Wave or Particle ?
Niels Bohr
1885-1962
Nobel Prize in Physics 1922
Werner Heisenberg
1901-1976
Nobel Prize in Physics 1932
Wave or Particle ?
• A particle is a localized
object, a wave is not
• How does “it” go from
waves to particles ?
• Wave-particle duality ?
Localization of a wave ?
Consider a single wave: y(x,t) = A cos(kx-ωt)
ħ
MOMENTUM:
The wave number k=2π/λ and the De Broglie wavelength λ = h/p k = 2πp/h = p/
In this case: k is well defined the momentum p = k is also very well defined
x (arbitrary units)
ħ
POSITION:
Where is the wave ? Everywhere (within the limit of the “definition” of the wave)
Localization of two superposed Waves ?
Consider 2 waves: y(x,t) = A cos(k1x-ω1t) + A cos(k2x-ω2t) with k1 close to k2 [BEATS!]
= 2 A cos(½[(k 1-k2)x-(ω1-ω2)t]) cos(½[(k 1+k2)x-(ω1+ω2)t])
Posing ∆k=k1-k2 and ∆ω=ω1-ω2: y(x,t) = 2 A cos(½[∆kx-∆ωt]) cos(½[(k 1+k2)x-(ω1+ω2)t])
POSITION:
Where is the “particle” ? Still not really localized, but now some “regions of space”
seem more likely to contain the particle than others.
λenvelope/2
The particle is between x1 and x2:
x2-x1 = λenvelope/2 = π/(½∆k)
Posing ∆x = x2-x1: ∆x∆k=2π
ħ
MOMENTUM:
The momentum is now only known
within ∆p= ∆k
x1
x2
We gained a better knowledge of x, but
we are losing our “perfect” knowledge of p
More Waves ?
3 Waves
Adding more waves:
∆x decreases
∆k increases
5 Waves
∆x.∆k~1
One can show similarly that:
∆ω.∆t~1
9 Waves
ħ
(Energy: E = hν = (2π)ν = ω)
ħ
7 Waves
Uncertainty relations between:
Position and Momentum
Energy and Time
Particles can be represented by
wave packets
The double slits
experiment with photons
From: Quantum (J. Al-Khalili)
• What about sending one photon at
a time ?
INTERFERENCE EFFECT !
The double slit
experiment with electrons
• 1961: C.Jonsson manages
to produce very narrow slits
to observe the interference
effects due to the wave-like
behavior of the electrons.
Which slit ? (I)
One slit blocked
No interference
Pictures from: Quantum (J. Al-Khalili)
Which Slit ? (II)
A apparatus is placed to
detect when one photon goes
through the top slit.
Detector On
NO INTERFERENCE !
Detector Off
INTERFERENCE !
Pictures from: Quantum (J. Al-Khalili)
Measurements and
Quantum systems
• To determine which slit the electron went through:
We set up a light shining on the double slit and use a
powerful microscope to look at the region. After the
electron passes through one of the slits, light bounces
off the electron; we observe the reflected light, so we
know which slit the electron came through.
•
Use a subscript “ph” to denote variables for light (photon). Therefore the
momentum of the photon is
•
The momentum of the electrons will be on the order of
•
The difficulty is that the momentum of the photons used to determine which
slit the electron went through is sufficiently great to strongly modify the
momentum of the electron itself, thus changing the direction of the electron!
The attempt to identify which slit the electron is passing through will in itself
change the interference pattern.
Principle of Complementarity
• Bohr’s principle of complementarity: it
is not possible to describe physical
observables simultaneously in terms
of both particles and waves
• Consequence: once you measure the
wave- (particle-) like behavior of a
phenomenon, you cannot measure a
property linked to its particle- (wave-)
like behavior.
– e.g. once you determine that the
photon/electron has made it through a
given slit, you reveal its particle-like
behavior, therefore you cannot observe
the interference phenomenon anymore
(linked to the wave behavior) !
Niels Bohr’s Coat of Arms (1947)
“Opposites are Complementary”
Heisenberg
Uncertainty (I)
• It is impossible to measure simultaneously, with no uncertainty,
the precise values of k and x for the same particle. The wave
number k may be rewritten as
• For the case of a Gaussian wave packet we have
Thus for a single particle we have Heisenberg’s uncertainty
principle:
Heisenberg Uncertainty (II)
with ∆x~l/2 [we know the particle is located between 0 and l ]
• If we are uncertain as to the exact position of a particle, for
example an electron somewhere inside an atom, the particle
can’t have zero kinetic energy.
• The energy uncertainty of a Gaussian wave packet is
combined with the angular frequency relation
• Energy-Time Uncertainty Principle:
The Copenhagen Interpretation
From: Quantum (J. Al-Khalili)
• “No one knows what happens behind the quantum
curtain… but we should not care!”
• Physics only depends on the outcomes of measurements
(in other words: only the results count… )
Other interpretations…
• De Broglie - Bohm
• Many-Worlds
• …