Tầm quan trọng của việc học ngoại ngữ

Tôi vẫn nhớ một câu thơ của Mayakovsky mà tôi đọc được từ rất lâu khi đang học tiếng Nga, dịch ra đại khái như sau (xin lỗi là dịch không vần):

Kể cả nếu tôi là người da đen cao tuổi
tôi cũng sẽ không chán nản lười biếng
học tiếng Nga chỉ vì đó là tiếng nói của Lênin

Trên thực thế, tôi không biết người nào (tất cả các màu da và tuổi tác) mà học tiếng Nga chỉ vì đó là tiếng nói của Lênin. Tuy vậy gần đây tôi đọc được một câu chuyện làm tôi nhớ tới câu thơ của Mayakovsky. Câu chuyện được nhà toán học Vladimir Voevodsky kể lại:

Năm 1984, Alexander Grothendieck nộp cho CNRS một đề án có tên là “Esquisse d’un Programme”. Ngay sau đó giới toán học bắt đầu chuyền tay nhau các bản sao của đề án này.

Vài tháng sau đó, thầy hướng dẫn khoa học đầu tiên của tôi, ông George Shabat, đưa đề án này cho tôi đọc. Lúc đó tôi là sinh viên năm thứ nhất trường Đại học Tổng hợp Mátxcơva.

Sau khi học một ít tiếng Pháp với mục tiêu duy nhất là để đọc được tài liệu này, tôi bắt đầu triển khai một số ý tưởng mà Grothendieck phác thảo trong đó…”

18 năm sau khi bắt đầu học tiếng Pháp, Voevodsky được huy chương Fields.

Tổng Ramanujan

Có một công thức thường được gắn với tên Ramanujan:

1+2+3+4+\cdots = - \displaystyle{\frac1{12}}

Về mặt toán học công thức này có vẻ không thể nào đúng, nhưng trong vật lý công thức này rất nổi tiếng. Nó liên quan đến lực Casimir và xuất hiện nhiều trong lý thuyết dây. Bình thường công thức này có thể giải thích được qua hàm zeta Riemann: \zeta(-1)=-\frac1{12}. Tuy nhiên ta có thể “chứng minh” nó chỉ dùng toán sơ cấp. Các bạn có thể xem video

Bài tập:

1+1+1+1+\cdots = ?

1 \times 2 \times 3\times 4\times \cdots = ?

Kerson Huang: Chen Ning Yang and I Ching

When I was a postdoctoral fellow in physics at the Institute for Advanced Study in Princeton, I worked with Chen Ning Yang on a problem of statistical mechanics. Every morning we would have heated arguments in his office, but rarely, if ever, did we speak about anything other than physics, so concentrated was our interest.

Earlier, Yang had collaborated with Tsung Dao Lee of Columbia University in an attempt to resolve an outstanding puzzle of the time concerning the so-called “weak interactions.” In a series of now-classic papers, they had made the bold proposal that nature is not left-right symmetric. Specifically, they suggested that left-right symmetry is violated because the neutrino, a spinning subatomic particle important for the weak interactions (which also happens to be indispensable in the nuclear process that causes the sun to shine), always “spins to the left,” like an advancing left-handed screw.

The proposal led to very specific experimental predictions, and Chien Shiung Wu, an experimental physicist at Columbia, set out to test it with a team at the National Bureau of Standards. After six months of hard work, she and her co-workers verified that left-right symmetry was indeed violated. The news sent shock waves through the physics community, and Lee and Yang were awarded the Nobel Prize in Physics the following year.

I remember the morning when Yang learned of the news of the downfall of parity. He was excited about the new outlook on physics the discovery brought. Then he said suddenly, “Let’s ask the I Ching.” We threw the coins in his office and got the hexagram 53 PROGRESS:

Favorable for a maiden’s marriage.
Auspicious omen.

The body of the hexagram emphasizes that progress comes only gradually.

I think Yang was a little disappointed, but the I Ching has proven to be prophetic. By knocking down a sacred cow, Lee and Yang had led physics across a threshold, beyond which an immense vista opened up. A long fuse was lit, which has been sputtering for thirty years, illuminating vast domains in particle physics and leading to furious attempts to probe matter at a deeper level, even to plumb dimensions beyond space-time. But why the neurino should be a “left-handed screw” still remains a deep mystery, and perhaps holds the key to further progress.

