Quantum Physics
Quantum Physics
As physicists began to probe the mysteries of the atom, they came across a number of unexpected results along the lines of Rutherford’s gold foil experiment. Increasingly, it became clear that things at the atomic level are totally unlike anything we find on the level of everyday objects. Physicists had to develop a whole new set of mechanical equations, called “quantum mechanics,” to explain the movement of elementary particles. The physics of this “quantum” world demands that we upset many basic assumptions—that light travels in waves, that observation has no effect on experiments, etc.—but the results, from transistor radios to microchips, are undeniable. Quantum physics is strange, but it works.
Electronvolts
Before we dive into quantum physics, we should define the unit of energy we’ll be using in our discussion. Because the amounts of energy involved at the atomic level are so small, it’s problematic to talk in terms of joules. Instead, we use the electronvolt (eV), where 1 eV is the amount of energy involved in accelerating an electron through a potential difference of one volt. Mathematically,
The Photoelectric Effect
Electromagnetic radiation transmits energy, so when visible light, ultraviolet light, X rays, or any other form of electromagnetic radiation shines on a piece of metal, the surface of that metal absorbs some of the radiated energy. Some of the electrons in the atoms at the surface of the metal may absorb enough energy to liberate them from their orbits, and they will fly off. These electrons are called photoelectrons, and this phenomenon, first noticed in 1887, is called the photoelectric effect.
The Wave Theory of Electromagnetic Radiation
Young’s double-slit experiment, which we looked at in the previous chapter, would seem to prove conclusively that electromagnetic radiation travels in waves. However, the wave theory of electromagnetic radiation makes a number of predictions about the photoelectric effect that prove to be false:
Predictions of the wave theory Observed result
Time lapse Electrons need to absorb a certain amount of wave energy before they can be liberated, so there should be some lapse of time between the light hitting the surface of the metal and the first electrons flying off. Electrons begin flying off the surface of the metal almost instantly after light shines on it.
Intensity The intensity of the beam of light should determine the kinetic energy of the electrons that fly off the surface of the metal. The greater the intensity of light, the greater the energy of the electrons. The intensity of the beam of light has no effect on the kinetic energy of the electrons. The greater the intensity, the greater the number of electrons that fly off, but even a very intense low-frequency beam liberates no electrons.
Frequency The frequency of the beam of light should have no effect on the number or energy of the electrons that are liberated. Frequency is key: the kinetic energy of the liberated electrons is directly proportional to the frequency of the light beam, and no electrons are liberated if the frequency is below a certain threshold.
Material The material the light shines upon should not release more or fewer electrons depending on the frequency of the light. Each material has a certain threshold frequency: light with a lower frequency will release no electrons.
Einstein Saves the Day
The young Albert Einstein accounted for these discrepancies between the wave theory and observed results by suggesting that electromagnetic radiation exhibits a number of particle properties. It was his work with the photoelectric effect, and not his work on relativity, that won him his Nobel Prize in 1921.
Rather than assuming that light travels as a continuous wave, Einstein drew on Planck’s work, suggesting that light travels in small bundles, called photons, and that each photon has a certain amount of energy associated with it, called a quantum. Planck’s formula determines the amount of energy in a given quantum:
where h is a very small number, J · s to be precise, called Planck’s constant, and f is the frequency of the beam of light.
Work Function and Threshold Frequency
As the wave theory correctly assumes, an electron needs to absorb a certain amount of energy before it can fly off the sheet of metal. That this energy arrives all at once, as a photon, rather than gradually, as a wave, explains why there is no time lapse between the shining of the light and the liberation of electrons.
We say that every material has a given work function, , which tells us how much energy an electron must absorb to be liberated. For a beam of light to liberate electrons, the photons in the beam of light must have a higher energy than the work function of the material. Because the energy of a photon depends on its frequency, low-frequency light will not be able to liberate electrons. A liberated photoelectron flies off the surface of the metal with a kinetic energy of:
Example
Two beams of light, one blue and one red, shine upon a metal with a work function of 5.0 eV. The frequency of the blue light is Hz, and the frequency of the red light is Hz. What is the energy of the electrons liberated by the two beams of light?
In order to solve this problem, we should translate h from units of J · s into units of eV · s:
We know the frequencies of the beams of light, the work function of the metal, and the value of Planck’s constant, h. Let’s see how much energy the electrons liberated by the blue light have:
For the electrons struck by the red light:
The negative value in the sum means that , so the frequency of the red light is too low to liberate electrons. Only electrons struck by the blue light are liberated.
The Bohr Model of the Atom
Let’s now return to our discussion of the atom. In 1913, the Danish physicist Niels Bohr proposed a model of the atom that married Planck’s and Einstein’s development of quantum theory with Rutherford’s discovery of the atomic nucleus, thereby bringing quantum physics permanently into the mainstream of the physical sciences.
