OBJECTIVES
OBJECTIVES
After you have finished this lesson you should understand and know about
* YoungÕs double slit experiment.
* the photo electric effect.
* quantum mechanics.
* the principles of uncertainty, and correspondence.
* complementary properties.
Throughout these commentaries I have utilized current physical theories and
concepts to describe and explain various phenomena. These theories are based on
a Quantum Mechanical view of the universe in contrast to a Newtonian or
Classical Mechanical view that held sway from the seventeenth century to the
turn of the twentieth century. The quantum mechanical view grew from
seemingly contradictory experimental evidence as to the nature of light. In 1704
Issac Newton described light as streams of particles or corpuscles, while a
contemporary of his, Christian Huygens, thought of light as waves. It seemed
evident that such views were mutually exclusive; waves were objects that
extended in space showing repeating patterns, while particles had definite sizes
and boundaries. Further they interacted with their environment in distinct
waysŃparticles as one localized cohesive unit, waves as delocalized phenomena
of many parts.
In 1801 Thomas Young performed his famous Double Slit experiment where
he seemed to show conclusively that light was constituted of waves. The
experiment was a simple one. He shined light through a screen that had two
parallel slits separated by a very short distance. When the light coming from the
two slits was shone onto a screen some distance away, a distinctive pattern
always appeared on the screen. The pattern was symmetrically made up of a
bright central band of light followed by alternating dark and bright bands, the
bright bands were of decreasing intensity as shown in the following diagrams.
Remembering that intensity of light corresponds to the amplitude of light waves,
we can easily explain such patterns as the result of constructive and destructive
interference of light waves emanating from the two slits. Such patterns can also
be produced by using surface water waves and a barrier with two closely spaced
openings in it and observing the effect some distance away.
In the latter part of the nineteenth century various researchers found that
when light was shone onto a metal surface, electrons were ejected from that
surface. This was called the photoelectric effect. The effect of the ejected
electrons was to produce an electric current in a circuit as shown below.
Meter to read current
The photoelectric effect had some very noticeable characteristics:
1. If the frequency of the incident light was below a certain threshold frequency,
no current would flow; therefore, no electrons would be ejected. The
threshold frequency was different for different metals.
2. If the frequency of light was above the threshold frequency, the current would
flow instantaneously.
3. The magnitude of the electric current was proportional to the intensity of
incident light when the frequency of light was above the threshold frequency.
4. The maximum energy of the emitted electrons was proportional to the
amount the incident frequency was above the threshold frequency.
If one assumes that light is constituted of electromagnetic waves and tries to
explain the photoelectric effect, one would come to the following conclusions:
a. The larger the amplitude (i.e., intensity) of the light wave, the higher the
probability that an electron will be ejected from the metallic surface, no
matter the frequency of the light; therefore, the greater the magnitude of
the electric current produced.
[This seems evident if one uses the analogy of water waves crashing onto
a pebble beach; a big wave will be more likely than a small wave to
dislodge a pebble.]
b. The effects of the incident waves will be cumulative, thus an electric
current will gradually build up in time, no matter the frequency, while
the frequency of the wave will have an effect on the time it takes to reach
an equilibrium current.
c. The maximum kinetic energy of the electrons being ejected from the
surface will depend on the amplitude of the incident light, no matter the
frequency.
These conclusions are totally at odds with the observed properties of the
photoelectric effect!
On the other hand if one postulates that light is constituted of particles whose
energy is related to the frequency of light through the Planck relationship,
E =hu
one can explain all the observed effects. Only particles of light of great enough
energy, therefore frequency, will eject electrons and the effect will be
instantaneous, thus the threshold frequency and instantaneous electric current.
If light is streams of particles, the intensity of light corresponds to the
number of particles arriving in a given area per second. Thus, no matter the
intensity, if the frequency is below the threshold frequency, it will have no
effect. However, if the
frequency is above the threshold, the greater the number of particles of light
arriving in a given time, the greater the number of electrons ejected, thus the
greater the magnitude of the electric current produced. Further the higher the
energy of the light particles, the greater the maximum kinetic energy of the
ejected electrons. Thus all the characteristics 1 through 4 above can be explained!
Einstein did this in 1905 and was rewarded with the Nobel prize. He postulated
that light was constituted of particle-like packets of electromagnetic energy that
he called photons, and the photon's energy was related to its frequency by
PlanckÕs relationship.
