Wave or Particle?
In the first few decades of the twentieth century, many of the greatest theoretical
physicists grappled tirelessly to develop a mathematically sound and physically sensible
understanding of these hitherto hidden microscopic features of reality. Compared with
the clear, logical framework of Newton's laws of motion or Maxwell's
electromagnetic theory, the partially developed quantum theory was in a chaotic
state. And now there is the Schizophrenic nature of light. Einstein’s resolution of the
photoelectric effect problem suggests that light consists of little bundles of minimal
energy (hf) called photons. Its almost like light was a little particle. However, throughout
this course, I have been assuming that light is a wave. Confirmation of the wave nature
of light relied on wave interference, a phenomenon that simply doesn’t happen with
particles.
A brief history of light
• Newton (in the mid-1600s) proposed that light consists of particles. He was able to
explain the phenomena of reflection, refraction, and color using his particle model.
• Christian Huygens (1600) proposed an alternative: that light consists of waves.
• Thomas Young (1800) provided conclusive evidence that light is a wave. His doubleslit experiment showed that light beams interfere, something that is possible only
with waves.
• Maxwell in the 1860s stated that light was a wave.
• Einstein (in 1905) explained the photoelectric effect by proposing that light behaves
as if it were a particle, in that light energy is concentrated in particle-like photons.
Applet two-slit interference
http://surendranath.tripod.com/Applets/Optics/Slits/DoubleSlitID/DblSltIntDifApplet.html
Water waves
http://www.ngsir.netfirms.com/englishhtm/Interference2.htm
Double-slit Interference
Everyone knows that water—and hence water waves—are composed of a huge number
of water molecules. So is it really surprising that light waves are also composed of a
huge number of particles, namely photons? It is. But the surprise is in the details. You
see, more than three hundred years ago Newton proclaimed that light consisted of a
stream of particles, so the idea is not exactly new. However,
Christian Huygens, disagreed with him and argued that light is a
wave. The debate raged but ultimately experiments carried out
by the Thomas Young in the early 1800s showed that Newton
was wrong. A version of Young's experimental setup—known as
the double-slit experiment—is shown on the right.
Particle Picture of Light
Light is shone on a thin solid barrier in which two slits are cut. A photographic plate
records the light that gets through the slits—brighter areas of the photograph indicate
more incident light. The experiment consists of comparing the images on photographic
plates that result when either or both of the slits in the barrier are kept open and the light
source is turned on.
• If the right slit is covered and the left slit is open, the photograph looks like that
shown on the left. This makes good sense, since the light that hits the photographic
plate must pass through the only open slit and will therefore be concentrated around
the left part of the photograph. Similarly, if the left slit is covered and the right slit
open, the photograph will look like that right.
1
•
If both slits are open, Newton's particle picture of light leads to the prediction that the
photographic plate will look like that below. In essence, if you think of Newton's
corpuscles of light as if they were little pellets you fire at the wall, the ones that get
through will be concentrated in the two areas that line up with the two slits.
Wave Picture of Light
The wave picture of light, on the contrary, leads to a very different prediction for what
happens when both slits are open. Let's see this. Imagine for a moment that rather than
dealing with light waves we use water waves. The result we will find is the same, but
water is easier to think about.
APPLET single diffraction with small opening
Applet #1: http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm
Applet # 2: http://projects.cbe.ab.ca/sss/science/physics/map_north/applets/waterdiffraction/waterdiffraction.html
•
Show single diffraction with small opening
When water waves strike the barrier, outgoing circular water waves emerge
from each slit, much like those created by throwing a pebble into a pond
•
Show double-slit interference with small opening
As the waves emerging from each slit overlap with each other, something
quite interesting happens. There are regions of constructive and destructive
interference. The sequence of light and dark bands is known as an
interference pattern.
This photograph is significantly different from that for the particle picture of light
