It is often said in Japan that teaching is learning. I am always convinced that this saying is true everytime I prepare for lectures and discussions with students. Last week it was the partial wave expansion of plane wave that reminded me of the saying again.
Obvioulsly, the Hamiltonian for a free particle has translational and rotational symmetries. So, linear and angular momenta are good quantum numbers. However, the linear and angular momentum operators do not commute with each other, which means we need to choose one of them for labelling the free-particle wave function. In the scattering problem, linear momentum is usually chosen because of the nature of the scattering experiment (you know, the incident beam is produced with a good linear momentum...). This means the wave function far from the target is a plane wave.
In the vicinity of the target, however, angular momentum plays more important role than linear momentum because the interaction between the target and the beam causes the trajectory of the beam bent. So, in many cases, the plane wave is expanded by wave functions labelled by angular momentum, which has a product form of the spherical Bessel function and the spherical Harmonics.
Then, I had a question. When the flux of the free-particle wave function is calculated with the plane wave, the result is well-known to be the constant velocity (in a non-relativistic limit). Even after a conversion into the spherical polar coordinates, the flux has only \theta (orientation) dependence, and there is no dependence of distance (from the origin). However, with the angular momentum representation ( that is, with spherical Bessel and sherical harmonics), the resultant flux seems r-dependent (that is, dependent on the distance). What is wrong with this?
After analysing several mathematical relations, I found that the spherical Bessel functions satisfies a kind of conditions similar to the completeness condition. But this completeness condition is realised only with the weight factor (2L+1), where L represents angular momentum. I have never seen this in the literatures.
I talked to J. on last Friday about this problem. His expertise is the nuclear reaction theory. He said that he was aware of this problem, but could not give the instant answer. "OK, so this problem is not so trivial." is my impression. He continued that he has seen an article in the Am. J. Phys. discussing this problem many years ago. Maybe, the answer can be obtained from the paper.... if J's information is correct and the paper really exists....
Monday, February 28, 2005
Thursday, February 24, 2005
Snow and Lecture
Last four or five days, we have had snowfall in the southeast England. Even today... Last three years since I came to Britain, I have never seen this kind of weather. I asked my colleague, who has lived here more than five years, if he has seen the similar weather before, but his answer was negative.
Nevertheless, thanks to the Gulf stream (probably), snow melts quickly this year. In fact, it is now at noon and I can still see, through my office windows, snow falling, but there is no snow on the road. When I came to the university this morning, the world was totally white, though.
So, the serious problem this time is the snow on the raod early in the morning, in particular on Thursdays. This is based on a totally selfish reason, that is, I have a lecture of Maths to give from 9:00 am to 11:00 am every Thursday in the Spring semster. So, I need to come to the university by 8:30 am. Around 8 o'clock, air temperature is still low and there is much snow on the road.
Last three years, I have at least one day per year when I needed to fight with snow to come to the university. Two years ago was the most serious. I could not drive a car on that day, but luckily I lived relatively near to the university at that time. So, I can walk to the university. Now, I am living far away from the University. Even by car, it takes about 30 minutes (using high-speed road, similar to freeways), so if heavy snow falls I have a great difficulty to come to Guildford.
They say the global warming is happening. Yes, there are many strange things happening in weather and climate, but sometimes not in a way we expect. Well, if you say "warming", then why so many snow in early spring or end of winter? Of course, I know things are not so easy and complicated because the dynamics that weather follows is highly non-linear. But still, simply from the selfish reason, I hope the "warming" is literally meant as "warming". What I am saying is that I don't like warm winter, but that I want real spring without much snowfall! (I believe today can be regarded as "early spring"...)
In a Korean drama I have recently seen, there was a scene for a couple who went out for a date for the first time to ask personal questions with each other. The questions contained:
"What is your favorite colour?" --- "White"
"What is your favorite season?" ---- "Winter"
Definitely, with 100% surely, I don't belong to that sort of humans. (The drama is "Kyour Yonka", or "冬恋歌", of course...)
