Section 5–7 Short distances

(Molecular sizes / Nuclear sizes / Uncertainty principle)

In this section, the three interesting points discussed are molecular sizes, nuclear sizes, and the uncertainty principle.

1. Molecular sizes:

“…Electron micrograph of some virus molecules. The ‘large’ sphere is for calibration and is known to have a diameter of 2 × 10⁻⁷ meter (Feynman et al., 1963, section 5.7 Short distances).”

In an optical microscope, we can measure a small object by using an eyepiece reticule which is a small piece of glass insert with a ruler scale inscribed in it. This method of measurement is limited by the wavelength of visible light (about 5 × 10⁻⁷ meter) that “see” microscopic objects. For smaller objects such as virus molecules, we need to use electron microscopes instead of optical microscopes. In short, an electron microscope utilizes electromagnetic lenses to focus electrons into a very thin beam as compared to an optical microscope that utilizes glass lenses to focus a light beam.

We can continue to measure sizes of microscopic objects to smaller scales by selecting electromagnetic waves that have shorter wavelengths. For x-rays, we need to first determine its very short wavelength by observing the diffraction angles from a plane ruled grating. Then, from a measurement of the pattern of the scattering of the x rays from a crystal, we can determine the atomic spacings that have a dimension of about 10⁻¹⁰ m. It is worth mentioning that the Nobel Prize in Physics 1914 was awarded to Max von Laue for his discovery of the diffraction of X-rays by crystals and the Nobel Prize in Physics 1915 was awarded to Sir William Henry Bragg and his son William Lawrence Bragg for their analysis of crystal structure by using X-rays.

2. Nuclear sizes:

“…we find that the radii of the nuclei are from about 1 to 6 times 10⁻¹⁵ meter. The length unit 10⁻¹⁵ meter is called the fermi, in honor of Enrico Fermi (Feynman et al., 1963, section 5.7 Short distances).”

Dr. Sands explains a measurement of nuclear sizes by passing high energy particles through a thin slab of material and by observing the number of particles which come out. This method is based on the chance that some small particles will hit the nuclei in a trip. Suppose in an area A of the slab that has N atoms, the fraction of the area “covered” by the nuclei is about Nσ/A in which σ is the apparent area of a nucleus (or the effective cross section). If the number of particles of a beam which arrive at the slab is n₁ and the number of particles which come out from the other end is n₂, then the fraction which is blocked is (n₁ − n₂)/n₁, which is possibly equal to the fraction of the area that is covered or blocked. Thus, we can deduce the radius of the nucleus from the equation Nσ/A = (n₁ − n₂)/n₁ by rewriting it as πr² = σ = (A/N)(n₁ − n₂)/n₁.

Note: This is a crude measurement of nuclear sizes. Dr. Sands elaborates the concept of probability that a particle will experience a collision in a slab in section 6.1 Chance and likelihood.

In his Nobel lecture titled The electron-scattering method and its application to the structure of nuclei and nucleons, Hofstadter (1961) mentions that “in the year 1919 the first vague ideas concerning the sizes of nuclei were worked out. By studying the deviations from Coulomb scattering of alpha particles Rutherford showed that a nuclear radius was of the order of 10⁵ times smaller than an atomic radius (p. 560).” He adds that “[w]e have used the method of high-energy electron scattering. In essence, the method is similar to the Rutherford scattering technique, but in the case of electrons it is presently believed that only a simple and well-understood interaction - the electromagnetic or Coulomb interaction - is involved between the incident electron and the nucleus investigated (p. 561).” Mathematically, quantum electrodynamics and Dirac theory can be used to calculate a differential elastic scattering cross section.

3. Uncertainty principle:

“…Perfectly precise measurements of distances or times are not permitted by the laws of nature (Feynman et al., 1963, section 5.7 Short distances).”

According to Dr. Sands, perfectly precise measurements of distance and time are not permitted by Heisenberg’s uncertainty principle. Interestingly, his explanation includes a perspective of the special theory of relativity such that measurements of distance and time are dependent on an observer’s frame of reference. He elaborates that the errors in a measurement of the position of an object must be minimally as large as Δx ≥ ℏ/2Δp where ℏ is the reduced Planck constant and Δp is the error in our knowledge of the momentum of the object whose position we are measuring. However, the term “error” should be avoided because it has a connotation of mistake. Thus, physicists prefer the word uncertainty which may mean a lack of knowledge in determining these physical quantities or experimental inaccuracy instead of an error.