The Problem with Rutherford’s Model
Light and other electromagnetic waves are emitted by accelerating charged particles. In particular, the electrons being accelerated in orbit about the nucleus of an atom release a certain amount of energy in the form of electromagnetic radiation. If we recall the chapter on gravity, the radius of an object in orbit is a function of its potential energy. If an electron gives off energy, then its potential energy, and hence the radius of its orbit about the nucleus, should decrease. But according to Rutherford’s model, any radiating electron would give off all its potential energy in a fraction of a second, and the electron would collide with the nucleus. The fact that most of the atoms in the universe have not yet collapsed suggests a fundamental flaw in Rutherford’s model of electrons orbiting nuclei.
The Mystery of Atomic Spectra
Another puzzling phenomenon unexplained by Rutherford’s model, or anything else before 1913, is the spectral lines we see when looking through a spectroscope. A spectroscope breaks up the visible light emitted from a light source into a spectrum, so that we can see exactly which frequencies of light are being emitted.
The puzzling thing about atomic spectra is that light seems to travel only in certain distinct frequencies. For instance, we might expect the white light of the sun to transmit light in an even range of all different frequencies. In fact, however, most sunlight travels in a handful of particular frequencies, while very little or no light at all travels at many other frequencies.
Bohr’s Hydrogen Atom
Niels Bohr drew on Rutherford’s discovery of the nucleus and Einstein’s suggestion that energy travels only in distinct quanta to develop an atomic theory that accounts for why electrons do not collapse into nuclei and why there are only particular frequencies for visible light.
Bohr’s model was based on the hydrogen atom, since, with just one proton and one electron, it makes for the simplest model. As it turns out, Bohr’s model is still mostly accurate for the hydrogen atom, but it doesn’t account for some of the complexities of more massive atoms.
According to Bohr, the electron of a hydrogen atom can only orbit the proton at certain distinct radii. The closest orbital radius is called the electron’s ground state. When an electron absorbs a certain amount of energy, it will jump to a greater orbital radius. After a while, it will drop spontaneously back down to its ground state, or some other lesser radius, giving off a photon as it does so.
Because the electron can only make certain jumps in its energy level, it can only emit photons of certain frequencies. Because it makes these jumps, and does not emit a steady flow of energy, the electron will never spiral into the proton, as Rutherford’s model suggests.
Also, because an atom can only emit photons of certain frequencies, a spectroscopic image of the light emanating from a particular element will only carry the frequencies of photon that element can emit. For instance, the sun is mostly made of hydrogen, so most of the light we see coming from the sun is in one of the allowed frequencies for energy jumps in hydrogen atoms.
Analogies with the Planetary Model
Because the electron of a hydrogen atom orbits the proton, there are some analogies between the nature of this orbit and the nature of planetary orbits. The first is that the centripetal force in both cases is . That means that the centripetal force on the electron is directly proportional to its mass and to the square of its orbital velocity and is inversely proportional to the radius of its orbit.
The second is that this centripetal force is related to the electric force in the same way that the centripetal force on planets is related to the gravitational force:
where e is the electric charge of the electron, and Ze is the electric charge of the nucleus. Z is a variable for the number of protons in the nucleus, so in the hydrogen atom, Z = 1.
The third analogy is that of potential energy. If we recall, the gravitational potential energy of a body in orbit is . Analogously, the potential energy of an electron in orbit is:
Differences from the Planetary Model
However, the planetary model places no restriction on the radius at which planets may orbit the sun. One of Bohr’s fundamental insights was that the angular momentum of the electron, L, must be an integer multiple of . The constant is so common in quantum physics that it has its own symbol, . If we take n to be an integer, we get:
Consequently, . By equating the formula for centripetal force and the formula for electric force, we can now solve for r:
Don’t worry: you don’t need to memorize this equation. What’s worth noting for the purposes of SAT II Physics is that there are certain constant values for r, for different integer values of n. Note also that r is proportional to , so that each successive radius is farther from the nucleus than the one before.
Electron Potential Energy
The importance of the complicated equation above for the radius of an orbiting electron is that, when we know the radius of an electron, we can calculate its potential energy. Remember that the potential energy of an electron is . If you plug in the above values for r, you’ll find that the energy of an electron in a hydrogen atom at its ground state (where n = 1 and Z = 1) is –13.6 eV. This is a negative number because we’re dealing with potential energy: this is the amount of energy it would take to free the electron from its orbit.