Nevertheless, thanks to the Gulf stream (probably), snow melts quickly this year. In fact, it is now at noon and I can still see, through my office windows, snow falling, but there is no snow on the road. When I came to the university this morning, the world was totally white, though.
So, the serious problem this time is the snow on the raod early in the morning, in particular on Thursdays. This is based on a totally selfish reason, that is, I have a lecture of Maths to give from 9:00 am to 11:00 am every Thursday in the Spring semster. So, I need to come to the university by 8:30 am. Around 8 o'clock, air temperature is still low and there is much snow on the road.
Last three years, I have at least one day per year when I needed to fight with snow to come to the university. Two years ago was the most serious. I could not drive a car on that day, but luckily I lived relatively near to the university at that time. So, I can walk to the university. Now, I am living far away from the University. Even by car, it takes about 30 minutes (using high-speed road, similar to freeways), so if heavy snow falls I have a great difficulty to come to Guildford.
They say the global warming is happening. Yes, there are many strange things happening in weather and climate, but sometimes not in a way we expect. Well, if you say "warming", then why so many snow in early spring or end of winter? Of course, I know things are not so easy and complicated because the dynamics that weather follows is highly non-linear. But still, simply from the selfish reason, I hope the "warming" is literally meant as "warming". What I am saying is that I don't like warm winter, but that I want real spring without much snowfall! (I believe today can be regarded as "early spring"...)
In a Korean drama I have recently seen, there was a scene for a couple who went out for a date for the first time to ask personal questions with each other. The questions contained:
"What is your favorite colour?" --- "White"
"What is your favorite season?" ---- "Winter"
Definitely, with 100% surely, I don't belong to that sort of humans. (The drama is "Kyour Yonka", or "冬恋歌", of course...)
Friday, February 18, 2005
Broadband and Physics
Finally, Broadband came to my home. I have waited for today for 5 years.... Before the introduction of Broadband, I cannot edit this blog from home. (I need to come to the university....) But now I can edit at home.
At the University, I feel I am working. I mean, I talk to students, answer the questions from them, printing papers to read, writing e-mails to colleagues and colloaborators... But I have no time to think at the university! So, I prefer to work at home because there is a plenty of time for thinking (but dangerously for sleeping too). When tired from calculations and debugging, I can take a walk around my home, which has nice hills and forests around with beauttiful views of the North Downs (Surrey Hills). Many wild animals such as rabits, foxes, deers, etc. and English wild flowers and trees are also refreshing my mind everytime. So, BT broadband, which allows me to work at home, is extremely useful to me.
I usually prepare my lecture at home. While preparing, many questions appear in mind for the things I thought I understood. These questions make me understand physics much more deeply. I agree, this is the experience of learning from teaching. So, giving lectures to studnets is very rewarding although it is very time consuming and sometimes very hard.
Today, I was preparing a lecture of quantum mechanics for next tuesday. The subject is about the scattering theory. I know there are two representations for the description of a free particle. When linear momentum is given, it is a plain wave. While, when angular momentum is given, it is the spherical Bessel functions with the spherical harmonics. Then, suddenly today, I have a question.... why in the presence of the central potential do we need to think about the partial wave expansion for the scattered wave function? Since the potential is rotationally symmetric, angular momentum is a good quantum number. We should then start from the wave function with good angular momentum! I thought for a while (about an hour?), but could not find the answer. So, I decided to take a bath. Then after 10 minutes or so (even after singing some Japanese old pop music), like Archimedes, the idea came down from the heaven (or somewhere)!
Well, it might be too expensive to take a bath for finding every new idea. But I manage to survive as a lecturer like this, and I like this kind of life very much. Thinking is what I like most, and surprisingly it is my job now! I thank God (and my employers) and the broadband for allowing me this.
