Firstenberg covers a wide range of issues in The Invisible Rainbow and let me begin by admitting that I’m not an expert in all of these subjects. For instance, I don’t know much about infectious diseases, such as influenza, and I’m not particularly knowledgeable about viruses in general. However, the Centers of Disease Control and Prevention gathers input from authorities on these topics and here is what it says about the causes of the flu.
“Most experts believe that flu viruses spread mainly by tiny droplets made when people with flu cough, sneeze, or talk. These droplets can land in the mouths or noses of people who are nearby. Less often, a person might get flu by touching a surface or object that has flu virus on it and then touching their own mouth, nose or possibly their eyes.”
Firstenberg, on the other hand, claims that the flu is an electrical disease not caused by a virus spread from person to person. He writes
In 1889, power line harmonic radiation began. From that year forward the earth’s magnetic field bore the imprint of power line frequencies and their harmonics. In that year, exactly, the natural magnetic activity of the earth began to be suppressed. This has affected all life on earth. The power line age was ushered in by the 1889 pandemic of influenza.
In 1918, the radio era began. It began with the building of hundreds of powerful radio stations at [low] and [very low] frequencies, the frequencies guaranteed to most alter the magnetosphere. The radio era was ushered in by the Spanish influenza pandemic of 1918.
In 1957, the radar era began. It began with the building of hundreds of powerful early warning radar stations that littered the high latitudes of the northern hemisphere, hurling millions of watts of microwave energy skyward. Low-frequency components of these waves rode on magnetic field lines to the southern hemisphere, polluting it as well. The radar era was ushered in by the Asian flu pandemic of 1957.
In 1968, the satellite era began. It began with the launch of dozens of satellites whose broadcast power was relatively weak. But since they were already in the magnetosphere, they had as big an effect on it as the small amount of radiation that managed to enter it from sources on the ground. The satellite era was ushered in by the Hong Kong flu pandemic of 1968.
No mechanism is offered to explain how electromagnetic fields might cause a flu pandemic. No distinction is made between power line frequency (60 Hz) and radio frequency (MHz) radiation, although their physical effects are distinct. No estimation of “dose” (the distribution and magnitude of electric and magnetic field exposure) is provided. No randomized, controlled, double-blind studies are cited. He merely lists anecdotal evidence and coincidences.
Perhaps we could just ignore such dubious claims, except that The Invisible Rainbow is often quoted as evidence supporting the assertion that the Covid pandemic is somehow related to 5G cell phone radiation. Why would anyone get a Covid vaccine if they erroneously believe that the disease is caused by electromagnetic radiation? Such misinformation is dangerous to us all.
Firstenberg describes old studies without critical analysis. For instance, on page 73 he writes
In 1923, Vernon Blackman, an agricultural researcher at Imperial College in England, found in field experiments that electric currents averaging less than one milliampere (one thousandth of an ampere) per acre increased the yields of several types of crops by twenty percent. The current passing through each plant, he calculated, was only about 100 picoamperes.
One hundred picoamperes is 10−10 amperes. We aren’t told what the crops were, but let’s assume they consist of a thin stalk that I’ll estimate has a cross-sectional area of one square centimeter (10−4 m2). That means the current density would be 10−6 A/m2. Furthermore, let’s assume an electrical conductivity on the order of saline, 1 S/m. The resulting electric field is 10−6 V/m, or one microvolt per meter. This is far less than the electric field that always surrounds us and is caused by thermal fluctuations. The proposition that one milliamp per acre has such an effect defies credulity.
It was the Schwann cells, Becker concluded—the myelin-containing glial cells—and not the neurons they surrounded, that carried the currents that determined growth and healing. And in a much earlier study Becker had already shown that the DC currents that flow along salamander legs, and presumably along the limbs and bodies of all higher animals, are of semiconducting type.
Firstenberg believes cell phones cause many health hazards. On page 176, he writes
[Allan Frey] discovered the blood-brain barrier effect, an alarming damage to the protective shield that keeps bacteria, viruses, and toxic chemicals out of the brain—damage that occurs at levels of radiation that are much lower than what is emitted by cell phones today.
I could go on. Firstenberg believes electromagnetic fields are responsible for diabetes, heart disease, and cancer. His views on the mechanism of hearing are at odds with what most researchers believe. He thinks the “qi” that supposedly underlies acupuncture is electric in nature (similar to Becker’s view).