and hence there is a concrete experiment to distinguish between the particle and the
wave pictures of light. Young carried out a version of this experiment and thereby
confirming the wave picture, Newton's particle view was defeated. The prevailing wave
view of light was subsequently put on a mathematically firm foundation by Maxwell.
APPLET light interference http://vsg.quasihome.com/interfer.htm
The Compton Effect (1923)
In the next step in our evolution of the understanding of light, we have Einstein, the man
who brought down Newton's revered theory of gravity, seems now to have resurrected
Newton's particle model of light by his introduction of photons. Light comes in these little
bundles called photons. The wave energy is not spread out but comes bundled in these
little energy packets. These photons are acting like particles (billiard balls) that can hit
electrons like particles and eject them out of a metal in the photoelectric effect
experiment.
Quantum theory as we know it today only really began with the acceptance of Einstein’s
idea of the light quantum (photons, and the realization that light had to be described both
in terms of particles and waves. And even through Einstein first introduced the light
quantum in his 1905 paper on the photoelectric effect, it was not until 1923 that the idea
became accepted and respectable. Einstein left off serious thinking about quantum
theory while he developed his General theory of Relativity. When he returned to the
quantum fray in 1916, after some work on the statistical behavior of blackbody radiation
and Bohr’s model, Einstein become even more convinced of the existence of the photon.
In 1917, Einstein stood alone in his belief in the reality of photons and it was
another six years before direct experimental proof of the reality of photons was obtained
by Compton. Historically, the Compton Effect was for many old-time physicists the final
convincing evidence for the reality of quanta. Through a series of experiments in the
early 1920’s, he was led inexorably to the conclusion that the interaction between x-rays
and electrons could only be explained if the x-rays where treated in some ways as
2
particles. The key experiments concern the way in which the x-rays are scattered by an
electron.
Wave picture: When an x-ray wave hits an electron, classical physics predicts that the
electron should absorb the energy from the x-ray wave, then re-emit at
the same frequency.
DEMO show two colliding carts http://www.lon-capa.org/~mmp/kap6/cd155a.htm
Billiard ball Analogy: the collision is like the impact of a moving billiard
ball on a stationary ball, and the transfer of momentum occurs in jus the
same way.
Particle picture: the x-ray light wave scatters off the electron with lower
frequency—just as if the light were a beam of particles that interacts
with electrons in the same way that two billiard balls collide. The incoming photon
bounces off an electron, giving up some of its energy and lowering its frequency
(since E=hf). However, the analogy is limited in that the ball in the applet slowed down
while for light it would not slow down.
You need descriptions, particle and wave, to get a complete explanation of the
experiment. When Compton made the experiments, he found the interaction behaving
exactly in accordance with this description all fitted perfectly with the idea that x-rays
comes in the form of particles with energy hf. After 1923, the reality of photons as
particles carrying both energy and momentum was established. As Einstein said,
“there are therefore now two theories of light, both indispensable … without any logical
connection.” In essence, since all the experiments designed to test the wave theory of
light showed light to be made up of waves, how could light be made of particles? So light
has acquired a schizophrenic personally – is a particle or a wave, which is it?
Particle or Wave?
In 1923, the young French nobleman Prince Louis de Broglie added a new element to
the quantum fray, one that would shortly help to usher in the mathematical framework of
modern quantum mechanics and that earned him the 1929 Nobel Prize in physics. It
sounds so simple, yet it struck to the heart of the matter. “If light waves also behave
like particles, why shouldn’t electrons also behave like waves?” That is, de Broglie
suggested that the particle-wave duality applied not only to light but to matter as well. In
his Ph.D. thesis, inspired by a chain of reasoning rooted from two equations that Einstein
had derived for light quanta he reasoned if E = mc2 relates mass to energy and E = hf
related energy to the frequency of waves, then by combining the two, mass should have
a wave-like incarnation as well:
1
E = mc 2 
=
m  c 2 h 
f ∝
 → 

