Quantum Theory 1

Planck & de Broglie's hypothesis:

We have already discussed Planck's hypothesis of photon quantization. Based on this hypothesis the energy of the electromagnetic wave of frequency f is quantized in units of hf, where h is known as Planck's constant and has a measured value of

h = 6.63 x 10 ^-34 J.s = 4.14 x 10^-15 eV.s

This means that electromagnetic energy can only interact in units of hf. To see some of the predicted consequence of this, let us consider ejection of electrons from a metal surface in the presence of light. When we shine light on a metal electrons could absorb the energy from light and get free. In this way, we expect to create an electric current by shining light on the metal. If we model light the same as heat, then we would expect it to be just a matter of time for the electron to leave the surface once we shine light on it; i.e. the brighter the light, the faster and more numerous the number of electrons ejected.

If we instead use the photon model of light, then an electron is released provided that it absorbs a single photon. (This is so, because the probability of two identical photons being absorbed is vanishingly small for reasonable brightness.) This model would then predict that for a given color of light (i.e. given f) if hf is not greater than the binding energy of the electron for that metal, then no electron will be ejected, even if we increase the brightness of light source. As you may have guessed, experimental evidence corroborates the photon picture. In terms of energy balance, we could write:

KE = hf - f

where KE is the kinetic energy of the ejected electron and f is the so called "work-function" of the metal - a bulk binding energy. Below is a table of work functions, in eV, for a sample of metals:

Metal	f (eV)
Cs	1.9
K	2.2
Na	2.3
Li	2.5
Ca	3.2
Cu	4.5
Ag	4.7
Pt	5.6

Notice that, for example, green light of 514.5 nm wavelength has a (per photon) energy of

E = hf = hc/l= (4.14 x 10^-15) (3x 10⁸) / (514.5 x 10^-9) = 2.4 eV

So, if we were to shine this light on the above metals, only the first three would create any photoelectrons; the rest would not, no matter how intense a light source we were to use!

Another type of experiment that could only be understood under this new "quantum" picture is the reverse case, i.e. photon production by bombarding the surface with energetic electrons. The photons that are emitted in this process have very short wavelength (0.01 - 10 nm). This is because it takes highly energetic electrons that are accelerated in kV range to create them. As you may have guessed, when these were first discovered it was not obvious that they were " high energy photons", so they were called X-rays (even higher energy photons than X-rays - in MeV range - are called g-rays). In X-ray production a beam of electrons are accelerated through several tens of kV and then impinged on a metal target. These electrons lose energy as they enter the metal and are eventually brought to rest in it. In the process of decelerating electrons give off their kinetic energy to the metal atoms, which in tern desiccates this energy by emitting X-rays. Since the metal's work function is in the range of eV, it can be neglected in this consideration. So, for X-ray production all that we need to balance is the energy of the electron and the emitted photon, i.e.

E = hf = hc/l = KE = eV

Here, V is the electric potential that is used to accelerate the electron beam. Please note that the above wavelength is the minimum wavelength of the X-rays emitted for a fixed accelerating voltage V. This, as in the case of photoelectric effect, is in agreement with experimental evidence.

The short wavelength of X-rays make them very useful for probing small entities, such as atoms and molecules. We saw how the wavelength of light limits our ability to resolve objects smaller than about half the wavelength, in optical microscopy. The same limiting phenomena, diffraction, can be used to get information about the structure of matter made with small dimensions of the order of X-ray wavelengths. Through X-ray diffraction we can learn about the structure of crystals. By cataloging this information, we could then learn about unknown substances through X-ray scattering analysis.

Another experiment that shook the foundation of 19^th century physics was that when electrons were scattered off of a thin target they behaved just as light did: they diffracted! This suggested that particles could also exhibit wave qualities. But why is this not observed in our every day macroscopic lives? Louis de Broglie solved this puzzle by hypothesizing that any matter object could act as a wave; its wavelength is related to its momentum, the same as the photon's, i.e.

l = h / p

Here again, h is Planck's constant.