I do have some sympathy for Firstenberg. He’s been plagued by a variety of symptoms that he associates with electromagnetic hypersensitivity. I have no doubt his suffering is real. Yet, the evidence from controlled, double-blind experiments does not support his claim that electromagnetic radiation causes his illness. Rubin et al. reviewed many experiments and concluded that “at present, there is no reliable evidence to suggest that people with [idiopathic environmental intolerance attributed to electromagnetic fields] experience
unusual physiological reactions as a result of exposure to [electromagnetic fields]. This supports suggestions that
[electromagnetic fields are] not the main cause of their ill health” (Bioelectromagnetics, Volume 32, Pages 593–609, 2011). The World Health Organizationconcludes
EHS [electromagnetic hypersensitivity] is characterized by a variety of non-specific symptoms that differ from individual to individual. The symptoms are certainly real and can vary widely in their severity. Whatever its cause, EHS can be a disabling problem for the affected individual. EHS has no clear diagnostic criteria and there is no scientific basis to link EHS symptoms to EMF [electromagnetic field] exposure. Further, EHS is not a medical diagnosis, nor is it clear that it represents a single medical problem.
I put Arthur Firstenberg in the same category as Martin Pall, Robert Becker, Paul Brodeur, and Devra Davis: well-meaning scientific mavericks whose hypotheses have not been confirmed. The Invisible Rainbow is an interesting read, but beware: as science it is flawed.
Listen to Arthur Firstenberg, author of The Invisible Rainbow, answer questions about the hidden dangers of wireless and cellular phone radiation (I post this video so you can hear his side of the story, not because I agree with him).
Robert Resnick is professor emeritus at Rensselaer and the former Edward P. Hamilton Distinguished Professor of Science Education, 1974–93. Together with his co-author David Halliday, he revolutionized physics education with their now famous textbook on general physics, still one of the most highly regarded texts in the field today.
He is author or co-author of seven physics textbooks, which appear in 15 editions and more than 47 languages.
Resnick introduced Rensselaer’s interdisciplinary science curriculum in 1973 and was its chair for 15 years. He was awarded the American Association of Physics Teachers’ highest honor, the Oersted Medal, in 1975, and served as its president, 1986–90. A Distinguished Service Citation issued in 1967 by the association said, “Few physicists have had greater or more direct influence on undergraduate physics students than has Robert Resnick.”
Rensselaer named its Robert Resnick Center for Physics Education in his honor.
Today is Edith Anne Stoney’s birthday; she was born on January 6, 1869. In an article that appeared in the December, 2013 issue of Scope (the quarterly magazine of the Institute for Physics and Engineering in Medicine), Francis Duck describes Stoney as “the first woman medical physicist.” This week’s blog post includes excerpts from Duck’s fascinating article.
Stoney began her education in math and physics, then later switched to medicine.
As a young woman, Edith demonstrated
considerable mathematical talent, gaining a scholarship
at Newnham College, Cambridge, where she achieved
a First in the Part I Tripos examination in 1893.
Extraordinarily, she was never awarded her Cambridge
degree: women were excluded from graduation, a
situation that would not change for another 50 years.
She was later awarded [bachelor’s and master’s] degrees from Trinity College Dublin, after they accepted women in 1904.
Career possibilities for university women were limited.
She carried out some difficult calculations on gas turbines and searchlight design for Sir Charles Parsons,
and then took a mathematics teaching post at
Cheltenham Ladies’ College.
The 1876 Medical Act had made it illegal for academic
institutions to prevent access to medical education on
the basis of gender. Anticipating this change in the law,
the London School of Medicine for Women was
established in 1874 as the first medical school for
women in Britain. It soon became part of the University of London, with clinical teaching at the Royal Free Hospital. Edith’s sister Florence studied there,
obtaining her [medical degree] in 1898. By this time, changed
regulations had embedded physics firmly into medical
training, and Edith gained an appointment as a physics
lecturer there in 1899.
She became interested in medical imaging through her sister, the first female radiologist in the United Kingdom.
In 1901, the Royal Free Hospital appointed Florence
into a new part-time position of medical electrician. The
two sisters set about selecting, purchasing and
installing x-ray equipment and, the following April, a
new x-ray service was opened in the electrical department.
Edith and Florence with their father George Johnstone Stoney.
During the next few years Edith actively supported
the women’s suffrage movement, though opposed the
direct violent action with which it was later associated.
The years from 1910–1915 did not go smoothly for her.