λ 
E = hf 
associated with mass
associated

frequency
associated
wavelength
The de Broglie equation is
=
wavelength
Planck ' s constant
=
λ
or
mass ⋅ velocity


h
mv
momentum
De Broglie suggested that just as light is a wave phenomenon, quantum theory shows to
have an equally valid particle description; an electron—which we normally think of as
being a particle—might have an equally valid description in terms of waves. De Broglie’s
great achievement was to take the idea of particle/wave duality and to carry it through
mathematically, describing how matter waves ought to behave and suggest ways in
which they might be observed. Nothing is a substitute for experimental proof and such
proof was soon to come from the work of Davisson and Germer.
3
In the mid-1920s, Davisson & Germer were studying how a beam of electrons bounces
off of a chunk of nickel. The only detail that matters for us is that the nickel crystals in
such an experiment act very much like the two slits, except that a beam of electrons is
used in place of a beam of light. They found something remarkable. A pattern very much
akin to interference patterns emerged.
Electron diffraction
Light diffraction
Their experiment therefore showed that electrons exhibit interference phenomena, the
telltale sign of waves. At dark spots, electrons were somehow "canceling each other
out." Even if the beam of fired electrons was "thinned" so that, for instance, only one
electron was emitted every ten seconds, the individual electrons still built up the bright
and dark bands—one spot at a time. Somehow, as with photons, individual electrons
"interfere" with themselves in the sense that individual electrons, over time,
reconstruct the interference pattern associated with waves. We are inescapably
forced to conclude that each electron embodies a wave-like character in conjunction with
its more familiar depiction as a particle.
Two-Slit Interference of de Broglie Waves
This was a bit hard and took longer to accomplish – it was done for the first time in 1976.
It is now fairly straightforward to show nowadays, the buildup of an interference pattern
by a beam of electrons. The picture below is actual electron interference pattern filmed
from a TV monitor as the electron beam densities increase. They start off by shooting
one electron, and increasing the numbers until the interference pattern is observed.
Once again, the electron’s must interference with itself in order to produce
interference patterns – that is, the electron spreads out like “ocean waves” and does
not have behave as a localized particle.
Wave Diffraction
To get an idea of when wavelike behavior is observed, I first need to talk about wave
diffraction. Suppose you walk out this lecture while I was lecturing on de Broglie waves.
The fact that you cannot see me around a corner implies light does not go around
corners very well. On the other hand, the fact that you can hear my sexy voice around
the corners implies that sound does go around corners well. However, both light and
sound have wave properties – so why the difference?
It turns out that the wave properties of a wave are most noticeable when it interacts
with objects (or systems) that are themselves roughly comparable to the size of a
wavelength. That is,
• if waves interact with objects that are much, much larger than the wavelength,
then you don’t notice the wavelike aspects of it
• if waves interact with objects comparable to the wavelength, then you do notice
the wavelike aspects of it
4
APPLET http://www.ngsir.netfirms.com/englishhtm/Diffraction.htm
small bending
(λ << opening)
large bending
(λ ≈ opening)
large bending
(λ ≈ opening)
small bending
(λ << opening)
Applying this understanding to how light sound interact with the doorway,
 size of 
λ visible light << 
→
 
 doorway 
 size of 
λ sound ≈ 
→
 
 doorway 
 does not bend well round corners 


 and cannot see around the corner 
 does bend well round corners 


 and can hear around the corner 
Historical aside: why was there a debate between Huygens and Newton whether light
was a wave or a particle? They could not see the wave aspect of light because everyday
objects are much, much larger than the wavelength of visible light. Therefore, light did
not diffract well and the wavelike aspects went unnoticed until Young’s Double-slit
experiment some 200 years later.
Physical Interpretation of de Broglie Waves
Let’s go back to de Broglie equation. Note that the de Broglie wavelength depends on
two things: (1) mass and (2) velocity.
Mass Affects
h

→λ =
large masses ⇒ small de Brloglie waves 

h

v
λ=
→ 
mv
small masses ⇒ large de Brloglie waves → = h

mv
“Big objects” like a tennis ball or your brave instructor have masses considerably larger
than the mass of an atom or electron (mtennis ball ≈ 1 kg >> melectron ≈ 10−31 kg). One can
use the de Broglie equation to estimate the matter wavelength of a tennis ball moving at
a speed of 1 m/s:
h 10−34
λ=
≈
≈ 10−34 m
mv
1⋅ 1
As you can see, this is a very tiny, tiny number and we will never notice the wavelike
aspects of this tennis ball, though in principle, according to quantum physics they exist.
Why? Planck’s constant is so small and ordinary masses are very large compared to
atomic masses. On the other hand, if we want to see wavelike aspects of matter, two
things must happen:
• we need very, very small masses in order to make the de Broglie wavelength as
large as possible.
• if the de Broglie wavelength becomes comparable to the size of the object then the
wavelike aspects will become very dramatic for those objects.
For example, for an electron in orbit in a typical atom, you’ll find the wavelength
associated with that electron is roughly comparable to the size of the atom itself. So if de
Broglie hypothesis is correct, we ought to notice wavelike aspects at the subatomic
λ
m
5
world but not in the macroscopic world. That is exactly what is observed in the Quantum
Corral.
The Orbits for the Bohr Atom
What do we gain from de Broglie’s wave hypothesis? We immediately gain an
understanding of the Bohr atom. In the Bohr atom electrons where stuck in certain
allowable quantized orbits, however, the model gave no explanation of why. Now with
the de Broglie hypothesis, there is a very good explanation of why these allowed orbits
occur.
Analogy: standing waves on a string
In order to form standing waves, two things must happen:
• a reflected wave must be inverted upon the fixed right hand end
• an integer number of half-wavelengths must “fit” in between the posts such that
length of string =whole number of half waves =21 λ,
2
2
λ,
3
2
λ,
4
2
λ, ...
As the wave is sent down to the fixed right hand end, it reflects back inverted and these
incoming and reflected wave interference to produce a series of standing waves. That is,
the wave on a string is reinforced by its successive reflections. One cannot create a
standing wave pattern such that the wave does not reflect inverted upon itself. It is just
not possible. The only phenomena involving integers in Physics were those of
interference and of normal modes of vibration.
Applying this to the Bohr orbitals, the electron wave would travel around the orbit,
reinforcing itself constructively at each turn, just as the wave on a string is reinforced by
its successive reflections. The de Broglie waves must fit evenly into the circumference of
the orbits.
The explanation of the allowed quantized electron orbits using de Broglie matter waves
is exactly analogous except these orbits are circular orbits around the nucleus and
therefore, we are talking about how many waves can fit in this circular type of orbit.
• Atomic orbits have an integer number of these half-de Broglie matter wavelengths
that fit around a particular orbit. That would be an allowed orbit.
•
whole number of
 circumference  