At the University, I feel I am working. I mean, I talk to students, answer the questions from them, printing papers to read, writing e-mails to colleagues and colloaborators... But I have no time to think at the university! So, I prefer to work at home because there is a plenty of time for thinking (but dangerously for sleeping too). When tired from calculations and debugging, I can take a walk around my home, which has nice hills and forests around with beauttiful views of the North Downs (Surrey Hills). Many wild animals such as rabits, foxes, deers, etc. and English wild flowers and trees are also refreshing my mind everytime. So, BT broadband, which allows me to work at home, is extremely useful to me.
I usually prepare my lecture at home. While preparing, many questions appear in mind for the things I thought I understood. These questions make me understand physics much more deeply. I agree, this is the experience of learning from teaching. So, giving lectures to studnets is very rewarding although it is very time consuming and sometimes very hard.
Today, I was preparing a lecture of quantum mechanics for next tuesday. The subject is about the scattering theory. I know there are two representations for the description of a free particle. When linear momentum is given, it is a plain wave. While, when angular momentum is given, it is the spherical Bessel functions with the spherical harmonics. Then, suddenly today, I have a question.... why in the presence of the central potential do we need to think about the partial wave expansion for the scattered wave function? Since the potential is rotationally symmetric, angular momentum is a good quantum number. We should then start from the wave function with good angular momentum! I thought for a while (about an hour?), but could not find the answer. So, I decided to take a bath. Then after 10 minutes or so (even after singing some Japanese old pop music), like Archimedes, the idea came down from the heaven (or somewhere)!
Well, it might be too expensive to take a bath for finding every new idea. But I manage to survive as a lecturer like this, and I like this kind of life very much. Thinking is what I like most, and surprisingly it is my job now! I thank God (and my employers) and the broadband for allowing me this.
Thursday, February 17, 2005
Dirac and Wheeler
Here are two textbooks about the general relativity.One was written by Dirac, and the other by Wheeler et al. Although the theme is the same between them, there is the big difference, which is thickness of the books. I think it is correct to assume that this difference is clearly reflecting the difference in the standing point to look at physics. (I mean, simply the difference in their personal characters.....maybe.)
I bought Dirac's book about 10 years ago. This book contains just 100 pages, with large letters and many pages with a lot of white blank. But, still I don't finish reading it. On the other hand, I already read about 50 pages of Wheeler's, which I bought last year. Interestingly, the ratio (R) of the pages I have ever read (P) and the total page (T) is the same for Dirac's and Wheeler's. (Namely, R_dr \simeq R_wh, where R=P/T.) This may explain the conservation of momentum to understand physics???
I bought Dirac's book about 10 years ago. This book contains just 100 pages, with large letters and many pages with a lot of white blank. But, still I don't finish reading it. On the other hand, I already read about 50 pages of Wheeler's, which I bought last year. Interestingly, the ratio (R) of the pages I have ever read (P) and the total page (T) is the same for Dirac's and Wheeler's. (Namely, R_dr \simeq R_wh, where R=P/T.) This may explain the conservation of momentum to understand physics???
Monday, February 14, 2005
Gravity and Wheeler
In Surrey University, BSc students in the final year are required to do a kind of research called the Final Year Project. This year, I took two students to supervise. I gave one of them the review project on the Dirac equation, which is not so difficult to supervise because it is a quantum theory. Topics from Quantum Mechanics are easy to handle to me because all of my research are from it. However, many students are interested in Black Holes for some reasons. Having considered this "demand from the market", I decided to give the other project on the theory of black hole this year.... which was my big mistake.
Personally, I did not spend much time to learn this subject. I think many physics students and academics have similar experiences as mine, except very few. This is because the gravitation theory cannot give us many jobs, unlike Quantum theory. In modern Universities, the emphasis is placed on mastering quantum mechanics and its applications. As a consequence, very few students seriously study the gravitation theory, or the general theory of relativity.