Where did Planck get his idea that energy could be related to frequency in the
above manner? In the late nineteenth century he, along with many other
scientists, was trying to explain the characteristic shape of incandescent spectra.
Recall that the form of this spectra does not depend on the type of material
emitting it, only on the temperature of the material. He considered that the
material could be modeled by closely packed harmonically oscillating molecules,
which could be considered as atoms joined to each other by springs. At this time it
was believed and taught at all colleges and universities that the energy of a
oscillator was proportional to the square of the amplitude of the oscillation. Thus
it would seem that its value could be any positive real number1. Planck
hypothesized that the energy of these molecular oscillators could only be certain
fixed values, and he further hypothesized that electromagnetic energy (i.e., light)
could only be absorbed and emitted by these oscillators in discrete bundles called
quantums of energy, thus changing the energy of the oscillator to a higher or
lower fixed value. The energy of the quanta of light thus absorbed and emitted
could have only certain discrete values, which he related to their frequency by his
famous formula. As the molecules were tightly packed together, light would be
continually absorbed and reemitted until some type of statistical equilibrium
mixture of frequencies would be present. By taking into account his various
hypotheses and postulates, he could then derive the correct shape for the plot of
the intensity of emitted light versus its frequency for an incandescent spectra.
Neither Plank nor Einstein could justify their various assumptions, except by
the fact that they worked and produced a useful scientific theory. In order to
describe discrete emission and absorption spectra, another theory (that also could
not be justified) was produced in 1912, this time by Neils Bohr. He postulated that
electrons circled nuclei in fixed orbits at certain frequencies, in a manner similar
to the earth circling the sun, to form atoms. The orbits and the energies of the
electrons could be changed by absorbing or emitting quanta of electromagnetic
radiation (i.e., photons). The frequencies of these photons corresponded to the
difference of the energies of the electrons in the initial and final orbits divided by
PlanckÕs constant. The trouble with this theory was that scientists in 1912 agreed
that accelerating charges (e.g., electrons) radiate away energy thus orbiting
electrons would continuously loose energy and fall into the nucleus! Bohr got
around this problem by postulating that electrons could only orbit at certain fixed
distances from the nucleus and at these distances the electrons did not radiate
energy even though they were accelerating. He called such orbits stationary
orbits. His theory could explain the absorption and emission spectra of hydrogen
atoms, but again we had a successful theory that could not be justified!
In 1924 Louis de Broglie began to solve the mystery by suggesting that if light
had both wave and particle properties, then there was no reason that electrons
should not. He postulated that the wavelength of such matter waves be given by
l = \F(h,p)
where h is PlanckÕs constant, p is the momentum of the particle, which is equal to
the particleÕs velocity times its mass, and l is the wavelength, now called the de
Broglie wavelength. De Broglie arrived at this formula by considering how the
wavelength of a photon was related to the momentum of a photon according to
EinsteinÕs Theory of Special Relativity. If electrons were waves and they were
constrained to exist in a closed volume of space, then they must reflect off the
boundaries of this volume. The only way incident and reflected waves can
maintain their existence in such a situation is if they constructively interfere to
form standing waves. Standing waves can only be of certain wavelengths,
determined by the dimensions of the volume of space, and at certain frequencies,
determined by the properties of that region of space. Thus BohrÕs stationary waves
must be de Broglie standing waves! The energies of these electronic standing
waves were then given by PlanckÕs relationship. Hence by asserting that matter
must have wave properties as well as particle properties just as light has particle
properties as well as wave properties, not only could BohrÕs picture of the atoms be
justified, but so could PlanckÕs assumptions in describing incandescent spectra.
In the seventeenth century Issac Newton produced an equation that
described the mechanical behavior of matter. This was based on the view that
matter was particle-like. His equation is very successful. It is regularly used by
the military to calculate the trajectories of shells, and when combined with
NewtonÕs theory of gravity, it describes very accurately the orbits of the planets
around the sun and aids in planning space missions.