When the electron jumps from its ground state to a higher energy level, it jumps by multiples of n. The potential energy of an electron in a hydrogen atom for any value of n is:
Frequency and Wavelength of Emitted Photons
As we said earlier, an excited hydrogen atom emits photons when the electron jumps to a lower energy state. For instance, a photon at n = 2 returning to the ground state of n = 1 will emit a photon with energy . Using Planck’s formula, which relates energy and frequency, we can determine the frequency of the emitted photon:
Knowing the frequency means we can also determine the wavelength:
As it turns out, this photon is of slightly higher frequency than the spectrum of visible light: we won’t see it, but it will come across to us as ultraviolet radiation. Whenever an electron in a hydrogen atom returns from an excited energy state to its ground state it lets off an ultraviolet photon.
Example
A hydrogen atom is energized so that its electron is excited to the n = 3 energy state. How many different frequencies of electromagnetic radiation could it emit in returning to its ground state?
Electromagnetic radiation is emitted whenever an electron drops to a lower energy state, and the frequency of that radiation depends on the amount of energy the electron emits while dropping to this lower energy state. An electron in the n = 3 energy state can either drop to n = 2 or drop immediately to n = 1. If it drops to n = 2, it can then drop once more to n = 1. There is a different amount of energy associated with the drop from n = 3 to n = 2, the drop from n = 3 to n = 1, and the drop from n = 2 to n = 1, so there is a different frequency of radiation emitted with each drop. Therefore, there are three different possible frequencies at which this hydrogen atom can emit electromagnetic radiation.
Wave-Particle Duality
The photoelectric effect shows that electromagnetic waves exhibit particle properties when they are absorbed or emitted as photons. In 1923, a French graduate student named Louis de Broglie (pronounced “duh BRO-lee”) suggested that the converse is also true: particles can exhibit wave properties. The formula for the so-called de Broglie wavelength applies to all matter, whether an electron or a planet:
De Broglie’s hypothesis is an odd one, to say the least. What on earth is a wavelength when associated with matter? How can we possibly talk about planets or humans having a wavelength? The second question, at least, can be easily answered. Imagine a person of mass 60 kg, running at a speed of 5 m/s. That person’s de Broglie wavelength would be:
We cannot detect any “wavelength” associated with human beings because this wavelength has such an infinitesimally small value. Because h is so small, only objects with a very small mass will have a de Broglie wavelength that is at all noticeable.
De Broglie Wavelength and Electrons
The de Broglie wavelength is more evident on the atomic level. If we recall, the angular momentum of an electron is . According to de Broglie’s formula, mv = h/. Therefore,
The de Broglie wavelength of an electron is an integer multiple of , which is the length of a single orbit. In other words, an electron can only orbit the nucleus at a radius where it will complete a whole number of wavelengths. The electron in the figure below completes four cycles in its orbit around the nucleus, and so represents an electron in the n = 4 energy state.
The de Broglie wavelength, then, serves to explain why electrons can orbit the nucleus only at certain radii.
Example
Which of the following explains why no one has ever managed to observe and measure a de Broglie wavelength of the Earth?
(A) The Earth is traveling too slowly. It would only have an observable de Broglie wavelength if it were moving at near light speed.
(B) The Earth is too massive. Only objects of very small mass have noticeable wavelengths.
(C) The Earth has no de Broglie wavelength. Only objects on the atomic level have wavelengths associated with them.
(D) “Wavelength” is only a theoretical term in reference to matter. There is no observable effect associated with wavelength.
(E) The individual atoms that constitute the Earth all have different wavelengths that destructively interfere and cancel each other out. As a result, the net wavelength of the Earth is zero.
This is the sort of question you’re most likely to find regarding quantum physics on SAT II Physics: the test writers want to make sure you understand the theoretical principles that underlie the difficult concepts in this area. The answer to this question is B. As we discussed above, the wavelength of an object is given by the formula = h/mv. Since h is such a small number, mv must also be very small if an object is going to have a noticeable wavelength. Contrary to A, the object must be moving relatively slowly, and must have a very small mass. The Earth weighs kg, which is anything but a small mass. In fact, the de Broglie wavelength for the Earth is m, which is about as small a value as you will find in this book.
Heisenberg’s Uncertainty Principle
In 1927, a young physicist named Werner Heisenberg proposed a counterintuitive and startling theory: the more precisely we measure the position of a particle, the less precisely we can measure the momentum of that particle. This principle can be expressed mathematically as:
where is the uncertainty in a particle’s position and is the uncertainty in its momentum.
According to the uncertainty principle, if you know exactly where a particle is, you have no idea how fast it is moving, and if you know exactly how fast it is moving, you have no idea where it is. This principle has profound effects on the way we can think about the world. It casts a shadow of doubt on many long-held assumptions: that every cause has a clearly defined effect, that observation has no influence upon experimental results, and so on. For SAT II Physics, however, you needn’t be aware of the philosophical conundrum Heisenberg posed—you just need to know the name of the principle, its meaning, and the formula associated with it.
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