Q5. How did the quantum theory of light come about?

It did not arise from any attempt to explain the behavior of light itself; by 1890 it was generally accepted that the electromagnetic theory could explain all of the properties of light that were then known.

Certain aspects of the interaction between light and matter that were observed during the next decade proved rather troublesome, however. The relation between the temperature of an object and the peak wavelength emitted by it was established empirically by Wilhelm Wien in 1893. This put on a quantitative basis what everyone knows: the hotter the object, the "bluer" the light it emits.

Q6. What is black body radiation?

All objects above the temperature of absolute zero emit electromagnetic radiation consisting of a broad range of wavelengths described by a distribution curve whose peak wavelength for a "perfect radiator" known as a black body is given by Wien's law

At ordinary temperatures this radiation is entirely in the infrared region of the spectrum, but as the temperature rises above about 1000K, more energy is emitted in the visible wavelength region and the object begins to glow, first with red light, and then shifting toward the blue as the temperature is increased.

This type of radiation has two important characteristics. First, the spectrum is a continuous one, meaning that all wavelengths are emitted, although with intensities that vary smoothly with wavelength. The other curious property of black body radiation is that it is independent of the composition of the object; all that is important is the temperature.

Q7. How did black body radiation lead to quantum physics?

Black body radiation, like all electromagnetic radiation, must originate from oscillations of electric charges which in this case were assumed to be the electrons within the atoms of an object acting somewhat as miniature Hertzian oscillators. It was presumed that since all wavelengths seemed to be present in the continuous spectrum of a glowing body, these tiny oscillators could send or receive any portion of their total energy. However, all attempts to predict the actual shape of the emission spectrum of a glowing object on the basis of classical physical theory proved futile.

In 1900, the great German physicist Max Planck (who earlier in the same year had worked out an empirical formula giving the detailed shape of the black body emission spectrum) showed that the shape of the observed spectrum could be exactly predicted if the energies emitted or absorbed by each oscillator were restricted to integral values of hn, where n is the frequency and h is a constant 6.626E­34 J s which we now know as Planck's Constant. The allowable energies of each oscillator are quantized, but the emission spectrum of the body remains continuous because of differences in frequency among the uncountable numbers of oscillators it contains.This modification of classical theory, the first use of the quantum concept, was as unprecedented as it was simple, and it set the stage for the development of modern quantum physics.

Q8. What is the photoelectric effect?

Shortly after J.J. Thompson's experiments led to the identification of the elementary charged particles we now know as electrons, it was discovered that the illumination of a metallic surface by light can cause electrons to be emitted from the surface. This phenomenon, the photoelectric effect, is studied by illuminating one of two metal plates in an evacuated tube. The kinetic energy of the photoelectrons causes them to move to the opposite electrode, thus completing the circuit and producing a measurable current. However, if an opposing potential (the retarding potential) is imposed between the two electrons, the kinetic energy can be reduced to zero so that the electron current is stopped. By observing the value of the retarding potential V_r, the kinetic energy of the photoelectrons can be calculated from the electron charge e, its mass m and the frequency of the incident light:

Q9. What peculiarity of the photoelectric effect led to the photon?

Although the number of electrons ejected from the metal surface per second depends on the intensity of the light, as expected, the kinetic energies of these electrons (as determined by measuring the retarding potential needed to stop them) does not, and this was definitely not expected. Just as a more intense physical disturbance will produce higher energy waves on the surface of the ocean, it was supposed that a more intense light beam would confer greater energy on the photoelectrons. But what was found, to everyone's surprise, is that the photoelectron energy is controlled by the wavelength of the light, and that there is a critical wavelength below which no photoelectrons are emitted at all.

Albert Einstein quickly saw that if the kinetic energy of the photoelectrons depends on the wavelength of the light, then so must its energy. Further, if Planck was correct in supposing that energy must be exchanged in packets restricted to certain values, then light must similarly be organized into energy packets. But a light ray consists of electric and magnetic fields that spread out in a uniform, continuous manner; how can a continuously-varying wave front exchange energy in discrete amounts? Einstein's answer was that the energy contained in each packet of the light must be concentrated into a tiny region of the wave front. This is tantamount to saying that light has the nature of a quantized particle whose energy is given by the product of Planck's constant and the frequency:

Einstein's publication of this explanation in 1905 led to the rapid acceptance of Planck's idea of energy quantization, which had not previously attracted much support from the physics community of the time. It is interesting to note, however, that this did not make Planck happy at all. Planck, ever the conservative, had been reluctant to accept that his own quantized-energy hypothesis was much more than an artifice to explain black-body radiation; to extend it to light seemed an absurdity that would negate the well-established electromagnetic theory and would set science back to the time before Maxwell.

Q10. Where does relativity come in?

Einstein's special theory of relativity arose from his attempt to understand why the laws of physics that describe the current induced in a fixed conductor when a magnet moves past it are not formulated in the same way as the ones that describe the magnetic field produced by a moving conductor. The details of this development are not relevant to our immediate purpose, but some of the conclusions that this line of thinking led to very definitely are. Einstein showed that the velocity of light, unlike that of a material body, has the same value no matter what velocity the observer has. Further, the mass of any material object, which had previously been regarded as an absolute, is itself a function of the velocity of the body relative to that of the observer (hence "relativity"), the relation being given by

in which m_o is the rest mass of the particle, v is its velocity with respect to the observer, and c is the velocity of light.

