Physicists' knowledge of differential equations

[mpresic]

<<Moderator's note: this is a spin-off of https://www.physicsforums.com/threads/how-long-to-learn-physics.891250/>>

To Zapper. Some differential equations have no analytical solutions but some do. I remember a case where a co worker was using a computer to find the solution to x dot = 1 / (1 + x squared). I asked him, when you finish, are you going to compare your answer with x = inv tan (t). He told me you cannot integrate this equation because the problem is non-linear. Can you beat that?

I was also told that integral exp -ax^2 from 0 to infinity could not be evaluated analytically (his teachers told him). When I showed him the evaluation, he told me make sure to write it down, in case I would forget it. Both these co-workers had advanced degrees in engineering.

These are egregious cases but I can cite at least a few more which are only slightly less egregious.

The computer has opened new vistas in the solutions to differential equations, and I agree only a small minority of problems can be solved in any other way. Nevertheless, I think solving (using separation of variables) e.g. The Hydrogen Atom, The QM harmonic oscillator, Legendre's differential equation, etc, (usually by power series), and understanding why the separat1ion constants are chosen (to terminate the series), should be bread and butter for newly minted physicists.

Knowing the solution to these idealized problems is often a first step in modeling actual problems efficiently.

 

[ZapperZ]

To Zapper. Some differential equations have no analytical solutions but some do.

Actually, MOST differential equations (at least the one dealing with real-world situation) don't, only some do!

I remember a case where a co worker was using a computer to find the solution to x dot = 1 / (1 + x squared). I asked him, when you finish, are you going to compare your answer with x = inv tan (t). He told me you cannot integrate this equation because the problem is non-linear. Can you beat that?

I was also told that integral exp -ax^2 from 0 to infinity could not be evaluated analytically (his teachers told him). When I showed him the evaluation, he told me make sure to write it down, in case I would forget it. Both these co-workers had advanced degrees in engineering.

These are egregious cases but I can cite at least a few more which are only slightly less egregious.

The computer has opened new vistas in the solutions to differential equations, and I agree only a small minority of problems can be solved in any other way. Nevertheless, I think solving (using separation of variables) e.g. The Hydrogen Atom, The QM harmonic oscillator, Legendre's differential equation, etc, (usually by power series), and understanding why the separat1ion constants are chosen (to terminate the series), should be bread and butter for newly minted physicists.

Knowing the solution to these idealized problems is often a first step in modeling actual problems efficiently.

But does this qualify you to claim that "...I many physicists today, have a limited background in solving partial differential equations analytically..."? That is what I am disputing.

"Newly minted physicists" know how to solve those problems that you described. They are part of a standard undergraduate course. However, I do not believe you are aware of the type of problems that many, if not most, physicists have to deal with. For example, have you seen the Green's function that one has to solve for a sharp, pointed tip in a uniform electric field? This is the typical problem that one has to deal with in finding the field-enhancement factor and subsequently the field-emission current from geometrical consideration. Or what about solving the Hamiltonian for the general solution in a n-body problem (n≥3)?

Let me repeat. In many of the real-world problems that we have to solve as researchers, in the overwhelming majority of the cases, you cannot solve the diff. eq. analytically, at least not without some kind of simplification. I challenge you to pick up any published papers that had to solve something numerically, and show that there is an analytical solution.

Zz.

 

[Dr. Courtney]

For much of my numerical work, the analytical solutions are important for validating the numerical code for known cases.

Then the perturbations are added and the analytical solutions are no longer valid.

Another point of interest is that analytical solutions often provide an upper or lower boundary for a numerical solution. In some cases, there are analytical solutions that provide both upper and lower bounds.

 

[ZapperZ]

For much of my numerical work, the analytical solutions are important for validating the numerical code for known cases.

Then the perturbations are added and the analytical solutions are no longer valid.

Another point of interest is that analytical solutions often provide an upper or lower boundary for a numerical solution. In some cases, there are analytical solutions that provide both upper and lower bounds.

I don't doubt the usefulness of such things. I use them myself as a "quick-and-dirty" reality check. But to claim that many physicists simply don't know how to find such solutions is unbelievably naive. I WISH I can solve many of the rf waveguides fields that we design analytically. It will save us a lot of licensing fees for our group's use of COMSOL.

Zz.

 

[Dr. Courtney]

But to claim that many physicists simply don't know how to find such [analytical] solutions is unbelievably naive. I WISH I can solve many of the rf waveguides fields that we design analytically. It will save us a lot of licensing fees for our group's use of COMSOL.

I took the claim as more of a concern regarding newly minted and future physicists than the pool of existing practicing physicists. As applied to practicing physicists it is silly.

But I think there is some legitimate concern that training to find analytic solutions may be in decline, both in terms of the propensity to approach problems numerically, as well as in the trend of using automated solvers (MMa, etc.) rather than thinking methods to find analytic solutions.