James Clarke Maxwell in the late nineteenth century found equations that
described the behavior of electricity and magnetism and the interaction of matter,
constituted of particles, with electric, magnetic, and electromagnetic fields (the
latter considered as waves). If matter and light have both wave and particle
properties, one must find, then, new equations to supersede NewtonÕs and
MaxwellÕs equations. In the mid-nineteen twenties Erwin Schrdinger developed
an equation that described the behavior of matter waves in various environments,
including the interaction of matter with light. This equation, now known as
SchrdingerÕs Equation, describes the behavior of all objects and processes in the
universe. Does this mean that NewtonÕs equation is now invalid? No; NewtonÕs
equation still works fine in the situations where it worked before, but
SchrdingerÕs equation gives a new interpretation to some of the Newtonian
concepts. The big difference is in situations where NewtonÕs equation does not
work, but SchrdingerÕs equation doesŃsuch situations as when the wave nature
of matter is particularly pronounced, which is the case for microscopic objects.
This is an example of The Correspondence Principle mentioned earlier in the
course, where the new, more general theory corresponds to elements of the old,
more restrictive theory.
In the years between 1924 and 1932 this new view of nature was refined into a
sophisticated mathematical theory called Quantum Mechanics or Quantum
Theory. Many famous scientists were involved with this effort, among them
Werner Heisenberg, Paul Dirac, Max Born, and John Von Neumen. However, not
all the glitches were ironed out. Quantum Mechanics and the theory of Special
and General Relativity had to be merged and in fact the General Theory still has
not been conclusively merged! Further, the Quantum Theory produced many
philosophical questions that are still being debated.
If objects have both particle and wave properties, how shall we create
pictures and models that display this? Clearly wave-like properties and particle-
like properties are qualitatively different and describe different aspects and
properties of a given object. Thus we need both descriptions to completely
characterize an object. The two sets of properties, wave-like and particle-like, are
called complementary. It seems that the best wave to visualize objects in the light
of this wave-particle duality is as wave packets.
Wave packets are made up of the interference of infinitely many waves, in
fact all waves with a wavelength belonging to a range
l0 ® l0 + Dl
For example, l0 could be 10m and Dl = 2m; then all wavelengths in the range of
10m to 12m could be found in the wave packet. Dl is called the spread or
uncertainty of the wavelength. Not all waves of wavelengths belonging to this
range occur with equal amplitude in the wave packet; for instance, the l = 10m
wave may have an amplitude of 1m, while the l = 11m wave may have an
amplitude of 6m. As amplitudes of waves are additive2, one could just as well
imagine that there are 6 of the l = 11m waves present. In this way one can talk
about the presence of a number of waves of certain wavelengths and one can
calculate the average wavelength. For instance, if one has 1 l=10m and 6 l=11m
waves, the average wavelength is
= \F(1x10 + 6x11,1+6) =
10.86m
The probability of the wavelength being 11m is \F(6,7), while the probability of the
wavelength being 1m is \F(1,7). One can describe the wave packet in terms of the
probability of finding a wave of wavelength l around the average wavelength ,
which is called the wavelength of the wave packet. The probability is generally
higher for l close to and decreases as one gets closer to l0 or l0+Dl. The
spread or uncertainty of the wavelengths, Dl, in the wave packet can also be
identified as the statistical quantity, dispersion, which is the average of the sum of
the squares of the deviation of the possible values of l from the average value .
What is the position of the wave packet? Clearly we can answer yes to the
question ŅIs the wave packet here?Ó if the here refers to any position belonging to
X0+DX, where DX is the width of the wave packet and X0 is the starting point of
the wave packet. Thus there is an uncertainty of DX in the position. The average
position can be calculated and has a definite value, but the actual ŅpositionÓ is
dispersed about this average value.
The intensity of a wave at any position in space and instant of time
corresponds to its amplitude squared, while the intensity of a stream of particles
corresponds to the rate at which particles are arriving at that position and time.
Thus at positions of high intensity one would expect to find the greatest number of
particles. A wave packet has both particle- and wave-like properties, and the
greatest intensity of the wave packet corresponds to its position of largest
amplitude, which is the position where there is the highest probability of finding
the wave packet. Properties relating to the wave packet as a whole can be viewed
as particle-like properties, while properties relating to the constituent waves can
be viewed as wave-like properties. Thus the greatest probability of finding the
particle that is associated with the wave packet is at the position of greatest
amplitude of the wave packet. If the wave packet is moving, one can track the
motion of the average position, which is then the motion of the particle associated
with the wave packet.