This explained why electrons and other small particles do exhibit diffraction from atomic structures, but that macroscopic objects do not.

Notice that the larger the momentum, the smaller is the object's de Broglie wavelength. If the object is macroscopic, then even at rest it has a vanishingly small de Broglie wavelength. But microscopic objects could have small enough momenta to end up with de Broglie wavelength that are comparable with atomic dimensions. To make the momentum a bit larger, we just need to add more mass; i.e. instead of using an electron, we could use a proton, or even an atom! Atoms are many thousands more massive than the electrons. So, their de Broglie wavelength could be much smaller.

This is the basis of Atom Optics!

Heisenberg Uncertainty Principle:

One of the problems with describing particles as waves is that waves cannot be completely fixed in place, but particles could. So, how do we resolve this apparent contradiction? One way out of this, is based on more familiar Fourier components that we have already examined in the context of waves. We could make a localized traveling pulse by combining many harmonic waves of different frequencies. In this picture, however, the frequency dependence of these harmonic waves and the localization of the summed pulse become interdependent. This interdependence, for example, can be viewed as the connection between the many different frequencies required with a highly localized pulse. In fact, if we include a continuum of frequencies in the make up of the pulse (i.e. a continuum of harmonic waves of differentially varying frequencies), then the range (width) of this frequency continuum is almost inversely proportional to the location of the pulse. This outcome, which is just a natural property of waves, is called the Heisenberg's Uncertainty Principle. Mathematically it is written as:

(Dx)(Dp)> or = h/4p

Where Dx, and Dp refer to the spread (uncertainty) in the value of x and p, respectively. Of course, h (you guessed it!) is the familiar Planck's constant!

This was one of the earliest consequences of wave mechanics applied to microscopic objects through Planck's quantization and de Broglie's hypothesis. This odd result, as it turns out, directly out of the formal treatments of Quantum Mechanics; and it is also observed in nature itself! So, evidently nature agrees with Heisenberg's Uncertainty Principle. To date, we have not found any violation of this intuitively odd principle.

To fully appreciate this principle, let us consider a hypothetical experiment through which we could determine the position of a traveling electron. For example, let us imagine that we place a very sensitive electron detector at position x = 0, i.e. origin of a fixed coordinate axis. This way, when the detector gives a signal then we know that an electron was located at the origin. How well could we also know the velocity (and therefore the momentum) of the electron? According to Heisenberg's Uncertainty Principle, the lower is our uncertainty if this position measurement, the less we would be able to know of the value of the electron's speed. Furthermore, it states that this vagueness is not due to our lack of imagination in creating a clever experiment, or better apparatus, but rather a consequence of nature - a limitation of some sort, or a way out; depending on your point of view!

Another aspect of this principle that is worth investigating is that as far as macroscopic objects are concerned, this principle is in a way irrelevant. This is because both Dx and Dp are Huge as far as the value of h is concerned. That is to say, in the case of a bird, for example, flying in a straight line (they some time do this!) a 0.001% uncertainty in position measurement (clearly an excellent precision) is still so large that results in an unmeasureable small value of Dp, using the minimum relation of (Dx)(Dp) = h. Therefore neither the position nor the velocity measurements are limited in their precision of measurement. So, this principle really matters when we are dealing with microscopic objects possessing "nanoscale" positions and equally small momenta.

Notice also, that using de Broglie's hypothesis we could rewrite Heisenberg's Principle as:

(Dx)(Dl) >= l²/4p

This is a consequence of the fact that p = h / l. So, to obtain a relation between Dp h and Dl we have to take the derivative of p with respect to l.

Alternatively, using the relationships between E, f, v and t we could connect the uncertainty in energy to the uncertainty in time interval measurement, i.e.

(DE)(Dt) >= h/4p

Last Modified January 17, 2006 malekis@union.edu