Strangely enough, the I Ching had never come up in our conversations until that morning. Yet, by the mere fact that we shared a certain Chinese cultural background, it was taken for granted that we both knew about the I Ching. Neither of us believed that the I Ching could predict the future, in the sense that physics predicts the future in certain systems, but there was the unspoken understanding that to consult it was to solemnize the moment.

From chapter 6, “I Ching and Physics” of the book I Ching by Kerson Huang and Rosemary Huang (Workman Publishing, New York, 1987).

Vận tốc âm thanh và vận tốc thoát ly

Để vượt ra ngoài trường hấp dẫn của của trái đất cần vận tốc 11 km/s (vận tốc thoát ly, còn gọi là “tốc độ vũ trụ cấp 2”). Vận tốc này lớn hơn nhiều vận tốc âm thanh trong không khí, 330 m/s.

Liệu trong vũ trụ có hành tinh nào mà ở đó tốc độ âm thanh (trong khí quyển của nó) lớn hơn vận tốc thoát ly ra khỏi hành tinh đó?

The Hellmann of the Hellmann-Feynman theorem

Portrait of Hans Hellmann by Tatjana Livshits (1999)

The Hellmann-Feynman theorem is well known in quantum mechanics, perhaps even more so in quantum chemistry. Feynman is of course Richard Feynman, who gave a proof of the theorem in his undergraduate thesis. But who is Hellmann?

Hans Hellmann was born in Germany in 1903. He received his PhD in 1929 from University of Stuttgart. He proved the famous theorem in 1933. Shortly after, in 1934 Hellmann escaped from Nazi Germany to the Soviet Union. It was in the USSR where he wrote the first textbook in quantum chemistry, predating Linus Pauling’s book by a few years. He tragically died in 1938 during Stalin’s Great Terror. Below is an excerpt from a biography written by his son Hans Hellmann Jr., which appeared in the new edition of Hellmann’s quantum chemistry textbook Einführung in die Quantenchemie (Springer Spektrum, 2015). Caution: this translation relies heavily on Goole Translate. All inaccuracies in translation belong to me.


… On November 1, 1931, at the age of 28, my father got a job as a lecturer in physics (Physik-Dozent) at the University of Veterinary Medicine Hanover, despite the fact that his Habilitation has not been completed yet. His mentor, Professor Fues, assured that it was imminent. In March and July of 1933, two important papers written by my father appeared in the journal Zeitschrift für Physik. In the first paper, he presented a method by which one can make quantitative statements about the energy of polyatomic molecules on the basis of spectroscopic data of their diatomic fragments. The other work highlights the role of the kinetic energy of the electrons in the covalent bond and contained the virial theorem and the theorem known today as the Hellmann-Feynman theorem. Both works would form the basis of my father’s Habilitation thesis.

But my father was denied the Habilitation. Following the appointment of Adolf Hitler as Chancelor on January 30, 1933, several new laws directed against political opponents, and especially against Jews, were passed. The first set of laws, including the “Law for the Restoration of the Professional Civil Service” of April 7, 1933, took aim at Jewish officials. They could not fulfill their new obligation – to prove their “Aryan origins” – and lost their job. A little later, with the Reich Civil Service Act of June 30, 1933, those in the so-called “mixed marriages” were also affected.

My parents faced hard times. With the Habilitation application, my father must declare the “racial origins” of his wife [who was from a Jewish family in Ukraine], but he refused to submit the requested information. My father was never a member of any political party, but he had political beliefs and never concealed his negative attitude towards National Socialism. In contrast, most students at the University of Veterinary Medicine welcomed the “new order” with enthusiasm. They would greet my father at his lectures with disruptive and hostile noise.

My parents had in their library some books that were banned at the time: Heine, Zweig, Fallada, and magazines with articles by progressive authors. Since keeping them was dangerous, they had to destroy them. My mother told me that once, I reported enthusiastically at my kindergarten: “Yesterday my parents burned a lot of red books!” (they were issues of the [leftist] magazine “Die Weltbühne”). The teacher became scared and brought me home. That was the first, but unfortunately not the last, political lesson in my life.

In the autumn of 1933, the Prussian Culture Ministry prohibited the University of Veterinary Medicine from conferring Habilitation on my father. On December 24, 1933, he was informed that his lecturer position was to be terminated effective March 31, 1934, as his Habilitation was no longer expected due to the “non-Aryan origins” of his wife. His doctoral adviser Erich Regener would be dismissed “only” in 1938.