Let us examine the relationship between the width of the wave packet, DX,
and the waves that make up the wave packet more closely. Clearly waves of
wavelengths greater than the width of the wave packet cannot be in the wave
packet, as the wave packet could not sense their presence (the whole wave packet
would fit into a trough of such a wave), so DX³l. This implies, further, that the
spread in the possible wavelengths cannot be greater than the width of the wave
packet (i.e., DX³Dl), otherwise there would be a wave present in which the wave
packet could fit, thus
\F(DX,Dl) ³
1
As the number of waves in the wave packet decreases, the width of the wave
packet increases until in the limit; when there is just one wave of just one
wavelength, the width of the wave packet becomes infinite. When this occurs,
Dl=0 and DX=„ (i.e., the value of the wavelength of the wave packet is totally
certain, while the position of the wave packet is totally uncertain.) Using de
BroglieÕs relationship between momentum and wavelength, we obtain
(DX) (DP) ³ h
This is called HeisenbergÕs Uncertainty Relationship, which shows that the
momentum and position of a wave packet cannot be simultaneously measured
with infinite precision. It follows from the above definition of complementary
properties that momentum and position are such properties. It is possible to
obtain various forms of HeisenbergÕs Uncertainty Relationships, such as the
energy time version \X((DE) (Dt)³h)3, where DE is the uncertainty in the energy of
a system during a process involving that system that takes a time Dt. If the
position and momentum of an object cannot be given exactly simultaneous values,
then applying the Principle of Reality, we come to the conclusion that objects do
not have precise values of both position and momentum at a given instant of time
or for any pair of complementary properties! We also see from above that objects
do not have exact positions as there is always a dispersion in value of the position!
How does that relate to everyday objects such as you and me? Well there must
always be an uncertainty in our positions. If my mass is 62.6 kg and my speed is 1
m/s, then my momentum is 66.26 kgm/s. My average de Broglie wavelength is
given by
l = \F(6.626x10-34,66.26)
= 10-35m
and the uncertainty of my position is physically limited to be greater than 10-35 of a
meter; that is, my position can only be determined within 10-35 of a meter, even
with perfect measuring devices. However, this is not very significant for everyday
events and we do not have to take this uncertainty into account when we are
describing the behavior of such macroscopic objects! Further, one will not observe
wave effects unless the waves interact with objects that have spacings in their
structure that are distances of approximately the wavelengthŃsuch distances as
those between slits (as in YoungÕs experiment) or distances between atoms in
molecules or wavelengths of other waves. Thus my de Broglie wave nature will
not interact with the environment on the surface of the earth; there is nothing in
it that I can interact with! Wavelengths as short as 10-35 correspond to very high
energies. In environments where such energies exist, such as the beginning of
the universe, my wave nature would be emphasized and definitely to my
disadvantage!!
Now that we have built up a pictorial model for objects with wave and particle
properties, let us consider atoms and molecules once more. The electronic wave
packets are trapped around the nuclei as three dimensional standing wave
patterns. These wave patterns form harmonic series of increasing frequencies,
and thus energies. The different members of the series are differentiated by the
number of nodes that the standing wave pattern exhibits. These nodes form two
dimensional surfaces. The amplitude of the standing waves is zero at the nodes;
thus, using the probabilistic interpretation suggested by Max Born, we see that
the probability of finding an electron at one of these nodes is zero! Each standing
wave pattern, however, is one electron and the amplitude squared at a point in
space represents the probability of finding an electron there. For the 2s standing
wave pattern one finds that the probability distribution with respect to distance
from the nucleus of find finding a single electron is given by
Thus one sees that there is a finite probability of finding the electron in region I
and a finite probability in region II, but zero probability of finding it between
these two regions. Therefore, the electron must go from region I to region II
without passing through the boundary to the regions!! We can resolve this magical
situation by observing that exact position does not have a physical meaning, an
electron surrounding a nucleus should be viewed as a fuzzy cloud of probabilities,
or more concretely, of negative electric charge, rather than a point-like object.
Why then do electrons sometimes act like point like objects? The aspect of the
electron that is observed is determined by the nature of the instrument interacting
with it. If this instrument is sensitive to wave properties, it will react to those
properties; if it is sensitive to particle properties, it will react to those.