Feynman’s explanation of uncertainty principle is related to the Young double slit experiment. In Feynman’s words, “If you make the measurement on any object, and you can determine the x-component of its momentum with an uncertainty Δp, you cannot, at the same time, know its x-position more accurately than Δx ≥ ℏ/2Δp. The uncertainties in the position and momentum at any instant must have their product greater than or equal to half the reduced Planck constant. This is a special case of the uncertainty principle that was stated above more generally. The more general statement was that one cannot design equipment in any way to determine which of two alternatives is taken, without, at the same time, destroying the pattern of interference (Feynman et al., 1963, Section 37–8 The uncertainty principle).”

Questions for discussion:

1. How do physicists measure the size of molecules?

2. How do physicists measure the size of nuclei?

3. Does the uncertainty principle simply mean a complete precise measurement of the location of an object will result in a complete uncertainty in its momentum?

The moral of the lesson: physicists are able to measure the sizes of molecules and nuclei, however, it is debatable whether it is possible to measure smaller dimensions as stipulated by the uncertainty principle.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Hofstadter, R. (1961). The Electron Scattering Method and its Application to the Structure of Nuclei and Nucleons. In Nobel Lectures in Physics 1942 – 1962. Singapore: World Scientific.

Section 5–6 Large distances

(Nearby stars / Distant stars / Sizes of galaxies and universe)

In this section, the three interesting points discussed are the distance of nearby stars, distant stars, as well as sizes of galaxies and universe.

1. Nearby stars:

“…The distance of nearby stars can be measured by triangulation, using the diameter of the earth’s orbit as a baseline (Feynman et al., 1963, section 5.6 Large distances).”

In general, we do not always measure distance by using a meter stick. Dr. Sands explains that we can also measure the horizontal distance between two mountains using a method of triangulation. This means that we have a different definition of distance depending on the method of measurement. Importantly, the distance measured by different methods should agree within an acceptable uncertainty. For example, we measure the distance from the earth to the moon by using two telescopes at different places on the earth that give us two angles subtended. Similarly, we can focus a telescope on a star in summer and in winter to determine the two angles. Thus, the distance of nearby stars can be measured using the method of triangulation in which the diameter of the earth’s orbit is used as a baseline.

According to Percy Williams Bridgman (a Nobel laureate), “[w]e never have under observation more than two angles of a triangle, as when we measure the distance of the moon by observation from the two ends of the earth's diameter. To extend to still greater distance our measures of length, we have to make still further assumptions, such as that inferences from the Newtonian laws of mechanics are valid. The accuracy of our inferences about lengths from such measurements is not high. Astronomy is usually regarded as a science of extraordinarily high accuracy, but its accuracy is very restricted in character, namely to the measurement of angles (Bridgman, 1927, pp. 16-17).” He also suggests that the concept of length which is measured by different methods should have different names.

The method of triangulation in determining the distance of nearby stars to earth is also known as “stellar parallax method.” In 1838, Friedrich Bessel was the first to successfully measure a stellar parallax for the star (61 Cygni) by using a Fraunhofer heliometer. In other words, he determined a trigonometric parallax in which a nearby star appears to change its position with respect to the background of distant (“fixed”) stars as the Earth moves from one side of the sun to the other in half a year.

2. Distant stars:

“… If one now measures the color of a distant star, one may use the color-brightness relationship to determine the intrinsic brightness of the star (Feynman et al., 1963, section 5.6 Large distances).”

According to Dr. Sands, physicists first determine the color of a distant star and then use the color-brightness relationship to deduce the intrinsic brightness of the star. By measuring the apparent brightness of the star, they could compute the distance of the distant star that is farther away. Essentially, the apparent brightness of the star decreases with the square of the distance. The correctness of this method of measuring stellar distances was confirmed by the results obtained for groups of stars known as globular clusters. By looking at the photograph of stars in globular clusters, one might be convinced that these stars are all together. Dr. Sands elaborates that the same result can be obtained from measurements of distance by using the color-brightness method.

In 1908, Henrietta Leavitt published her findings of Cepheid variables in which their periods of brightness are related to their apparent brightness. A Cepheid variable is a type of pulsating star that is varying in both its diameter and temperature and producing changes in brightness with a well-defined period and amplitude. In 1913, Ejnar Hertzsprung realized the significance of this discovery and established an important distance measuring tool for distant stars and it is sometimes known as the standard candle method. Specifically, Cepheid variables can be used to measure distances of distant stars from about 1 kpc to 50 Mpc by using the distance modulus equation: m - M = 5 log d - 5. A parsec (or a pc) is equal to about 31 ´ 10¹⁵ m.