The continuation of scientific collaboration with colleagues at the Technical University of Hanover became impossible. There, my father had had very intensive discussions with Wilhelm Jost, Privatdozent of Physical Chemistry, on the “problem of the nature of the chemical forces.” They still managed to publish, as two papers in the journal Zeitschrift für Elektrochemie (1934/1935), the clear understanding based on quantum mechanics that they worked out, but the second paper had Wilhelm Jost listed as the sole author. Later, these two papers would form the basis of the first chapter of my father’s two textbooks of quantum chemistry.

Like many scientists, particularly those from Göttingen, a center of the development of quantum mechanics, my father had to emigrate with his family. But to where? A fateful decision was made in favor of the Soviet Union. Why? There were probably two deciding factors. First, my father had a certain sympathy for socialist ideas. Second, my mother came from there (she did not lose her citizenship) and still had relatives there. Later I learned from her that my father also had other invitations, including some from America. From about 1930, he had been looking for a job in the Soviet Union. Perhaps he talked about it with his sister back then.

Hans Hellmann and his sister Greta (c. 1930)

Through the mediation of Victor Weisskopf, still in Göttingen at the time, he got two invitations. The first was from the Ukrainian Physico-Technical Institute in Kharkov (now Kharkiv), where several well-known physicists (among others Alexander Weissberg and Lev Shubnikov, a few years later also Lev Landau) were working at that time. The second invitation was from the Physics Institute of the University of Dnepropetrovsk, where Boris Finkelstein was very interested in problems of quantum chemistry. But in both occasions the Soviet authorities refused to issue the necessary entry documents. Then came 1932 when, again with the help of colleagues in Göttingen, he made contact with the Karpov Institute in Moscow, then a leading center of physico-chemical research in the Soviet Union. After a meeting in Berlin with academician Alexander Frumkin, Deputy Director of the Karpov Institute, my father received an official invitation to Moscow and an attractive job offer. In March 1934 we got the necessary entry and exit documents. We bid farewell to our dear relatives and left Germany. Traveling by train from Berlin, we arrived in Moscow on April 31, 1934. My mother’s aunt Maria Minchina picked us up at Belorussian-Baltic Station. Her first words were: “How did you decide to come here? You are crazy!” [“Сумасшедшие, куда вы приехали?”, according to a Russian source].

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*     *

The Karpov Institute in Moscow was at that time very well financed by the government. The research carried out there were important both for the economy and the military. The directors of the Institute were two academicians, Alexei Bach, a biochemist, and Alexander Frumkin, a physicist. In the “Department of the Structure of Matter” under the leadership of Yakov Syrkin, my father was hired as the “Head of the Theory Group.” He was well received at the Institute and, which was particularly important for him, could now fully devote his knowledge and work to science. At that time, foreign scientists in the Soviet Union had some privileges. Of course, complete loyalty was demanded in return. My father had the freedom to plan his working hours, and he was also able to work at home a lot. In his letters to his mother, who was then living in Hamburg, he described the atmosphere at the Institute and the local working conditions in enthusiastically positive terms. Not a single word about the small two-room apartment or the lack of certain foods. He talked about his work and his contacts with foreign colleagues, for example, at an international conference in Kharkov in 1934.

This photo from the AIP archive was mislabeled. The person to the left of Niels Bohr, under the red arrow, is Hans Hellmann, and to the left of him, Yuri Rumer.

After numerous insults and humiliations in Hanover during the first year of the Nazi regime, he was very happy with his new life in Moscow, and the whole family was also mostly happy. In the summer of 1935 my grandmother came to the Soviet Union to visit us in the for the first and only time. She spent a short vacation in the Crimea with her son during this visit.

Early in 1935, the doctorate, the Russian equivalent of Habilitation, was conferred on my father. Approximately every two months he would publish a scientific paper. At a meeting in Dnepropetrovsk he was elected to the organizing committee of a quantum chemistry conference held in 1936. He was honored several times with monetary awards for his research and results. In June 1936 my father became a Soviet citizen, in November his starting salary of 700 rubles was increased to 1200 rubles, and in December he was invited to give a lecture at the Academy of Sciences. Shortly afterwards, on January 1, 1937, he was appointed a “Full Member of the Karpov Institute” (which corresponds to the title of a professor at a university), and in autumn 1937 became a “Senior Scientist.” In this period of about three and a half years in Moscow, my father had advised and mentored a number of young PhD students and postdocs: V. Kasatochkin, K. Maevsky, M. Mamotenko, S. Pshezhetsky, N. Sokolov, and M. Kovner.