After her father’s death in 1911 she no longer had his
guidance to call on. As student numbers increased so
did her staff, but they often did not stay long,
finding her difficult to work with. Finally, in March
1915, she left [her teaching position at the University of London].
Edith was now free from other commitments and
could make her own contribution to the war. She
contacted the Scottish Women’s Hospitals (SWH), an
organisation formed in 1914 to give medical support in
the field of battle, financed by the women’s suffrage
movement. In May she set off to Europe, and would be
away for most of the next 4 years… She established
stereoscopy to localise bullets and shrapnel and
introduced the use of x-rays in the diagnosis of gas gangrene… [The war resulted in] traumatically
injured soldiers and difficult working conditions. It
could have crushed a weaker character…
It was hard physical work for the women
to pack up the whole tented hospital, weighing three or
four hundred tons.
In March 1918, and for the third time, she had to
supervise a camp closure and retreat, when Villers-Cotterets was overrun by the advancing front. During
the final months of the war the fighting intensified and
there was a huge increase in workload. In the month of
June 1918 alone the x-ray workload peaked at over 1,300,
partly resulting from an increased use of fluoroscopy... However, [fluoroscopy] also resulted in an increased
incidence of radiation burns to Edith’s staff, two of
whom had to take sick leave to recover.
After the war ended, her work supporting the troops was honored by government awards, but not with an appropriate job.
Her war service was recognised by the medals that she
was awarded: from France, the Médaille des épidémies
and the Croix de Guerre; from Serbia, the Order of St Sava; and the Victory [Medal] and [the] British War Medal from
Britain. Returning to England and with no pension and
no medical qualification she took a post as lecturer in
physics in the Household and Social Science department
at King’s College for Women, which she held until 1925.
She retired in 1925, but remained active supporting women in science.
After leaving King’s she retired to Bournemouth
where she lived with Florence who was by then
terminally ill with spinal cancer. She supported the
British Federation [of] University Women (BFUW) for
which she had acted as the first treasurer before the war.
She travelled widely, first with her ailing sister, and then
alone after Florence died in 1932.
Stoney passed away just as Europe was hurtling toward another world war.
Edith Stoney died, aged 69 years, on 25th June 1938.
Obituaries were printed in Nature, The Lancet and The
Times…. She was not noted as a
creative scientist: this was not her forte. She was a tough
and single-minded woman with high academic ability.
Her organisational skills established physics laboratories
and courses in two institutes of higher education. She
showed considerable bravery and resourcefulness in the
face of extreme danger, and imagination in contributing
to clinical care under the most difficult conditions of war.
She was a strong advocate of education for women... At a
time when medical physics was still struggling to become
an identified profession, Edith Stoney stands out as one
of its most able pioneers.
Anyone searching for a female role model in medical physics need look no further. What an amazing life.
Since a changing magnetic field generates an induced electric
field, it is possible to stimulate nerve or muscle cells
without using electrodes. The advantage is that for a given
induced current deep within the brain, the currents in the
scalp that are induced by the magnetic field are far less than
the currents that would be required for electrical stimulation.
Therefore transcranial magnetic stimulation (TMS) is
relatively painless.
The method was invented in 1985 and when I arrived at NIH in 1988 the field was new and ripe for analysis. I spent the next seven years calculating electric fields in the brain and determining how the electric field couples to a nerve.
This review describes the development of transcranial magnetic stimulation in 1985 and the research related to this technique over the following 10 years. It not only focuses on work done at the National Institutes of Health but provides a survey of other related research as well. Key topics are the calculation of the electric field produced during magnetic stimulation, the interaction of this electric field with a long nerve axon, coil design, the time course of the magnetic stimulation pulse, and the safety of magnetic stimulation.
I like magnetic stimulation because it's a classic example of how a fundamental concept from physics can have a major impact in biology and medicine. If you combine this review of transcranial magnetic stimulation together with my earlier review of the bidomain model of cardiac tissue, you get a pretty good summary of my most important research.
I encourage students to build their qualitative problem solving skills by recasting equations in dimensionless variables, analyzing the limiting behavior of mathematical expressions, and sketching plots showing how functions behave. “Think Before You Calculate!” is my mantra. But how, specifically, do you do this? Let me show you an example.
Fig. 2.16 from IPMB. A plot of the solution of the logistic equation when y0 = 0.1, y∞ = 1.0, b0 = 0.0667. Exponential growth with the same values of y0 and b0 is also shown.