3. Sizes of galaxies and universe:

“…Knowing the size of our own galaxy, we have a key to the measurement of still larger distances—the distances to other galaxies (Feynman et al., 1963, section 5.6 Large distances).”

Dr. Sands explains that knowing the size of our galaxy provides a key to the measurement of even larger distances, including the distances to other galaxies. According to him, there were information and evidence to support the idea that the sizes of galaxies are all in the same order of magnitude. (This is not exactly true because galaxies can be classified as giant galaxies and dwarf galaxies.) Thus, by using the method of triangulation again, one might measure the angle subtended by a galaxy in the sky and deduce its size. By assuming its size is similar to size of our galaxy, some astronomers estimated the distance from its apparent size to be 30 million light-years from the earth.

Dr. Sands mentions that some of the galaxies are about halfway to the limit of the universe’s size (10²⁶ meters). Currently, based on an interpretation of the 7-year Wilkinson Microwave Anisotropy Probe (WMAP) data, the diameter of the observable Universe is deduced to be 28.3 ´ 10⁹ parsecs (Bielewicz & Banday, 2011). In other words, the distance between the earth and the edge of the observable universe is about 14 billion parsecs. In a sense, the distance travelled by the light from the edge of the observable universe is close to the age of the universe times the speed of light, However, this is not accurate because the edge of the observable universe and the earth are moving further apart. More important, we are unable to observe light beyond the edge of the observable universe, and thus, the size of the universe could be far larger.

Questions for discussion:

1. How do physicists measure the distance of nearby stars?

2. How do physicists measure the distance of distant stars?

3. How do physicists measure the sizes of galaxies and universe?

The moral of the lesson: the distance of stars can be measured by using the method of triangulation and the standard candle method.

References:

1. Bielewicz, P., & Banday, A. J. (2011). Constraints on the topology of the Universe derived from the 7-yr WMAP data. Monthly Notices of the Royal Astronomical Society, 412(3), 2104-2110.

2. Bridgman, P. W. (1927). The Logic of Modern Physics. New York: Macmillan.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Section 5–5 Units and standards of time

(Units of time / Standards of time / Atomic clock standards)

In this section, the three interesting points discussed are units of time, standards of time, and atomic clock standards.

1. Units of time:

“…Until very recently we had found nothing much better than the earth’s period, so all clocks have been related to the length of the day, and the second has been defined as 1/86,400 of an average day (Feynman et al., 1963, section 5.5 Units and standards of time).”

Dr. Sands explains that it is convenient to have a standard unit of time, such as a day or a second, and we can refer any period of time as a multiple or a fraction of this unit. However, the experimental determination of a unit of time has been central to the history of time-keeping and time measurement. Specifically, it was a challenge to fit the two units of time, namely month and day, into a calendar year. First, a year was the time for the earth to rotate around the sun that is measured between two successive vernal equinoxes. Second, a month was the time for the moon to rotate around the earth such that it passes a “fixed” star again. Third, a day was the time for the earth to rotate about its axis such that the “high noon” is observed repeatedly. Currently, a day is divided into 24 hours based on the ancient Egyptians’ observations of stars.

Historically, we used to define a second as 1⁄86,400 of a solar day or “the fraction 1/31,556,925.9747 of the tropical year for 1900 January 0 at 12 hours ephemeris time.” (One day being 24 hours X 60 minutes X 60 seconds = 86,400 seconds). The subdivision of hours and minutes into 60 is based on the ancient Babylonians’ preference of using numbers to the base 60. Our current definition of a second is based on the 1967 Thirteenth General Conference on Weights and Measures held during 10-16 October 1967. The SI second of atomic time was defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

2. Standards of time:

“…For a long time, the rotational period of the earth has been taken as the basic standard of time (Feynman et al., 1963, section 5.5 Units and standards of time).”

Importantly, a human pulse is not reliable as a basic standard of time because the pulse rate of a person may vary throughout a day. Although the earth’s rotational period was used as a basic standard of time, it has been found that the rotation of the earth is not exactly periodic when it is compared with more accurate clocks. Furthermore, the rotational speed of the earth is not constant and gradually slowing because of atmospheric circulations, geophysical phenomena and tidal friction (due to the moon’s gravity). As a result, a “second” based on this astronomical standard of time is gradually longer than the “second” that is defined by an atomic standard of time. In a sense, the definition of a second is a historical and an arbitrary choice.