At least since 1933 my father had been planning to write a monograph on his field of quantum chemistry. The above-mentioned join work with Wilhelm Jost in Hanover would form part of this monograph. The first version of the manuscript was finished before emigration, but remained with Jost, who tried in vain to find a publisher in Germany. Based on the manuscript, by now translated into Russian, my father in 1935-1936 gave a lecture course at the Karpov Institute, attended also by young researchers from other institutes in Moscow. Since my father’s Russian was not perfect, his PhD students sometimes had to help him find appropriate terms for notions that were new also in German. The eager Russian listeners offered their criticism and proposed corrections, for which my father expressed gratitude in the preface of “Quantum Chemistry” (Квантовая Химия, Volume 1 in the series “Physics in Monographs” ONTI, Moscow and Leningrad, 546 pages), which appeared in early 1937. More specifically, it includes an acknowledgment to his friend and colleague Yuri Rumer.

Even before completing the Russian version, my father had started revising and tightening the German version. It carried the title “Einführung in die Quantenchemie” (Deuticke, Leipzig and Vienna, 350 pages) and appeared in late 1937. While the Russian version was written for a largely unprepared reader, the shorter German version, with about the same content, placed significantly higher requirements on the reader’s preparation. But while the Russian book was selling well and was soon out of print, the German book had far fewer buyers. The reasons are possibly the contemporary historical circumstances on the one hand and the events which occurred after the publication of the books on the other hand.

*
*     *

In 1937, mass arrests of the “enemies of the people” began in the Soviet Union. Among those arrested were German and Russians, writers and farmers, engineers and artists, officers and soldiers. No one could sleep peacefully any more. The total number of innocent victims in those years amounts to over twenty million. In a letter to his mother from December 1937, my father wrote that “the current international situation has become complicated” and that he did not want to write her more often. In the night from the 9th to the 10th of March 1938, my father was arrested. I was eight and a half years old and I can still remember this event. They woke me up and searched my bed for anti-Soviet writings and evidence of espionage activities.

My father’s doctoral student M. Kovner, who often visited our house, came to Moscow from Voronezh a few days later. He wanted to visit us, but a neighbor warned him against it and told him about my father’s arrest. As the result he had to leave immediately. Later, M. Kovner published two articles about his dear teacher.

After my father’s arrest, my mother tried several times to get information on his fate from the People’s Commissariat for Internal Affairs (NKVD), the predecessor of the KGB, all in vain. She was forced, by the threats from local officials, to end her inquiries. We had to leave Moscow. My mother found a job as a German language teacher at a middle school in a village 120 km west of Moscow (near Volokolamsk). We knew nothing about my father’s fate. Former friends disappeared likewise or avoided us. Only a few people maintained friendly contact with us: the Livshits family (my mother’s relatives) and the translator Nadezhda Volpina.

Several months after the start of the Russian campaign of the German Wehrmacht, on September 9, 1941, when German troops were already on the march to Moscow, my mother was arrested. We found each other again until after the war. She was accused of “anti-Soviet propaganda,” and it was claimed that she, a Jew who had fled Nazi Germany, had been waiting for the German troops to work for them as a translator. After several months in Moscow prison, she was exiled to the Semipalatinsk area in Kazakhstan.

Only after Stalin’s death, and the beginning of the politics of “the Thaw,” my mother was “fully rehabilitated.” She then requested information about her husband. Initially she received a certificate, which later turned out to be wrong, that he died in prison from a disease (peritonitis). Then my mother applied for a certificate of rehabilitation for my father, which she received in 1957. Now my father was “fully rehabilitated.” Sadly, this happened only after his death. Only in 1989, during the “perestroika,” did we get the real death certificate. The documents showed that my father was convicted to “high treason” and “espionage in favor of Germany” in accordance with Article 58 of the Criminal Code and was shot dead on May 29, 1938.

