Standing on the Shoulder of Richard Feynman to Teach Physics Better

By Shui Yin Lo and 盧遂顯

Richard Feynman gave his two years undergraduate physics in Caltech in a completely new way from the traditional undergraduate physics course in early nineteen sixties.

Now it is fifty years after Feynman gave his famous Lecture on Physics. It is time to present a completely new way of teaching physics. Feynman said, "We prefer to take fir

• Language: English
• Publisher: EHGBooks
• Release date: Mar 1, 2019
• ISBN: 9781647849603

Book preview

Part One: Standing on the shoulder of Richard Feynman to teach Physics better

Shui Yin Lo, Ph.D.

Chapter 1. Probability

Chapter 1. Probability

1.1: Quantum Framework (QF)

Quantum field theory was invented in the 1930s and matured in the 1940s and 1950s. It is the pinnacle of human intellectual achievement in the 20th century. It has passed the scrutiny of the most arduous experiments. It is able to explain the beginning of the biggest objects, the universe, and the properties of the smallest particles: the atomic nucleus, pions, and quarks. It is obvious that it should be able to explain objects of all sizes in the universe. So how can we do it?

We need to extract some general features of quantum field theory so that it can be applied easily to all subjects in our universe. We propose that there are seven features of quantum field theory that are universal. We call them our quantum framework (QF). Here, we specifically apply quantum framework (QF) to classical mechanics. Then, they become the seven important aspects of classical mechanics. To be even more specific, we apply these seven QF to the case of small oscillations in classical mechanics. The seven features of QF are:

By going through the simple example of small oscillations, we intend to illustrate the most fundamental and the most profound aspects of modern physics. We also use three simple examples to illustrate the practical situations of small oscillations apparent to everyone:

1.2: The Experiment – The Small Oscillation of a Pendulum

We begin an experiment of small oscillation by constructing a simple pendulum using a stand, a string, and a metal ball. Use the string with length l to hang the ball from the stand as shown in Figure 1.1. Push the ball, and, using a stopwatch, count the number of periods that occur within a specific time frame when the ball is swinging.

Figure 1.1. Pendulum

Historically, Galileo Galilei of Italy was the first person to start using experiments as a means to discover scientific truths. Previously, the dominant thought was based on Aristotle, which was the Greek school that studied only the ideal world and believed that the practical world would follow the idealistic world. Galileo reportedly sat in a church, watching the chandelier. After it was lit and displaced, the chandelier would swing. Galileo observed the relations that the longer the length, the longer the period of the swing, and if two chandeliers have the same length but different weights, they still swing with the same period. So we may say that this small pendulum experiment is the beginning of modern physics, where experiments form the basis to decide the truth.

1.3: The Application – The Grandfather Clock

How can small oscillations with a pendulum be applied to real-life? A significant example of this is the grandfather clock (shown in Figure 1.2), which utilizes a built-in pendulum to keep time with much greater precision compared to its predecessors.

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Figure 1.2. Grandfather Clock

1.4: Example in Daily Life – A Swing in a Playground

Another example of small oscillations can be seen in a playground swing for a child’s entertainment, shown in Figure 1.3.

Figure 1.3. Playground Swing

We want to show in these two examples that pure science, such as the simple pendulum, and applications in the grandfather clock and entertainment are all important to the advancement of knowledge. We cannot have pure science advance without applications and vice versa; applications cannot advance without someone figuring out the scientific reasons behind them. Applications can then improve much further. These three examples serve as intrinsically related in the world rather than artificially separated in academia.

1.5: Probability and Error Analysis

A fundamental basis of quantum mechanics is the ever-present uncertainty. This uncertainty leads to the presence of error in experiments, which can be addressed and analyzed using probabilities.

A common practice to experimentation is the use of multiple trials; the goal of this is to improve the accuracy of results and is a means to properly address the occurrence of experimental error. Probability is then used to analyze the error. To use our pendulum experiment as an example, one should measure the length of the string several times and take an average of these measurements. Why? Because the measurements will always vary, even minimally, due to random or systematic error, which is intrinsic in any real life situation. Ultimately, some of the error can be traced to the uncertainty principle of quantum mechanics. The same applies for the count of the periods; multiple counts are needed. Standard deviation is also an important probability equation for multiple measurements. The calculations needed for the error analysis of our pendulum experiment are shown by the following equations:

Error bar:

: length of the string      (1.5.1)

: period of time      (1.5.2)

Standard Deviation:

(1.5.3)

Each measurement will never be definite; instead, the probability of each measurement can be calculated by the following equation and is shown in Figure 1.4.

Probability Distribution:

(1.5.4)

Figure 1.4. Probability Distribution

In summary, to account for measurement uncertainties, measure the length of the string many times to get the average and the standard deviation . If the error is random, i.e. not biased, the distribution is symmetric and Gaussian (Figure 1.5).

Figure 1.5. Gaussian Distribution

1.6: The Professor

Another example of Gaussian distribution can be seen by the professor’s assignment of grades for a lab report. The probability distribution of grades A, B,... to F falls under a normalized curve, commonly known as the bell-curve. 