In 1967, Jocelyn discovered the first pulsar (also known as a neutron star) that rotates at a high speed and it emits radio pulses at relatively regular intervals. Currently, it is possible to create a pulsar clock (or an astronomical standard of time) by using a radio telescope which receives signals from designated pulsars. These radio signals appear like a lighthouse beacon due to the rotation of the pulsar. Using the radio telescope, astronomers can measure the arrival times of successive radio pulses to a precision of 100 to 500 nanoseconds. In other words, pulsars could be considered as a more precise astronomical standard of time.

3. Atomic clock standards:

“…since it has been possible to build clocks much more accurate than astronomical time, there will soon be an agreement among scientists to define the unit of time in terms of one of the atomic clock standards (Feynman et al., 1963, section 5.5 Units and standards of time).”

According to Sands, an atomic clock’s internal period is based on an atomic vibration which is very insensitive to the temperature or any other external effects. In the 1960s, these clocks could keep time to a precision of one part in 10⁹ or better. During that time, Professor Norman Ramsey at Harvard University designed and improved an atomic clock which operates based on the vibration of the hydrogen atom. He believed that this hydrogen clock could be 100 times more accurate. Thus, physicists predicted that atomic clocks could be much more accurate than astronomical time, and there would be an agreement among scientists to define the unit of time in terms of an atomic clock standard.

Modern atomic clocks are based on Caesium 133 atoms because they are relatively massive and more stable as compared to hydrogen clocks and ammonia clocks. An important principle of the caesium clock is that caesium atoms are flipped into a higher energy state by using microwave radiations. Furthermore, these atoms will reach a detector (or a wire) and result in an electric current that can help to calibrate (or tune) quartz crystal clocks. Interestingly, the definition of a second was made even more specific in 1997 with the stipulation that caesium atoms are at rest at a temperature of 0 Kelvin (or an environment whose thermodynamic temperature is 0 K). However, one may question how this temperature can be achieved exactly.

Note: There are at least 13 Nobel (physics) laureates that have contributed in the physics of time-keeping since the 1940s: Otto Stern, Isidor Rabi, Polykarp Kusch, Nikolai Basov, Aleksander Prochorov, Charles Townes, Alfred Kastler, Norman Ramsey, Hans Dehmelt, Wolgang Paul, Steven Chu, Claude Cohen-Tannoudji, and Williams Phillips (Jones, 2000). Einstein, of course, has also contributed in providing an operational concept of time.

Questions for discussion:

1. How would you define a unit of time?

2. How would you select a standard of time?

3. Why do metrologists prefer an atomic time standard instead of an astronomical time standard?

The moral of the lesson: the unit of time can be defined by using an atomic time standard or an astronomical time standard.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Jones, T. (2000). Splitting the Second: The Story of Atomic Time. Bristol: Institute of Physics.

Section 5–4 Long Times

(Natural time / Radioactive time / Astronomical time)

A measurement of long times is relatively easier by means of counting. In this section, the three main ideas discussed are a natural time, radioactive time, and astronomical time.

1. Natural time:

“…we can use these natural time markers to determine the time which has passed since some early event (Feynman et al., 1963, section 5.4 Long times).”

According to Dr. Sands, measurements of longer times are easy because we can just count the number of days as long as there is someone around to do the counting. Similarly, we can count the number of years for a physical phenomenon based on the natural periodicity of earth’s revolutions. Sands elaborates that nature has also provided a counter for years, in the form of tree rings or river-bottom sediments. In other words, we can use these “natural time” clocks to determine the age of a tree or compare the age of a river.

Dendrochronology (dendro = tree, chronology = time) is a scientific method to determine the age of a tree by counting the characteristic patterns of tree rings in tree trunks. Tree rings, also known as growth rings or annual rings, can be seen in a horizontal cross section of the trunk of the tree. They are due to the growth in diameter or layers of cells near the bark over the four seasons of a year. The rings are more visible where there are greater variations of temperature. They can be used as a calibration or verification check for radiocarbon dating. Similar seasonal patterns or “cycle of erosion” can be observed in layers of sediment deposition in a lake, river, or seabed.