λ, 2λ, 3λ, ...

=

=
 of allowed orbit   de Broglie wavelengths 
In other words, the electron wave would travel around the orbit, reinforcing itself
constructively at each turn, just as the wave on a string is reinforced by its
successive reflections.
If there are not an integer number of whole waves, that is, the wave does not come
back on itself, then that orbit is not a possible situation and destructive interference
eliminates that orbit.
6
De Broglie matter waves explain Bohr model of the atom in terms of which allowed orbits
can exist versus those that cannot exist. The energy is related to the frequency of
vibration and these frequencies are quantized according to the standing waves formed
by de Broglie waves.
Velocity Affects
h

→λ =
large velocities ⇒ small de Brloglie waves 

h

m
λ=
→ 
mv
small velocities ⇒ large de Brloglie waves → = h

mv
Another way to make the de Broglie wavelength really large (even if the object had a
substantial mass) is to reduce the velocity of the particle to a very small value. For
example, if I could bring this tennis ball truly to rest then the de Broglie wavelength of the
tennis ball would be comparable to the size of the tennis ball and the wavelike aspects of
the ball would be apparent to you and me (that’s crazy!). However, because the ball is
composed of zillions and zillions of atoms that are bouncing around hitting the sides, I
can’t bring it to rest and therefore, the de Broglie wavelength is too small. But if I could
bring it to rest I could see quantum effects even in macroscopic size systems.
There are situations where regularly, one sees quantum effects of almost macroscopic
size systems. This is the exciting branch of low temperature physics. If I cool matter
down to almost absolute zero then that frenzy bouncing (vibrations) due mainly to heat is
dramatically reduced and comes as close to a stop as possible. However, a principle of
quantum mechanics (the uncertainty principle) does not allow it to come to a completely
stop and so the de Broglie wavelength is still very small but not infinitesimally small and
is observable in laboratory situations. So these almost macroscopic systems (such as
Buckyballs) will exhibit very bizarre behavior which is directly attributed to the fact that
matter does have a quantum wavelike aspect to it. That is one reason why low
temperature physics is such an interesting field and is used to confirm quantum
mechanics.
λ
v
Image Liquid helium
http://www.youtube.com/watch?v=2Z6UJbwxBZI
Summary of de Broglie Waves
The complete break with classical physics comes with the realization that not just
photons and electrons but all “particles” and all “waves” are in fact a mixture of
wave and particles. It just happens that in our everyday world the particle component
overwhelmingly dominates the mixture in the case of, say, a bowling bal, or a house.
The wave aspect is still there, in accordance with the relation λ = h/mv, although it is
totally insignificant. In the world of the very small where particle and wave aspects of
reality are equally significant, things do not behave in any way that we can
understand from our experience of the everyday world. It isn’t just that Bohr’s atom
with its electron “orbitals” is a false picture; all pictures are false, and there is no physical
analogy we can make to understand what goes on inside atoms. Atoms behave like
atoms, nothing else.
7
No familiar conceptions can be woven around the electron and our best description of
the atom boils down to “something unknown is doing we don’t know what.”
Historical comment
By 1925 quantum theory was in a mess. There was no great highway of progress, but
rather many individuals each hacking a separate path thought the jungle. The two great
authorities were Einstein and Bohr, but they had begun to differ markedly in their
scientific views. First, Bohr was one of the strongest opponents of the light quantum;
then, as Einstein began to be concerned about the role of probability in quantum theory
Bohr became its great champion. The statistical methods (ironically, introduced by
Einstein) became the cornerstone of quantum mechanics. Question – where next?
What’s waving? – The Schrödinger Equation
Schrödinger discovered an equation that described how de Broglie waves propagated
from one location to another. When Schrödinger applied this equation to the early
quantum problems, almost all the puzzles where resolved by this equation. When one
solves the Schrödinger equation, the solution or mathematical representation of the
wave is called the “wavefunction ψ.” There are several wavefunctions that we have
already encountered; electromagnetic waves have waving or oscillating E- and B-fields
whereas a standing wave has waving or oscillating string. So, what is it that is waving in
Schrödinger equation? In other words, de Broglie stated that all matter has a wave
component to its nature but that still didn’t tell us what was waving – so what is it?
Initially, Schrödinger thought that the wavefunction of an object was the smeared out
