
V (r ) = 1 2 =
r
r
Nucleus:
(Z-2) protons
V=0 for r ∞
Energy
Wave function picture:
Exponential decay in the barrier
V(r)
~1-10MeV of KE
outside
Wave function of the free particle:
Small KE Large wavelength
~100MeV
of KE inside
the nucleus
Wave function of the particle
inside the potential well: Large
KE small Wavelength
Observations show Alpha-particles from the same
chemical element exit with a range of energies.
V(r)
Energy
Coulomb repulsion:
kq1q2 k ( Z − 2)(e)(2e)
=
r
r
9 MeV KE
4 MeV KE
Different KE in different isotopes
Isotope: Different types of atoms of the
same chemical element (same number of
protons but different numbers of neutrons.)
We have already seen isotopes of hydrogen
in this class: Hydrogen(1p, 0n), Deuterium
(1p, 1n) and Tritium (1p, 2n).
# neutrons influence nuclear potential
Solving Schrodinger
equation for this
potential energy is hard!
V(x)
Nuclear Physics Sim
Square barrier is much easier
and get almost the same answer!
V(x)
phet.colorado.edu/simulations/sims.php?sim=Alpha_Decay
Application of quantum tunneling: Scanning
Tunneling Microscope 'See' single atoms!
Use tunneling to measure very(!) small changes in distance.
Nobel prize winning idea: Invention of “scanning tunneling
microscope (STM)”. Measure atoms on conductive surfaces.
STM (picture with reversed voltage, works exactly the same)
end of tip always
atomically sharp
Measure current
between tip and
sample
Crystal of
Ni atoms
How sensitive to distance?
Need to look at numbers.
Tunneling rate: T ~ (e-αd)2 = e-2αd
How big is α?
d
2m(V0 − E)
α=
h
If V0-E = 4 eV, α = 1/(10-10 m)
Fe atoms on Cu surface
So if d is 3 x 10-10 m, T = e-6 = .0025
add 1 extra atom (d ~ 10-10 m),
how much does T change?
T = e-4 =0.018
Decrease distance by
diameter of one atom:
Increase current by factor 7!
In typical operation, STM moves
tip across surface, adjusts
distance to keep tunneling
current constant. Keeps track of
how much tip moves up and
down to keep current constant.
Scan in x+y directions.
Draw a 2D map of surface
Another application:
Quantum Tunnel Transistor
(‘Controlled tunneling’)
One last 1D example:
The harmonic oscillator
Classical harmonic oscillator:
Mass ‘m’ experiences restoring force F = k⋅x
(x: displacement from equilibrium point).
Simple harmonic oscillator
All we have to do is to solve the Schrödinger equation
with a parabolic potential:
−
h 2 d 2ψ ( x ) 1 2
+ kx ψ ( x ) = Eψ ( x )
2m dx 2
2
In QM: don’t want to deal with forces. What can
we do?
Can derive corresponding PE function:
x
F = k⋅x
1
V ( x ) = ∫ F ( x ' )dx ' = kx 2
2
0
(“Hermite polynomials”)
Now put in Schrödinger equation and solve!
Simple harmonic oscillator (cont.)
Energy levels
Simple harmonic oscillator (cont.)
1
k
For n = 0: E0 = hω, with ω =
2
m
ψ 0 ( x ) = A0e
−
x2
2 L2
, with L =
‘ Ground state energy’
ψ(x)
h
mω
A0/e
L
Heisenberg: ∆x⋅∆p ≥ ħ/2
For ψ0 we find: ∆x = L =
And: ∆p ≈ 2 hmω
h
mω
∆x⋅∆p ≈ ħ/2
x