Physics 215
Introduction to Modern Physics
Winter 2003
Prof. Ioan Kosztin
Lecture #9
The Wave Nature of Matter
• de Broglie matter waves
• The Davisson-Germer experiment
• Matter wave packets
• Heisenberg uncertainty principle
• Particle wave duality
The Wave Properties of Particles
de Broglie’s hypothesis (1923)
The photon has a dual character:
wave and corpuscle
How about the electron ?
(After all, Nature is full of
symmetries!)
Electron has a dual
character too!
The associated de Broglie
wave length and frequency:
λ = h / p,
f = E/h
The electron of the
Bohr atom forms a
standing wave around
the nucleus.
2 πr = nλ
⇒ L = rmv = n!
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Davisson-Germer Experiment (1927)
direct experimental proof that electron has wavelength
λ = h/ p
Scattered intensity
Polar plot
Scattering of electrons from a
crystalline Ni target leads to
electron diffraction.
Scattered electrons energy
E = 54 eV
Confirmation of de Broglie’s hypothesis
Diffraction grating
1. Experiment: Condition
for diffraction maximum
AB = d sin φ = nλ
d=2.15Å, φ=50°, n=1 ⇒ λ=1.65Å
2. Theory:
X-ray diffraction
Electron diffraction
λ = h / mv = h / 2meU
U=54V ⇒ λ=1.67Å
2
Free Particle Wavefunction
is a monochromatic plane wave
Ψ ( x, t ) = ψ 0 exp[i ( kx − ωt )]
2π
E
p2
ω= , E=
!
2m
p
= ,
k=
λ !
a realistic quantum particle is
described by a wave packet,
which is a superposition of
plane waves
Space-time extent of a wave packet
A( ω, k ) = ∫ d t dx ψ(t , x ) exp[ −i ( ∆kx − ∆ω t )]
Since A(ω,k) is non zero only in the vicinity of ω0 and k0, the
temporal (spatial) extent ∆t (∆x) of the wave packet is
subject to the conditions
∆ω ⋅ ∆t ≥ 1
∆k ⋅ ∆x ≥ 1
Exp:
2∆x
A(k)
1
k0 + ∆k
k 0 − ∆k
| ψ( x ) |=| ∫
1
k0-∆k
k0
k0+∆k
dk exp(ikx ) |
sin( ∆k x )
=
∆k
ψ( ∆x ) = 0 ⇒ ∆k∆x = π ~ 1
0.8
0.6
0.4
0.2
0
-0.2
-7.5
-5
-2.5
0
2.5
5
7.5
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The Heisenberg Uncertainty Principle
(1927)
∆p x ⋅ ∆x ≥ ! / 2
Position and conjugate momentum can not be measured
simultaneously with any degree of accuracy (i.e., the concept
of trajectory looses its meaning in quantum mechanics).
Energy-time uncertainty:
∆E ⋅ ∆t ≥ ! / 2
∆t = duration of measurement;
∆E = precision of measured energy
⇒ Excited energy levels with finite lifetime are broadened!
The Heisenberg Uncertainty Principle
∆pe ~ (h / λ )sin θ
∆
xe ~ λ / sin θ
"###$###%
⇓
∆xe∆pe ~ h
in the microscopic
world the concept
of trajectory is
meaningless !!!
⇒ the position and
momentum of
electrons cannot
be measured
simultaneously w/
arbitrary accuracy
4
Double-Slit Experiment with Electrons
Condition for diffraction
minimum:
D sin θ = λ / 2
λ = h / px
θ ≈ sin θ = h / 2 px D
⇓
Electrons behave like either particles or waves, depending on
the experimental circumstances. However, it is impossible to
measure both the particle and wave properties simultaneously.
Particle-Wave Duality
P= Ψ
2
probability of
detecting an
electron
Ψ = Ψ1 + Ψ 2
P =| Ψ |2 =| Ψ1 |2 + | Ψ 2 |2 +2Ψ1Ψ *2
interference term
Electrons are detected as particles at localized spots, at
certain time, BUT their distribution function |Ψ|2 is
determined by superposition of waves.
5
Through which slit the electron passes ?
∆y << D
• For detecting the
electron as a particle:
∆y<<D
• Diffraction pattern
will be observed if
∆py~pxθ~h/2D
Thus,
∆y ∆py<<h/2
Which contradicts
Heisenberg uncertainty
principle
At the same time (simultaneously) it is impossible to see
the interference (wave property) and to determine which
slit the electron goes through (particle property),
without violating the uncertainty principle.
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