object itself, which turned out to be incorrect. The reason why this is incorrect is when
one find an electron, one finds either the whole electron or no electron at that spot, never
a piece of an electron. Warning: it is most likely you are not going to like what I am
going to tell you. In fact, the most difficult conceptual part of this course is the
interpretation of what the wavefunction is. As Bruce tells us, it’s difficult because it’s so
hard to believe. What’s waving in Schrödinger’s equation is the probability! In other
words, the waviness in a region is the probability of finding the object in that
region. Mathematically, the probability is written as
Pr obability= ψ
2
→ psi squared
The wavefunction tells us where something is. Let’s look at some wavefunctions to
get a feel for what we are looking at and interpret the phenomena in terms of
probabilities.
Electromagnetic waves
The double-slit interference pattern of light produces a series of bright
and dark fringes. One interpretation is where a bright spot is formed; we
are most likely to find a photon whereas where a dark spot is formed
there is no chance of finding a photon. Let look at some wavefunctions to
get a feel for what we are looking at. Its not unrealistic to interpret the
double-slit results in this manner since it makes good sense since the
greatest light intensity is directly proportional to the number of photons
landing at a particular location.
Standing waves on a string
Wavefunctions Probabilities
Sound Beats
8
Quantum Wavefunctions tell us where a particle is most likely to be where the amplitude
is largest. Wavefunctions or wave packets can be either spread out or compact as
shown below.
Very localized
Most likely
Least likely
In quantum theory the wavefunction describe everything about an object (matter or light).
In fact, in quantum theory there is no atom in addition to the wavefunction of the atom.
This is so crucial that I’ll state it again in other words: The atom’s wavefunction and
the atom is the same thing; “the wavefunction of the atom” is a synonym for “the
atom.”
Quantum Billiard Ball Analogy
If one was able to buy on the black market “quantum billiard balls (h = 1 J.s),” they
indeed would display very bizarre behavior. When a pool-hall hustler playing quantum
billiards hits a quantum billiard ball, its wavefunction would look something like that
below.
The quantum billiard ball would appear as many “ghost balls” moving in
the same general direction. However, as soon as one makes a
measurement to determine the location of the quantum billiard ball one is
said to “collapse” the wavefunction. Accordingly, before a look collapses
a widely spread-out wavefunction to the particular place where the billiard
ball is found, the billiard ball did not exist there prior to the look. The look
brought about the billiard ball’s existence at that particular place –
for everyone. At this point you may be mystified by quantum theory. (If so,
you join many experts.) And it is now time for us to display this via the
double-slit experiment.
Revisit of the Double-Slit Experiment
Richard Feynman stated that the heart of quantum mechanics is in the double-slit
experiment. Why? – Because this is “a phenomenon which is impossible, absolutely
impossible, to explain in any classical way. In reality, it contains the only mystery … the
9
basic peculiarities of all quantum mechanics.” If you can come to terms with the doubleslit experiment then the battle is more than half over, since “any other situation in QM, it
turns out, can always be explained by saying, “you remember the case of the experiment
with the two holes? It’s the same thing.” Now let’s revisit the double-slit experiment and
interpret it in terms of quantum mechanics, that is, in terms of probabilities. First, step
away from the quantum world of photons and electrons and look at what happens in the
everyday world. Suppose we carried out a double-slit experiment using large particles in
the everyday world.
A Double-slit experiment with bullets
If a machine gun with terrible aim shoots bullets at two holes made in armor plate, after
we had fired a large number of bullets through the holes, there are two situations: with
one hole blocked off and both holes open. If we block off one of the holes (hole B) in
the wall, the largest number of bullets is nearest the hole that is unblocked and
therefore, the highest probability of finding a bullet is there. The same pattern is found if
we block off this hole and open up the one that was previously blocked. The probability
of detecting a bullet on the screen is
2
2
=
Pr
ob(A)
A
=
and
Pr
ob(B)
B






only hole A open
only hole B open
Large particles
But with both holes open, the pattern of bullets found in different locations is just the
sum of the two effects from the two separate holes – most bullets in the region just
behind the two holes, and therefore, just behind each hole the probability of finding a
bullet is equal such that
2
Pr
ob(A +
= A2 + B