Sometimes a growing population will level off at some constant value. Other times the population will grow and then crash. One model that exhibits leveling off is the logistic model, described by the differential equation
dy/dt = b0y (1 – y/y∞) , (2.28)
where b0 and y∞ are constants….
If the initial value of y is y0, the solution of Eq. 2.28 is
y(t) = 1 / [1/y∞ + (1/y0 – 1/y∞) e−b0t] . (2.29)
Below is a new homework problem, analyzing the logistic equation in a way to build insight. Consider it an early Christmas present. Santa won’t give you the answer, so you need to solve the problem yourself to gain anything from this post.
Section 2.10
Problem 36 ½. Consider the logistic model.
(a) Write Eq. 2.28 in terms of dimensionless variables Y and T, where Y = y/y∞ and T = b0t.
(b) Express the solution Eq. 2.29 in terms of Y, T, and Y0 = y0/y∞.
(c) Verify that your solution in part (b) obeys the differential equation you derive in part (a).
(d) Verify that your solution in part (b) is equal to Y0 at T = 0.
(e) In a plot of Y(T) versus T, which of the three constants (y∞, y0, and b0) affect the qualitative shape of the solution, and which just scale the Y and T axes?
(f) Verify that your solution in part (b) approaches 1 as T goes to infinity.
(g) Find an expression for the slope of the curve Y = Y(T). What is the slope at time T = 0? For what value of Y0 is the initial slope largest? For what values of Y0 is the slope small?
(h) The plot in Fig. 2.16 compares the solution of logistic equation with the exponential Y = Y0 eT. The figure gives the impression that the exponential is a good approximation to the logistic curve at small times. Do the two curves have the same value at T = 0? Do the two curves have the same slope at T = 0?
(i) Sketch plots of Y versus T for Y0 = 0.0001, 0.001, 0.01, and 0.1.
(j) Rewrite the solution from part (b), Y = Y(T), using the constant T0, where T0 = ln[(1−Y0)/Y0]. Show that varying Y0 is equivalent to shifting the solution along the T axis. What value of Y0 corresponds to T0 = 0?
(k) How does the logistic curve behave if Y0 > 1? Sketch a plot of Y versus T for Y0 =1.5.
(l) How does the logistic curve behave if Y0 < 0? Sketch a plot of Y versus T for Y0 = –0.5.
(m) Plot Y versus T for Y0 = 0.1 on semilog graph paper.
If you solve this new homework problem and want to compare you solution to mine, email me at roth@oakland.edu and I’ll send you my solution.
Hallett came to NIH in 1984 and worked there for almost 40 years. I collaborated with him in the early 1990s, when I was working in NIH’s Biomedical Engineering and Instrumentation Program. My role was to calculate the electric field induced in the brain during transcranial magnetic simulation.
For a long time, Hallett was the clinical director for the National Institute of Neurological Disorders and Stroke intramural program. According to Google Scholar, his papers are cited about ten times every day, and his h-index is over 100, meaning he has published over 100 papers that have each been cited over 100 times. He has had an enormous impact on the field of neurophysiology. In particular, he trained an amazing number of young scholars who have gone on to be leaders in the field themselves, many of who spoke at the event.
Once Hallett told me that he started out college studying physics, but when his instructor explained to his class that a magnetic field is just a relativistic effect of an electric field (see Problem 5 in Chapter 8 of IPMB) he switched to a premed program!
At the end of his Festschrift, Hallett spoke and honored his many mentors. His final words were "I will be retiring, but not too much."
Enjoy your retirement, Mark Hallett, but not too much. Working with you was a delight.
We are now at the first of three chapters in which we explore the physics of the interface between water and air. As we will soon discover, the interface is a bizarre and fascinating place. To begin with, it is truly two-dimensional: it has neither outside nor inside. It is not contained in the water nor is it contained in the air; it is simply the place at which they meet. As such, its properties are not those of air or water alone, but of their mutual interaction, and these can be both surprising and nonintuitive…
We begin this exploration with an examination of the phenomenon known as surface tension. This is the force that keeps water droplets spherical, and it has a variety of biological consequences. For instance, we will see how surface tension allows trees to grow to majestic heights and flies to adhere to glass; how surface tension allows some insects to breathe under water and others to walk upon it.
Surface tension arises because a water molecule is usually hydrogen bonded to several other water molecules surrounding it. On the water surface, however, there is at least one hydrogen bond missing, so a higher surface energy is required to produce additional surface area. The surface energy of water in air is denoted γ and has a value of 0.07 J/m2. This is a fairly high surface energy compared to most other liquids (for instance, everyone’s favorite, ethanol, has a surface energy of only 0.02 J/m2).