Already in May 1937, shortly after the beginning of mass arrests, Albert Einstein sent a letter to Stalin expressing his great concern for the fate of many well-known scientists. A similar letter from three Nobel laureates Irène Joliot-Curie, Frédéric Joliot-Curie, and Jean Baptiste Perrin was sent to Stalin in June 1937. But their voices were not heard. When Wilhelm Jost noticed in 1938 that the name “Hellmann” no longer appeared among the authors in the journal Acta Physicochimica URSS where my father worked and regularly published until October 1937, he asked his British colleague John Lennard-Jones for help. Lennard-Jones sent a request for reprints to my father’s address at the Karpov Institute, but there was no reply.

Due to the historical and political circumstances and the tragic fate caused by them, for decades my father’s name disappeared almost completely from science. Although his book “Einführung in die Quantenchemie” appeared in the US in 1944 as a war booty, it hardly found the wide dissemination which it deserves. The name “Hellmann” is now remembered mostly through the term “the Hellmann-Feynman theorem”…

A discussion on symmetry in physics in the aftermath of the Cultural Revolution

The text below is from T.D. Lee, Symmetries, Asymmetries, and the Word of Particles, University of Washington Press, Seattle, 1988. (Disclaimer: this post is by no means an endorsement of Mao Zedong or the Cultural Revolution.)

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*     *

“Tell me, why should symmetry be of importance?” asked Chairman Mao Zedong.

That was on May 30, 1974, when China was still in the turmoil of the Cultural Revolution and the Gang of Four was at the zenith of its power. I was especially depressed to find, in that ancient land of civilization, that education had been almost totally suspended. I hoped desperately that somehow there would be a way to improve, however slightly, the course of events.

At about six o’clock that morning, the phone in my room at the Beijing Hotel had rung unexpectedly. I was told that Mao would like to see me in one hour at his residence in Zhong Nan Hai, inside the former imperial palace. I was even more surprised that when he saw me the first thing he wanted to find out was about symmetry in physics.

According to Webster’s dictionary, symmetry means “balanced proportions” or “the beauty of form arising from such balanced proportions.” In Chinese, symmetry is 对称, which carries an almost identical meaning. Thus it is essentially a static concept. In Mao’s view, the entire evolution of human societies is based on dynamic change. Dynamics, not statics, is the only important element. Mao felt strongly that this also had to be true in nature. He was, therefore, quite puzzled that symmetry should be elevated to such an exalted place in physics.

During our meeting, I was the only guest. A small end table was placed between our chairs, on which there were pads, pencils, and the ever present green tea. I put a pencil on the pad and tipped the pad toward Mao and back toward me. The pencil rolled one way and then the other. I pointed out that at no instant was the motion static, yet as a whole the dynamic process had a symmetry. The concept is by no means static; it is far more general than its common meaning indicates and is applicable to all natural phenomena from the creation of our universe to every microscopic subnuclear reaction. Mao appreciated the simple demonstration. He then asked more questions about the deeper meaning of symmetry, and also about other physics topics. He expressed regret that he had not had the time to study science, but he remembered a set of science books by J. Arthur Thomson which he had enjoyed reading when he was young.

Our conversation gradually shifted from natural phenomena to human activities. In the end, Mao accepted my limited proposal that the education of at least the very brilliant young students should be maintained, continued, and strengthened. This led, with the strong support of Zhou Enlai, to the elite “youth class,” a special intensive education program for talented students form the early teens through college. It was established first at the University of Science and Technology in Anhui and later, because of its success, also at other Chinese universities.

The next day, at the airport, I received a farewell present from the Chairman: a four-volume set of the original 1922 edition of The Outline of Science by J. Arthur Thomson.

To the general chaos produced by the Cultural Revolution, this meeting brought only a minute amount of order. Nevertheless, in a very limit way perhaps it does indicate a correlation between man’s intrinsic urge to search for the symmetry in nature and his desire for a society that is both meaningful and more balanced.

 

Giải Nobel của Einstein, hay là sóng điện thoại có gây ung thư hay không

Công chúng thường biết đến Albert Einstein như người khám phá ra thuyết tương đối, làm thay đổi quan niệm của chúng ta về không gian và thời gian. Chắc ai cũng biết thuyết tương đối bao gồm thuyết tương đối hẹp, được Einstein tìm ra năm 1905, và thuyết tương đối rộng, được ông tìm ra 10 năm sau. Tuy nhiên có thể không phải ai cũng biết là giải thưởng Nobel về vật lý năm 1921 của Einstein lại nhắc đến một khám phá khác của ông: hiệu ứng quang điện. Đây là công trình Einstein viết cũng vào năm 1905, cùng năm với công trình về thuyết tương đối hẹp và một công trình nữa, cũng rất nổi tiếng, về chuyển động Brown. Hiệu ứng quang điện là đóng góp lớn nhất của Einstein vào thuyết lượng tử, lý thuyết mà sau này được Bohr, Heisenberg, Schrödinger và nhiều người khác phát triển lên nhưng lại bị Einstein nghi ngờ đến cuối đời.