B)


both holes open
This is a no interference probability, as it will become apparent in a short period.
A Double-slit experiment with water waves
It is easy to see how water waves produce interference patterns. The waves spread
through the two holes and form a regular pattern of constructive (maximum height of
waves) and destructive (minimal height of waves) interference at the screen. If we block
off one of the holes in the wall, the height of the waves on the screen varies in a
simple, regular way. The biggest waves are the ones nearest the hole that is unblocked.
The same pattern is found if we block off this hole and open up the one that was
previously blocked. The curves are
2
2
=
Pr
ob(A)
A
=
and
Pr
ob(B)
B






only hole A open
only hole B open
10
Water waves
But when both holes are open, the pattern is much more complex – interference
patterns are produced. For these waves, the probability of detecting a maximum is
2
Pr ob(A + B) ≠ A 2 + B2 → Pr ob(A + B) = ( A + B ) = 
A 2
+ B
+ 
2AB

2
probabilities for
one hole closed
interference
term
A Double-slit experiment with electrons
Just like light, the electrons too show the double-slit interference patterns. So what?
Isn’t this just the particle/wave duality that we have learned to
live with? The point is that we leaned to live with it but we did
not look deeply into the implications. The time has come to do
so. What happens to each individual electron? We can
understand easily enough that a wave – a water wave – can
pass through both holes in the screen. A wave is a spread-out
thing. But an electron still seems to be a particle even if it
has associated wavelike properties. It is natural to believe that
each individual electron must surely, go through one hole
or the other. We can try experimentally the equivalent.
•
•
If we block off one hole at a time, we get the usual pattern
on our screen for single-hole experiments.
When we open both holes together, however, we do not get the pattern produced by
adding up those two patterns, as we would for bullets. Instead, we get the pattern for
interference by waves.
Remarks
1. The rules of wave behavior are needed to assign probabilities to the appearance of
an electron at A and B; yet when we look at A or B we either see an electron
(particle) or not. We don’t see a wave. We cannot say what the electron is “really”
doing during its passage through the apparatus.
2. Probability waves seem to decide where each “particle” in the beam goes, and
probability waves interfere just as water waves do.
3. Furthermore, we still get this pattern if we slow down our electron gun so much that
only one electron at a time goes through the whole setup. One electron goes through
only one hole, we would guess, and arrives at our detector; then another electron is
let through, and so on. If we wait patiently for enough electrons to pass through (so
that thousands has gone through), the pattern that builds up on our detector screen
is the interference for waves. A single electron, or a single photon, on its way
through one hole in the wall, obeys the statistical laws which are only
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appropriate if it “knows” whether or not the other hole is open. This is the
central mystery of the quantum world and is the so-called quantum enigma.
4. We can try cheating – shutting or opening one of the holes quickly while the electron
is in transit through the apparatus. It doesn’t work – the pattern on the screen is
always the “right” one for the states of the holes at the instant the electron was
passing through. We can try peeking, to “see” which hole the electron goes through.
When the equivalent of this experiment is carried out, the result is even more
bizarre. Imagine an arrangement that records which hole an electron goes through
but lets it pass on its way to the detector screen. Now the electrons behave like
normal, self-respecting everyday particles. We always see an electron at one hole or
the other, never both at once. And now the pattern that builds up on the detector
screen is exactly equivalent to the pattern for bullets, with no trace of interference.
Here are some experiments carried with electrons and photons.
APPLET Double-slit with electron build up http://phys.educ.ksu.edu/vqm/index.html
Electron interference build-up (http://www.hqrd.hitachi.co.jp/em/doubleslit.cfm)
MOVIE http://www.hqrd.hitachi.co.jp/em/movie/doubleslite-n.wmv
MOVIE Dr Quantum http://www.youtube.com/watch?v=DfPeprQ7oGc
Single electron events build up to from an interference pattern in the double-slit
experiments. The number of electron accumulated on the screen. (a) 8 electrons;
(b) 270 electrons; (c) 2000 electrons; (d) 160,000. The total exposure time from the
beginning to the stage (d) is 20 min
Key Point: the electrons not only know whether or not both
holes are open, they know whether or not we are watching
them, and they adjust their behavior accordingly. There is no
clearer example of the interaction of the observer with the experiment.
Same thing happens with photons
This sequence of photographs of a girl’s face shows that photography is a quantum
process. The probabilistic nature of quantum effects is evident from the first photographs
in which the number of photons is very small. As the number of photons increases the
photograph becomes more and more distinct until the optimum exposure is reached.
The number of photons involved in these photographs ranges from about 30000 in the
lowest exposure to about 30 million in the final exposure.
When we try to look at the spread-out electron wave, it collapses into a definite particle,
but when we are not looking it keeps its options open. In terms of Born’s probabilities,
the electron is being forced by our measurement to choose one course of action out of
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an array of possibilities. There is a certain probability that it could go through one hole,
and an equivalent probability that it may go through the other; probability interference
produces the interference pattern at our detector. When we detect the electron, though,
it can only be in one place, and that changes the probability pattern for its future
behavior – for that electron, it is no certain which hole it went through. But unless
someone looks, nature herself does not know which hole the electron is going through.
In the simplest experiment with two holes, the interference of probabilities can be
interpreted as if the electron that leaves the gun vanishes once it is out of sight, and is