After explaining the origin of surface energy, Denny adds an important caveat.
In nature it is extremely rare to find an air-water interface that is not fouled to some extent with an ill-defined organic film. Most biological molecules (fatty acids in particular) can lower the surface energy to a fraction of that found in pure water. As a result, the surface of all but the cleanest bodies of water is likely to have a lower surface energy than reported here.
Denny then relates the concept of surface energy to that of surface tension.
To this point we have discussed the air-water interface in terms of its surface energy. Why, then, is this chapter entitled “Surface Tension”? It turns out that surface tension is just another way of expressing surface energy.
In the abstract, this is easily seen by comparing the units of the two expressions. Surface energy is expressed as J m−2. But a joule is one newton meter, so J m−2 is the same as N m−1, that is, force per distance, a tension.
Surface tension is related to the concept of capillarity. Water tends to adhere to a clean glass surface. Therefore, water will rise in a hollow glass tube until it reaches a height at which the weight of the column is balanced by the adhesion force. This height is proportional to the surface tension and is inversely proportional to the radius of the tube. Denny observes that
Because the water in the tube is, in essence, hanging from the air-water interface, it is at a lower pressure than that of the surrounding air. As a result, if one were to poke a hole in the side of the tube, air would be drawn in rather than water being forced out.
Consider, for instance, the problems faced by insects. These animals rely on their tracheae and tracheoles to deliver oxygen to their muscles and to remove carbon dioxide… This system works only because the tracheae are tracheoles are filled with air. If these small tubes become filled with water, the rate at which they transport O2 and CO2 decreases 10,000-fold [because of the 100-fold difference of the diffusion constant in air and water]. How does the respiratory system of insects keep from filling up with water via capillarity?
The answer is likely to be that the inner surface of the respiratory system is coated with some substance that is not wetted by water… If the tracheae and tracheoles are coated with a waxy substance similar to that found on the external cuticle of many insects, water has no tendency to fill the system, and effective respiration is possible.
Denny then analyzes the law of Laplace, relating the air pressure inside a bubble to its radius and water’s surface tension. Russ and I analyze the case of a spherical bubble in Homework Problem 60 of Chapter 3 in IPMB.
Denny concludes the chapter by examining animals that can walk on water.
The surface of lakes and streams provides a unique opportunity for terrestrial organisms. An animal that can walk on water has available to it a flat substratum from which to hunt aquatic prey and a refuge on which to escape from predators. There is just one problem: How does an animal manage to walk on water?
Surface tension, of course, provides the answer. If the animal contacts the water with a nonwettable structure of sufficient perimeter, the upward force of surface tension can support the organism’s weight.
This only works for small animals.
The prospect of walking on water becomes less likely the larger the animal becomes. The crux of the problem is that an animal’s weight increases as the cube of a linear measure of its size, whereas the force due to surface tension increases in direct proportion to a linear measure (in this case, perimeter). For example, a mouse with a mass of 100 g weights 1 N. If the animal had feet coated with the same wax available to a water strider, it would need a perimeter of 40.3 m in contact with the water, roughly 10 m per foot. Feet of this size would clearly be impractical, and for this reason animals the size of mice do not walk on water.
First, let’s review the decibel. If the pressure amplitude of two sound waves are p1 and p2, their relative pressure can be written as
20 log10(p2/p1).
When expressed in this way, the pressure difference is said to be in decibels (dB). If p2 is ten times p1, then the pressure difference is 20 log10(10) or 20 dB.
Often we express sound in terms of intensity rather than pressure. The intensity is proportional to the pressure squared, so the relative intensity difference of I1 and I2 is
10 log10(I2/I1).
The leading factor of 20 in the expression containing the pressures is replaced by 10 in the expression containing intensities. If you don’t like that factor of ten out front, you could not use it, in which case your intensity difference is expressed in the rarely used unit of bels rather than decibels.
Notice that the decibel is defined using of a logarithm with base ten, also known as the common logarithm. Alternatively, we could use the natural logarithm, with base e = 2.718..., which leads to the neper. However—and this is the confusing part—instead of having the leading factor of 20 in the expression for decibels in terms of pressure, the expression for nepers has no leading factor at all. A factor of 2 is removed because the neper is defined in terms of the pressure and not intensity, and a factor of 10 is removed because nepers are like bels and not decibels. So,
ln(p2/p1)
is the pressure difference in nepers. If you insist on using intensity rather than pressure, you must use the ugly-looking expression
½ ln(I2/I1) .