According to this formula, the mass of an object increases without limit as the velocity approaches that of light. Where does the increased mass come from? Einstein's answer was that the increased mass is that of the kinetic energy of the object; that is, energy itself has mass, so that mass and energy are equivalent according to the famous formula

The only particle that can move at the velocity of light is the photon itself, due to its zero rest mass.

Q11. Can the mass-less photon have momentum?

Although the photon has no rest mass, its energy, given by , confers upon it an effective mass of

and a momentum of

Q12. If waves can be particles, can particles be waves?

In 1924, the French physicist Louis de Broglie proposed (in his doctoral thesis) that just as light possesses particle-like properties, so should particles of matter exhibit a wave-like character. Within two years this hypothesis had been confirmed experimentally by observing the diffraction (a wave interference effect) produced by a beam of electrons as they were scattered by the row of atoms at the surface of a metal.

de Broglie showed that the wavelength of a particle is inversely proportional to its momentum:

Notice that the wavelength of a stationary particle is infinitely large, while that of a particle of large mass approaches zero. For most practical purposes, the only particle of interest to chemistry that is sufficiently small to exhibit wavelike behavior is the electron (mass 9.11E­31 kg).

Q13. Exactly what is it that is "waving"?

We pointed out earlier that a wave is a change that varies with location in a periodic, repeating way. What kind of a change do the crests and hollows of a "matter wave" trace out? The answer is that the wave represents the value of a quantity whose square is a measure of the probability of finding the particle in that particular location. In other words, what is "waving" is the value of a mathematical probability function.

Q14. What is the uncertainty principle?

In 1927, Werner Heisenberg proposed that certain pairs of properties of a particle cannot simultaneously have exact values. In particular, the position and the momentum of a particle have associated with them uncertainties x and p given by

As with the de Broglie particle wavelength, this has practical consequences only for electrons and other particles of very small mass. It is very important to understand that these "uncertainties" are not merely limitations related to experimental error or observational technique, but instead they express an underlying fact that Nature does not allow a particle to possess definite values of position and momentum at the same time. This principle (which would be better described by the term "indeterminacy" than "uncertainty") has been thoroughly verified and has far-reaching practical consequences which extend to chemical bonding and molecular structure.

Q15. Is the uncertainty principle consistent with particle waves?

Yes; either one really implies the other. Consider the following two limiting cases: · A particle whose velocity is known to within a very small uncertainty will have a sharply-defined energy (because its kinetic energy is known) which can be represented by a probability wave having a single, sharply-defined frequency. A "monochromatic" wave of this kind must extend infinitely in space:

But if the peaks of the wave represent locations at which the particle is most likely to manifest itself, we are forced to the conclusion that it can "be" virtually anywhere, since the number of such peaks is infinite! · Now think of the opposite extreme: a particle whose location is closely known. Such a particle would be described by a short wavetrain having only a single peak, the smaller the uncertainty in position, the more narrow the peak.

To help you see how waveforms of different wavelength combine, two such combinations are shown below:

It is apparent that as more waves of different frequency are mixed, the regions in which they add constructively diminish in extent. The extreme case would be a wavetrain in which destructive interference occurs at all locations except one, resulting in a single pulse:

Is such a wave possible, and if so, what is its wavelength? Such a wave is possible, but only as the sum (interference) of other waves whose wavelengths are all slightly different. Each component wave possesses its own energy (momentum), and adds that value to the range of momenta carried by the particle, thus increasing the uncertainty p. In the extreme case of a quantum particle whose location is known exactly, the probability wavelet would have zero width which could be achieved only by combining waves of all wavelengths-- an infinite number of wavelengths, and thus an infinite range of momentum dp and thus kinetic energy.

Q16. Are they particles or are they waves?

Suppose we direct a beam of photons (or electrons; the experiment works with both) toward a piece of metal having a narrow opening. On the other side there are two more openings, or slits. Finally the particles impinge on a photographic plate or some other recording device. Taking into account their wavelike character, we would expect the probability waves to produce an interference pattern of the kind that is well known for sound and light waves, and this is exactly what is observed; the plate records a series of alternating dark and light bands, thus demonstrating beyond doubt that electrons and light have the character of waves.

Now let us reduce the intensity of the light so that only one photon at a time passes through the apparatus (it is experimentally possible to count single photons, so this is a practical experiment). Each photon passes through the first slit, and then through one or the other of the second set of slits, eventually striking the photographic film where it creates a tiny dot. If we develop the film after a sufficient number of photons have passed through, we find the very same interference pattern we obtained previously.

There is something strange here. Each photon, acting as a particle, must pass through one or the other of the pair of slits, so we would expect to get only two groups of spots on the film, each opposite one of the two slits. Instead, it appears that the each particle, on passing through one slit, "knows" about the other, and adjusts its final trajectory so as to build up a wavelike interference pattern.

It gets even stranger: suppose that we set up a detector to determine which slit a photon is heading for, and then block off the other slit with a shutter. We find that the photon sails straight through the open slit and onto the film without trying to create any kind of an interference pattern. Apparently, any attempt to observe the photon as a discrete particle causes it to behave like one.

The only conclusion possible is that quantum particles have no well defined paths; each photon (or electron) seems to have an infinity of paths which thread their way through space, seeking out and collecting information about all possible routes, and then adjusting its behavior so that its final trajectory, when combined with that of others, produces the same overall effect that we would see from a train of waves of wavelength = h/mv.

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