replaced by an array of ghost electrons that each follows a different path to the detector
screen. The ghosts interfere with one another, and when we look at the way electrons
are detected by the screen we then find the traces of this interference, even if we deal
only with one “real” electron at a time. However, this array of ghost electrons only
describes what happens when we are not looking; when we look, all of the ghosts except
one vanish, and one of the ghosts solidifies as a real electron. In terms of Schrödinger’s
wave equation, each of the “ghosts” corresponds to a wave, or rather a packet of waves,
the waves that Born interpreted as a measure of probability. The observation that
crystallizes one ghost out of the array of potential electrons is equivalent, in terms of
wave mechanics, to the disappearance of all o the array of probability waves except for
one packet of waves that describes one real electron. This is called the “collapse of the
wave function,” and, bizarre though it is, it is at the heart of the Copenhagen
interpretation, which is itself the foundation of quantum mechanics. What’s worse, as
soon as we stop looking a the electron or whatever we are looking at, it
immediately splits up into a new array of ghost particles, each pursuing their own
path of probabilities through the quantum world. Nothing is real unless we look at
it, and it ceases to be real as soon as we stop looking.
Interpretation of the quantum box solutions
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Quantum Uncertainty: Farewell to Determinism
Quantization places severe limits on our ability to observe nature at the atomic scale,
because it implies that the act of observation necessarily disturbs that which is being
observed. The fact that the amount of energy in a light beam cannot be less than that of
a single photon means that for a given color of light, there is a minimum amount of
energy we can use to observe the world—namely, the energy of one photon. Going to
redder (lower frequency and, therefore, lower photon energy) light doesn’t help, because
the wave nature of light limits our ability to know where the photon is. The result is the
Heisenberg uncertainty principle, which says that we can never measure simultaneously
and with arbitrarily good precision both the velocity (strictly speaking, the momentum)
and position of a particle. If we measure one of those quantities more precisely, the
value of the other necessarily becomes less certain.
The philosophical interpretation of the uncertainty principle goes further still. Most
physicists subscribe to the so-called Copenhagen interpretation of quantum physics.
Based in logical positivism’s view that it makes no sense to talk about what cannot
be measured, the Copenhagen interpretation asserts that it makes no sense to say that
a particle even has a precisely determined velocity and position. Because precise
velocity and position are required for the determinism of Newton’s laws and the
“clockwork universe,” the Copenhagen interpretation rules out strict determinism.
Quantum physics tells us only the probability that an experiment will have a given
outcome, rather than that the outcome will definitely occur. Not all physicists accept the
Copenhagen interpretation. Einstein remained all his life one of its staunchest critics.
Today, a small number of physicists are exploring alternatives, including hidden variable
theories that would restore determinism at a level hidden from us by the uncertainty
principle. Recent experiments put severe constraints on such theories.
Quantization means that we cannot observe the universe without affecting it. This,
in turn, limits our ability to make measurements with arbitrary precision. Thus, we
must say farewell to the “clockwork universe.” The least obtrusive way to observe
something is to see it—that is, to bounce light off it. To understand this statement,
we need speak in detail about three things.
1. Photons and wave packets.
First, consider how to prepare the light. Recall that the probability of finding a photon is
proportional to the intensity of the associated light wave at that point. If we want to
know with precision where a photon is likely to be, then we need a wave packet, with
the “wiggles” of the wave confined to a small region. We can do this by producing, for
example, a very short pulse of laser light. But note that making a localized wave such as
this requires a short wavelength and, correspondingly, a high frequency.
2. Heisenberg’s Quantum Microscope
Heisenberg’s quantum microscope “thought experiment” explores an attempt to measure
simultaneously the position and velocity of an electron with high precision, by bouncing
light (i.e., minimum one photon) off the electron.
To get accurate position information, we need a localized photon. There’s a problem,
though: The localized photon has high frequency and, therefore, high energy (recall the
quantization condition E=hf). As it bounces off the electron, the photon transfers a lot of
energy to the electron, altering its velocity substantially. The observation destroys some
of the information—the velocity—that we sought to measure. Note the crucial role of
quantization here: The requirement for a minimum amount of light energy—one
photon’s worth—causes the problem. We can’t observe a system without
interacting with it, and when energy is quantized, that means disturbing the
system. Surely there’s a way out of this problem: We can make the photon energy
lower, thus reducing the disturbance. But lower photon energy means lower frequency
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(again, E=hf), longer wavelength—and a less localized photon. Now our measurement
of the electron’s position is less precise.
3. The Heisenberg Uncertainty Principle
The quantum microscope thought experiment reveals a tradeoff between our ability to
measure a particle’s position and its velocity simultaneously. If you make the
velocity measurement more precise, you lose information about position and vice versa.
The Heisenberg uncertainty principle is the formal statement of this tradeoff.
• The uncertainty principle states that it is impossible to measure simultaneously
and with arbitrarily high precision both a particle’s position and its velocity
(actually its momentum, the product of mass and velocity).
• Quantitatively, the uncertainty principle says that the product of a particle’s mass,
the uncertainty in its position, and the uncertainty in its velocity cannot be less than
Planck’s constant h:
∆x 
m
∆v  ≥ 21 