It is, again, J. Wheeler, who motivated me to go for the theory of Black hole... I mean I don't know him personally,
but I have been influenced by his papers, works, books and articles, directly and indirectly. For instance, my current research as a theoretical nuclear physicist is to solve the Hill-Wheeler equation through a numerical approach for the nuclear wobbling motion; It is when I was a high-school student that I learned to know he was the supervisor of R. Feynman, by reading the famous book "surely, you're..."; and as a fatal blow, my former supervisor in Tokyo recommended me to read Wheeler's recent self-biography book "Geons....", which I've read it a few months ago.
You can tell easily from this book that Wheeler is a genius and a real theoretical physicist: his carrer started from nuclear physics, then particle physics (the famous collaboration with his "PhD student" about the QED) and gravitation theory (naming of the "black hole" was given by Wheeler, and his predictions of gravitational wave and worm holes are famous), and finally recently he arrived at quantum computation and quantum information theory! (David Deutsch is his fomer student!)
Well, I am a nuclear physicist. Particle physics is the thing I am already interested in and sometimes apply the concepts into nuclear physics, but the gravitation theory?? I took a lecture of Prof. Ezawa in Tohoku and of Prof. Eguchi in Tokyo. It was nearly ten years ago when I took these lectures.... So, naturally, I don't remeber what the Einstein's equation is as well as the Christoffel symbol... But If Wheeler did before, maybe I can ..... is the starting point of my mistake. It is true, however, that I enjoy relearning this "new" subject a lot. I hope my FYP student also enjoy this subject.
Personally, I did not spend much time to learn this subject. I think many physics students and academics have similar experiences as mine, except very few. This is because the gravitation theory cannot give us many jobs, unlike Quantum theory. In modern Universities, the emphasis is placed on mastering quantum mechanics and its applications. As a consequence, very few students seriously study the gravitation theory, or the general theory of relativity.
It is, again, J. Wheeler, who motivated me to go for the theory of Black hole... I mean I don't know him personally,
but I have been influenced by his papers, works, books and articles, directly and indirectly. For instance, my current research as a theoretical nuclear physicist is to solve the Hill-Wheeler equation through a numerical approach for the nuclear wobbling motion; It is when I was a high-school student that I learned to know he was the supervisor of R. Feynman, by reading the famous book "surely, you're..."; and as a fatal blow, my former supervisor in Tokyo recommended me to read Wheeler's recent self-biography book "Geons....", which I've read it a few months ago.
You can tell easily from this book that Wheeler is a genius and a real theoretical physicist: his carrer started from nuclear physics, then particle physics (the famous collaboration with his "PhD student" about the QED) and gravitation theory (naming of the "black hole" was given by Wheeler, and his predictions of gravitational wave and worm holes are famous), and finally recently he arrived at quantum computation and quantum information theory! (David Deutsch is his fomer student!)
Well, I am a nuclear physicist. Particle physics is the thing I am already interested in and sometimes apply the concepts into nuclear physics, but the gravitation theory?? I took a lecture of Prof. Ezawa in Tohoku and of Prof. Eguchi in Tokyo. It was nearly ten years ago when I took these lectures.... So, naturally, I don't remeber what the Einstein's equation is as well as the Christoffel symbol... But If Wheeler did before, maybe I can ..... is the starting point of my mistake. It is true, however, that I enjoy relearning this "new" subject a lot. I hope my FYP student also enjoy this subject.
Friday, February 11, 2005
Physics and Rain
It is raining today, nearly all day from the morning. Because there was no lecture to give today, I wanted to take a walk in the hill of Abinger Hammer, where my home is. Yes, there was no lecture today and raining... Theoretical Physicists need a calm and comfortable environment for thinking. Today looked a best day. So, I stayed at home until evening.
There is one thing disturbing my mind recently. It is a relation between symmetry and collectivity. I learned when I was a PhD student that a quantum object with perfect roational symmetry cannot rotate. Therefore, the ground states of the magic nuclei such as 16-O and 208-Pb have zero angular momentum I=0. In many (probably, "most of") textbooks, this is explained by words, and with no mathematics. Well, Dirac said that we need to understand physics without relying on mathematics, and Faraday realized what Dirac said in fact... But, As many other students, it is true that I forced myself to swallow this "well-known" fact as a common sense. However, deep in my heart, there have been something which I cannot digest completely.