If Feynman were to deliver this lecture, he might elaborate that “[t]he Maya Indians were interested in the rising and setting of Venus as a morning ‘star’ and as an evening ‘star’—they were very interested in when it would appear. After some years of observation, they noted that five cycles of Venus were very nearly equal to eight of their ‘nominal years’ of 365 days (they were aware that the true year of seasons was different and they made calculations of that also). To make calculations, the Maya had invented a system of bars and dots to represent numbers (including zero) and had rules by which to calculate and predict not only the risings and settings of Venus but other celestial phenomena, such as lunar eclipses… (Feynman, 1985, p. 11).”

2. Radioactive time:

“…One of the most successful is the use of radioactive material as a ‘clock’ (Feynman et al., 1963, section 5.4 Long times).”

Dr. Sands says that one of the most successful clocks for longer times is based on the use of radioactive materials. He explains that these so-called “clocks” do not have a periodic occurrence during the decay of radioactivity material. For example, the carbon dioxide molecules in the air contain a certain small fraction of the radioactive carbon isotope C¹⁴ that has a half-life of 5000 years and they are replenished continuously by the action of cosmic rays. If we measure the total carbon content of an object, there is a certain fraction of which that was originally the radioactive carbon. By careful measurements, we can measure the amount left after 20 half-lives and can, therefore “date” organic objects which grew as long as 100,000 years ago.

Radioactive clocks, such as the use of carbon-14 or uranium, help to determine intervals of natural time or geological time. They are sometimes described as non-cyclic clocks because the disintegration of a nucleus does not repeat periodically. More important, the disintegrations of nuclei are based on Rutherford and Soddy’s theory of radioactive decay in which the number of disintegrations in unit time is proportional to the total number of undecayed nuclei. However, radioactive clocks are subject to errors due to contaminations from other radioactive sources. These errors cannot be completely eliminated by calibration with the use of tree rings.

3. Astronomical time:

“… It is now believed that at least our part of the universe had its beginning about ten or twelve billion years ago. We do not know what happened before then. In fact, we may well ask again: Does the question make any sense? (Feynman et al., 1963, section 5.4 Long times).”

Dr. Sands explains that the age of the earth is found to be approximately 4.5 billion years and it is the same as the age of the meteorites which land on the earth, as determined by the uranium method. He hypothesizes that the earth was formed out of rocks floating in space and that the meteorites are likely come from some of these materials. Currently, it is questionable to say that the universe had its beginning about ten or twelve billion years ago (as mentioned by Sands). Based on Planck, a space observatory operated by the European Space Agency, the universe is estimated to be 13.82 billion years old. This is deduced using the concept of Hubble time or the time it would take for all the galaxies to converge at one point if they continue traveling at the same speed but in the opposite direction.

In the last few sentences of this section, Sands asks whether an earlier time has any meaning. In short, one possible answer is “no clock exists”. Interested students should read how Wilczek discusses this question as follows: “a question that vexed Saint Augustine: ‘What was God doing before He created the world?’ Saint Augustine gave two answers.

First answer: Before God created the world, He was preparing Hell for people who ask foolish questions.

Second answer: Until God creates the world, no ‘past’ exists. So the question doesn’t make sense… (Wilczek, 2008, pp. 103-104).”

Questions for discussion:

1. How is a natural time such as the age of a tree or river related to the revolution of the earth (astronomical time)?

2. How is the age of the earth determined by using radioactivity materials?

3. How is the age of the universe determined?

The moral of the lesson: the age of a tree is determined by counting the number of tree rings, the age of the earth is determined by the proportion of radioactive materials, and the age of the universe is determined by measuring the intensity of photons.

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Wilczek, F. (2008). The lightness of being: Mass, ether, and the unification of forces. New York: Basic Books.

Section 5–3 Short Times

(Mechanical pendulum / Electronic oscillator / Nuclear vibration)

It is possible to have to count time in smaller pieces and we may continue this process further, and learn to measure even smaller intervals of time. In this section, the three main interesting points discussed are the mechanical pendulum, electronic oscillator, and nuclear vibration.

1. Mechanical pendulum:

“…Galileo decided that a given pendulum always swings back and forth in equal intervals of time so long as the size of the swing is kept small (Feynman et al., 1963, section 5.3 Short times).”

According to Dr. Sands, Galileo determined that a pendulum always swings to and fro in equal intervals of time if the amplitude of the swing is kept small. Sands also claims that a test comparing the number of swings of a pendulum in one “hour” shows that this is indeed the case. He explains that the pendulum clock of our grandfathers is a mechanical device that counts the swings. In essence, he opines that we can divide the second into smaller and smaller intervals by using the same principles of comparison. However, it is not practical to make mechanical pendulums which go arbitrarily fast such that they can have a very short period of swing.