uncertainty uncertainty
in position in velocity
Because h is so small, the uncertainty principle has a negligible effect on measurements
of normal-sized objects, such as planets, baseballs, and even bacteria. At the atomic
scale, however, where particle masses are tiny, the uncertainty principle severely
limits our simultaneous knowledge of particles’ positions and velocities.
∆x
∆x m∆v ≥ 21  =
constant → 
∆v
high uncertainty
in position
⋅ 
m
∆v  = 
∆x
⋅ 
m

low uncertainty
in speed
Analogy: a pool rack in a quantum billiard table
∆x
low uncertainty high uncertainty
in speed
in speed
∆v
∆v  = 
∆x  ⋅ m


 ⋅ m


billiard ball somewhere
on the table
billiard ball moves
along a straight line
billiard ball is localized
within the rack
billiard ball is moving very
fast within the rack
What does it mean? Let’s consider the philosophical interpretation and implication.
Most physicists subscribe to the Copenhagen interpretation of quantum physics. This
view grows out of logical positivism, with its claim that it makes no sense to talk about
what cannot be measured.
1. In the Copenhagen interpretation, not only can one never measure the velocity
and position of a particle simultaneously, but it also makes no sense to say
that the particle has a velocity and a position.
2. Under the Copenhagen interpretation, such particles as electrons and protons
simply can’t be thought of as miniature bowling balls, whizzing around in precise
orbits. Rather, they’re fuzzy, statistical things describing paths that are only
vaguely determined.
3. Because precise velocity and position are required to use Newton’s laws to predict
future motion, the uncertainty principle and the Copenhagen interpretation abolish
the strict determinism of the Newtonian “clockwork universe.”
Not all physicists accept the Copenhagen interpretation. For all his life, Einstein was
among its staunchest critics. His famous remark, loosely paraphrased, “God does not
play dice with the universe,” expresses his rejection of quantum indeterminism.
(Einstein’s actual words are “But that He [God] would choose to play dice with the
world…is something that I cannot believe for a single moment.”) Today, a small group of
physicists is pursuing alternatives to the Copenhagen interpretation. Among these are
hidden variable theories that posit an underlying deterministic reality hidden from our
measurement by the uncertainty principle. However, recent experiments, to be described
in Lecture Twenty-One, place severe constraints on such theories.
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