A ten-fold difference in intensity is 1.15 nepers (Np), so 10 dB is the same as 1.15 Np, or 1 Np = 8.7 dB.
If a sound wave attenuates at a rate of 1 neper per meter that means for every meter traveled the pressure falls by a factor of e and the intensity falls by a factor of 7.4. In tissue, attenuation is usually proportional to frequency, so as a rule of thumb the attenuation is about 100 dB per meter per megahertz or roughly 12 neper per meter per megahertz.
Died: Merchiston Castle, near Edinburgh, April 4, 1617
... Napier’s solid reputation rests upon a new method of calculation that first occurred to him in 1594… It occurred to Napier that all numbers could be expressed in exponential form. That is, 4 can be written as 22, while 8 can be written as 23, and 5, 6, and 7 can be written as 2 to some fractional power between 2 and 3. Once numbers were written in such exponential form, multiplication could be carried out by adding exponents, and division by subtracting exponents. Multiplication and division would at once become no more complicated than addition and subtraction.
Napier spent twenty years working out rather complicated formulas for obtaining exponential expressions for various numbers. He was particularly interested in the exponential forms of the trigonometric functions, for these were used in astronomical calculations and it was these which Napier wanted to simplify. His process of computing the exponential expressions led him to call them logarithms (“proportionate numbers”) and that is the word still used.
Finally, in 1614, Napier published his tables of logarithms, which were not improved on for a century, and they were seized on with avidity. Their impact on the science of the day was something like that of computers on the science of our own time. Logarithms then, like the computers now, simplified routine calculations to an amazing extent and relieved working scientists of a large part of the noncreative mental drudgery to which they were subjected.
In Section 14.4 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss molecular energy levels. In particular, we examine translational, rotational, and vibrational levels. When analyzing rotational levels, we consider a simple diatomic molecule and divide the motion into two parts: a uniform translation of the center of mass, and a rotation about the center of mass. We show that the rotational energy can be written as ½Iω2, where ω is the rotational angular frequency and I is the moment of inertia. The moment of inertia is I = [m1m2/(m1 + m2)] R2, where m1 and m2 are the mass of the two atoms making up the diatomic molecule, and R is the distance between them. In quantum mechanics, the spacing of rotational energy levels depends on I.
Later in the same section, Russ and I consider vibrational motion. However, we don’t do a detailed analysis for a diatomic molecule, like we did for rotational motion. In this blog post, I will remedy that situation and present the analysis of vibrations of a diatomic molecule. Our goal is to derive an expression for the vibration frequency in terms of the masses of the two atoms and the spring constant connecting them.
Let’s do the analysis in one dimension. Consider two atoms with mass m1 and m2 connected by a spring with spring constant k. The position of m1 is x1, and the position of m2 is x2.
Next, define two new variables, as we did for rotational motion: x, the position of the center of mass, and X, the distance between the two masses
x =[m1/(m1+m2)] x1 + [m2/(m1+m2)] x2 ,
X = x1 − x2 .
Then, rewrite Newton’s second law in terms of x and X. After some algebra, we get two equations
(m1+m2) d2x/dt2 = 0
[m1m2/(m1+m2)] d2X/dt2 = − kX
The first equation represents a free particle of mass M, where
M = m1+m2 ,
and the second equation represents a bound particle with spring constant k and mass m (often called the reduced mass)
m = m1m2/(m1+m2)
.
The angular frequency of the vibration is therefore
ω = √(k/m)
(If you don’t follow that last step, see Appendix F of IPMB).
We have reached our goal: the angular frequency of the vibration, ω, written in terms of k, m1, and m2
ω = √[k (m1+m2)/m1m2]
In quantum mechanics, the energy levels depend on ω, and therefore on the reduced mass m.
If m1 >> m2 then m is approximately m2. Likewise, if m2 >> m1 then m is approximately m1. For example, if you want the vibration frequency of hydrogen chloride (HCl), the reduced mass is close to the mass of the hydrogen atom.
If m1 = m2 = μ (like for molecules such as O2 and N2), then the reduced mass m is equal to μ/2. It’s that factor of two in the denominator that’s the surprise.
Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, by Eisberg and Resnick.
The relationship between m, m1, and m2 can be written
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