The conscientious professor, of course, wishes that all his students earn A’s rather than have grades spread throughout a bell curve as in the figure above. But in reality, every professor and every student knows that it is impossible for every student to get 100% or for every student to fail. Probability distribution is what happens in real life and, empirically, it is the truth.

1.7: The Student

The student may wonder if he has time to complete the professor’s experiment in time for a given lab period. The completion time of the experiment by a student will also fall under a probability distribution. In other words, the likelihood for completing the experiment will vary. Consider the following data, shown in Table 1.1, as an example:

The student may want to hurry the pace of the experiment so that he can finish the lab report. For example, if he wants to see a movie at an earlier showing time after the lab, there is a small chance, say 50%, that he can finish in 1.5 hours. Since the time of the experiment is estimated to be longer, he can realistically finish in 4 hours, with say 99% probability, if he catches a late showing.

If the experiment goes well – he measures the string easily, puts up the stand with no problem, and everything is in his favor – then it is possible that 1.5 hours is enough. On the other hand, if he is not as lucky or something breaks, then it is possible that even 4 hours is not enough to finish the lab. Therefore, it is hard to have 100% certainty.

1.8: The Manufacturer

It is often said that classical mechanics is deterministic and not probabilistic. This actually means that in ideal situations, all classical objects are predicted precisely as the equations of motion - the way the Newton’s equations said it to be. The solution of a simple pendulum is clearly one solution. There is no ambiguity of its answer. However, in real life, when real objects are being manufactured, probability becomes important. The manufacturer of a grandfather clock does not have a luxury of a deterministic view with the clocks absolutely obeying the solution of small oscillation as calculated by the Newtonian equations.

For the manufacturer, there are at least two uncertainties for the future. First is the uncertainty of how much the clock runs, say, in one year. In other words, the manufacturer of a grandfather clock must specify the accuracy of the clocks he produces. Is the clock off by one second, one hour, or one month after it runs for a year? This uncertainty Δt is governed by the Gaussian distribution discussed previously (Figure 1.5). No clock will run perfectly accurately due to many factors in real life. If we put Δt to be the uncertainty of the clock, then:

  (1.8.1)

The uncertainty of a clock is one second in a day, which amounts to 10-5 or 0.001%. It is really a remarkable mechanical achievement.

The second uncertainty that a manufacturer must deal with is the probability of a clock malfunctioning. The clock just does not run, and it is a significant percentage. Currently in the market, the return rate of a guaranteed object is roughly 3%. However, the biggest uncertainty is none of these two; it is the uncertainty of whether the manufacturer makes money or not. This uncertainty belongs to the realm of the social sciences and not to the realm of the physical sciences. We just want the student to understand that the uncertainty in the social domain also originates from the quantum behavior of human beings.

Chapter 1. Question and Answers

Most textbooks in classical mechanics do not discuss experiments at all; however, physics is an experimental science. Experimentation is the basis of theory. Any textbook in classical mechanics should reflect this.

Pure science does not stand alone. For pure science to have ongoing life, it is necessary to have applications. Any textbook in physics should reflect this.

The fundamental reason is, of course, that classical mechanics is now considered to be derivable from quantum mechanics for macroscopic objects. But in practice, when we actually deal with classical mechanics properly, probability immediately comes into it. For example, when we measure the length of a string in a simple pendulum, it is not possible to measure it once. It is necessary to do multiple measurements to be as identical as we can be. Then statistical analyses of these multiple measurements are used to obtain the average and the standard deviation.

So in the purest theoretical classical mechanics, we may seem to be able to neglect probability. However, when we touch reality, it is immediately necessary to do multiple experiments or to take multiple measurements or to produce many identical products; then this probability enters in a fundamental way. Thus, we include probability as the number one importance in classical mechanics. Essentially, when we talk about the experimental basis of classical mechanics or its applications, probability becomes vital.

Let us expand more on the probability aspect of classical mechanics in the simple pendulum experiment performed by the student. First, there is definitely no guarantee or 100% certainty that when the student performs the pendulum experiment, he will get the answers predicted by Newton’s equations. Psychologically, the student is prepared to be unsuccessful. Psychology is not a subject described by classical mechanics; therefore, we have a genuine uncertainty from the student’s viewpoint. Any experiment cannot just be isolated as an idealistic experiment without the participation of a human being. Whenever a human observer or performer or experimentalist is involved, uncertainty is there. Probability is fundamental in designing an experiment or performing an experiment. The uncertainty for a simple experiment can specifically be stated as follows:

Small oscillations occur everywhere. When the breeze blows, leaves and branches oscillate. When people walk on the floor, the floor undergoes small oscillations. When a truck runs on a highway, the road’s surface oscillates. Bridges oscillate when they are subject to traffic, wind, and earthquakes. Small oscillations repeat themselves and do not cause damage easily. Large oscillations may spell disaster and are often much more difficult to treat.

We could pick another topic, such as planetary motions, launching of a satellite, a space station orbiting the Earth, etc. as the topic to teach classical mechanics.

It is well-known by teachers, students, and professionals that most of the things you learn at universities are not relevant to real