There is a misrepresentation of Galileo’s isochronic pendulum because he did not state the condition in which “the size of the swing is kept small.” In Galileo’s words, “[e]ach vibration, whether of ninety, fifty, twenty, ten, or four degrees occupies the same time: accordingly, the speed of the moving body keeps on diminishing since, in equal intervals of time, it traverses arcs which grow smaller and smaller. Precisely the same things happen with the pendulum of cork, suspended by a string of equal length, except that a smaller number of vibrations is required to bring it to rest, since on account of its lightness it is less able to overcome the resistance of the air; nevertheless the vibrations, whether large or small, are all performed in time intervals which are not only equal among themselves but also equal to the period of the lead pendulum (Galilei, 1638/1914, p. 85). To be accurate, the pendulum always swings to and fro in equal periods of time if the size of the swing theoretically approaches zero.

2. Electronic oscillator:

“…In these electronic oscillators, it is an electrical current which swings to and fro, in a manner analogous to the swinging of the bob of the pendulum (Feynman et al., 1963, section 5.3 Short times).”

In an electronic oscillator, the movements of charge-carriers are similar to the swing of a pendulum bob. In addition, it is possible to make a series of electronic oscillators that have periods about 10 times shorter. Thus, Sands explains that electronic oscillators can be built with modern electronic techniques and calibrated by using comparison methods. Interestingly, he elaborates that time shorter than 10⁻¹² second can be measured by determining the distance between two points (or happenings) of a moving object. This definition of time is essentially based on a measurement of distance, however, the SI unit of meter is defined in terms of time. (The meter is the distance traveled by light in vacuum in a time interval of 1/299792458 of a second.)

Alternatively, physics teachers should explain that a quartz clock is an electronic oscillator that is regulated by a quartz crystal to measure time. If you squeeze a crystal of quartz, it will produce an electrical voltage because of its “piezoelectric” qualities. The quartz crystal can also expand or contract depending on the voltage applied across it. Furthermore, an oscillating crystal produces an alternating electrical signal that in turn can be fed back to keep the crystal oscillating. In a modern quartz clock, the oscillating frequency is depending on the shape and size of a quartz crystal and it can be set as 32768 hertz. This frequency is also equal to 2¹⁵ Hz, and thus, the output electrical signal can be halved 15 times easily by using digital electronics such that a frequency of exactly one pulse per second is obtained.

3. Nuclear vibration:

“…By extending our techniques—and if necessary our definitions—still further we can infer the time duration of still faster physical events. We can speak of the period of a nuclear vibration (Feynman et al., 1963, section 5.3 Short times).”

Dr. Sands mentions the period of a nuclear vibration and relate it to the lifetime of newly discovered strange resonances (particles). The complete life of these particles occupies a time span of only 10⁻²⁴ second that is about the time it would take light to move through the nucleus of hydrogen. Importantly, nuclear vibrations refer to oscillations of nucleons (protons and neutrons) in a nucleus. Nevertheless, these vibrations involve collisions between nucleons that could cause damping quickly (Bertsch, 1983). As the mechanisms behind damping were not well understood, the discussion could be focused on atomic vibrations instead of nuclear vibrations.

Note: In his seminal lecture titled Simulating physics with computers, Feynman (1982) talks about simulating time as follows, “[w]e’re going to assume it's discrete. You know that we don’t have infinite accuracy in physical measurements so time might be discrete on a scale of less than 10^-27 sec. (You'd have to have it at least like to this to avoid clashes with experiment—but make it 10^-41 sec. if you like, and then you've got us!) One way in which we simulate time in cellular automata, for example -- is to say that ‘the computer goes from state to state’ (p. 469).” Essentially, he proposes to use computer simulations to understand the nature of time.

Questions for discussion:

1. Does a pendulum’s period depend on its mass, length, and amplitude of oscillation?

2. How do we define time by using electronic oscillators?

3. Should we define time by using atomic vibrations or nuclear vibrations?

The moral of the lesson: the period of a pendulum clock depends on the length of the pendulum, the period of a quartz clock depends on the size of the quartz crystal, and the period of a nuclear vibration depends on the size of the particle (size matters).

References:

1. Bertsch, G. F. (1983). Vibrations of the atomic nucleus. Scientific American, 248(5), 62-73.

2. Feynman, R. P. (1982). Simulating physics with computers. International journal of theoretical physics, 21(6), 467-488.

3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

4. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences. New York: Dover.