I think it is very important to understand (or definte carefully) the term "collectivity". For example, what does the collective wave function mean? In nuclear physics, the particle-plus-rotor model answers this question in a phenomenological manner. So, I decided to use this model to think this problem today.
Out of the window, I saw grey clouds come and go with a great speed. Sometimes, the sound of raindrops was magnified on the roof. Although the problem was hard to solve, I enjoyed this time of thinking under this natural rhythm and melody.
There is one thing disturbing my mind recently. It is a relation between symmetry and collectivity. I learned when I was a PhD student that a quantum object with perfect roational symmetry cannot rotate. Therefore, the ground states of the magic nuclei such as 16-O and 208-Pb have zero angular momentum I=0. In many (probably, "most of") textbooks, this is explained by words, and with no mathematics. Well, Dirac said that we need to understand physics without relying on mathematics, and Faraday realized what Dirac said in fact... But, As many other students, it is true that I forced myself to swallow this "well-known" fact as a common sense. However, deep in my heart, there have been something which I cannot digest completely.
I think it is very important to understand (or definte carefully) the term "collectivity". For example, what does the collective wave function mean? In nuclear physics, the particle-plus-rotor model answers this question in a phenomenological manner. So, I decided to use this model to think this problem today.
Out of the window, I saw grey clouds come and go with a great speed. Sometimes, the sound of raindrops was magnified on the roof. Although the problem was hard to solve, I enjoyed this time of thinking under this natural rhythm and melody.
Thursday, February 10, 2005
Physics and Blog
I didn't know what "blog" means just a week ago.
And suddenly, I noticed that many "diaries" by physicists are being replaced by "blogs".So, I decided to try by myself, too.
I know many Japanese physicists have their "research diaries" or "research blogs",and I found also some American physicists do the same thing.It is strange, but I haven't heared of anybody in the UK who is doing this. Maybe there are, maybe not.
I have a feeling that the Japanese and Americans are similar in this point, that is, writing diaries for other people.
What is the reason for this "diary"? They are supposed to be private by definition, aren't they?
I myself think why I want to write this kind of thing in the web, over and over again. I have not yet had the answer, but I can say one thing. I like to write and want them to be read since my elementary school days.
From the fact that my papers are sometimes cited by other physicists,my scientific papers are read by somebody,
which is quite rewarding to me. Unfortunately, it is rare to send messages about the contents of other people's paper
inside the nuclear physics community. (I think the reason is that nuclear physics does not have a central and common big problems everybody tries to attack, unlike particle physics. ) Probably, because of this, I want to show what I am doing and trying to do in my life of research, to other people. I don't know...
漢字もつかえるのだろうか？
And suddenly, I noticed that many "diaries" by physicists are being replaced by "blogs".So, I decided to try by myself, too.
I know many Japanese physicists have their "research diaries" or "research blogs",and I found also some American physicists do the same thing.It is strange, but I haven't heared of anybody in the UK who is doing this. Maybe there are, maybe not.
I have a feeling that the Japanese and Americans are similar in this point, that is, writing diaries for other people.
What is the reason for this "diary"? They are supposed to be private by definition, aren't they?
I myself think why I want to write this kind of thing in the web, over and over again. I have not yet had the answer, but I can say one thing. I like to write and want them to be read since my elementary school days.
From the fact that my papers are sometimes cited by other physicists,my scientific papers are read by somebody,
which is quite rewarding to me. Unfortunately, it is rare to send messages about the contents of other people's paper
inside the nuclear physics community. (I think the reason is that nuclear physics does not have a central and common big problems everybody tries to attack, unlike particle physics. ) Probably, because of this, I want to show what I am doing and trying to do in my life of research, to other people. I don't know...
漢字もつかえるのだろうか？
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