Hiệu ứng quang điện là hiện tượng khi ta chiếu ánh sáng vào một tấm kim loại thì thỉnh thoảng điện tử bị bứt ra khỏi kim loại. Ta có thể đoán là ánh sáng càng mạnh thì càng nhiều điện tử bị bứt ra. Phán đoán này hoá ra là không hoàn toàn đúng: có những nguồn ánh sáng rất mạnh không gây ra hiệu ứng quang điện, nhưng có những nguồn yếu hơn lại gây ra hiệu ứng này. Thực nghiệm cho thấy rằng hiệu ứng quang điện phụ thuộc vào tần số của ánh sáng. Ví dụ, với cùng một mẫu kim loại, ánh sáng đỏ hoặc tia hồng ngoại không gây ra hiệu ứng nhưng ánh sáng tím hoặc cực tím lại có tác dụng.

Einstein giải thích điều này bằng cách áp dụng và mở rộng giả thuyết lượng tử của Planck. Einstein giả thuyết rằng ánh sáng bao gồm các hạt photon, mỗi hạt mang một năng lượng tỉ lệ thuận với tần số của ánh sáng;

E=h\nu

Ở đây E là năng lượng của hạt photon, \nu là tần số của ánh sáng, và h là hằng số Planck. Công thức trên có tên là công thức Planck, công thức mà theo tôi đáng lẽ ra phải nổi tiếng hơn công thức E=mc2.

Hiệu ứng quang điện là quá trình một hạt photon truyền năng lượng cho một hạt điện tử. Để bứt một điện tử ra khỏi mảnh kim loại ta cần một năng lượng tối thiểu nhất định, ta gọi là \Delta. Như vậy chỉ khi \nu >\Delta/h ánh sáng mới có thể bứt được điện tử ra khỏi khối kim loại. Nếu \nu < \Delta/h thì nguồn sáng có mạnh thế nào cũng không có photon đủ năng lượng để gây ra hiệu ứng quang điện.

Bạn có thể hỏi liệu có khi nào hai hạt photon, hoặc nhiều hơn, cùng hợp sức để bứt ra một điện tử hay không. Điều này về nguyên tắc có thể xảy ra, nhưng xác suất rất thấp, có thể bỏ qua.

Hiệu ứng quang điện có liên quan trực tiếp đến một câu hỏi hay được đặt ra hiện nay: điện thoại di động có gây tác hại cho sức khoẻ hay không? Một trong những điều làm nhiều người lo lắng là khả năng gây ung thư của sóng điện thoại (ví dụ xem bài này). Nhiều người còn nói là sóng điện từ trong lò vi sóng cũng có thể gây ra ung thư.

Nếu ta nhớ lại công thức E=h\nu của Einstein thì ta sẽ thấy những lo lắng này không có cơ sở. Đó là do tần số sóng của các thiết bị điện tử quá thấp để có thể gây ra những biến đổi của phân tử ADN. Tần số sóng trong lò vi sóng là 2500 MHz, tần số của điện thoại di động là 800 MHz hay 1900 MHz. Hằng số Planck là 4×10-9 eV/MHz, như vậy 2500 MHz tương đương với năng lượng 10 phần triệu eV, trong khi các quá trình hoá học hay sinh hoá cần năng lượng cỡ ít nhất 0.1 eV, nếu không phải là 1 eV. Sự chênh lệch đến 10-100 nghìn lần giữa hai cỡ năng lượng làm cho lò vi sóng hay điện thoại di động không thể làm biến đổi gien của miếng thịt để trong lò, hay cơ thể chúng ta. (Tia cực tím thì lại khác, vì tần số của tia cực tím cao hơn tần số của điện thoại di động đến cả triệu lần, nên nó có đủ năng lượng để gây tác hại cho tế bào).

Tất nhiên là điện thoại di động hay các thiết bị điện tử có thể có những tác hại khác, ví dụ cho tâm lý hay giấc ngủ của người dùng, nhưng chúng ở ngoài khuôn khổ của bài viết này.