A number of schools (usually the math departments) are reducing the emphasis of analytic methods even as applied to first year integration and differentiation techniques. Students who get to courses in differential equations (or physics courses that use them) having gotten through first year Calculus without a solid background in pencil and paper integration and differentiation techniques are going to need crutches to survive and keep moving forward. It will be awfully tempting to offer numerical approaches and automated solvers as these crutches.

 

[ZapperZ]

But I think there is some legitimate concern that training to find analytic solutions may be in decline, both in terms of the propensity to approach problems numerically, as well as in the trend of using automated solvers (MMa, etc.) rather than thinking methods to find analytic solutions.

A number of schools (usually the math departments) are reducing the emphasis of analytic methods even as applied to first year integration and differentiation techniques. Students who get to courses in differential equations (or physics courses that use them) having gotten through first year Calculus without a solid background in pencil and paper integration and differentiation techniques are going to need crutches to survive and keep moving forward. It will be awfully tempting to offer numerical approaches and automated solvers as these crutches.

But at some level, this isn't a question based on a matter of tastes. Is there an analytical solution to such-and-such a problem, or not? This is not an ambiguous question with an ambiguous answer.

If the answer is no, then the question is, to what extent is an "accurate" numerical solution is needed? And to what extent can simplification of the problem that allows for an analytical solution is useful? This cannot be answered in general because it depends on the situation. I cannot resort to simplification when designing a $300,000 linac waveguide system because I only have ONE shot at getting it right or I've wasted a boatload of money and the darn thing becomes an expensive paper weight (try explaining that to the funding agency).

The issue here isn't the technique. The issue here is the final outcome. I'd use a gerbil running in a wheel if it gets me what I want.

Zz.

 

[mpresic]

T zapper: We shouldn't hijack the thread. I conceded the computer has opened new vistas in research.

Dr Courtney also supported me so I no longer feel defensive to my earlier statement. Maybe "many physicists" is too strong, but if say 10% of the physicists couldn't solve a partial differential equation in spherical coordinates using separation of variables, (and I suspect the 10% estimate is conservative), wouldn't that qualify as many.

I am lucky enough to work in an area where we are encouraged (even required) to learn e.g. potential theory from university professors and other researchers. Many of us "covered" the material (superficially), before in about 5-10 weeks in a graduate course, usually years before completing our degree. Often, these specialists "re-educate" us in special points we would (or should) have learned earlier.

I do think graduate programs in physics have de-emphasized mathematical physics, especially special functions, and classical mechanics in favor of areas of specialization, such as solid state, plasma, space, high energy, gravitation, etc. In my particular line, this de-emphasis is unfortunate.

Returning to the OP: Perhaps an applied mathematician (such as my friend whom I mentioned) already has or could develop tools in his toolbox, apart from the physicist.

 

[mpresic]

To Zapper again:

I hope I did not imply my mathematician friend (mentor) formerly in wave propagation (he is retired now) nor I do not see the value of using commercially available software and computer approximation. He used and promoted them extensively (especially MATLAB, and I forgot what he used for EM wave propagation in ducts).

He also criticized me at time for putting too much emphasis on special functions, though we had many discussions of their properties over lunch.

I also agree only a small minority of differential equations have analytic solutions.


Reconsidering, I think we may be closer than my earlier message.

 

[epenguin]

At the University I went to mnayh mnyah to do biochemistry 'A' level maths (involved a lot of integration) was an entry requirement.
On the one hand it was a relay effect. Biochemistry thought we had to have proper chemistry, including physical chemistry; then the chemists said they couldn't teach us quantum mechanics unless we had proper math. So additionally to A-levels we had a year of more advanced maths. After reaching the second year they decided not to teach us quantum mechanics anyway.
That had been one of the rationales though. The other I was told was that when they let the Profs at other universities know that they were thinking of dropping the maths entry requirement, these protested for heaven's sake don't do that because then we who don't have the requirement will never get any good students! In other words it was a selection mechanism: mine was a high prestige institution, there were not that many students yet attracted by or who had even heard of biochemistry, so they didn't easily get the numbers of good applicants they would have liked. But some were forced to apply to them because of their lack of A levels. People tended to specialise in either Biology with some chemistry or the physical sciences. In biology they were often in aiming at medicine. Anyway beyond aged 16 they didn't think they would ever need math. And I think for many years in Britain rather few people doing biology had math beyond age 16. I think often the universities put on remedial courses, not strictly in the syllabus, because it's a big disadvantage to a biochemist to not understand logs for example. For all I know this may have changed by now - the government seems to think that future prosperity depends on everybody having maths with everything and thinks that Singapore is the model.

About differential equations and integration I think that if I were teaching to this category of students what I would tell them about integrating expressions dx, and differential equations is - don't. If your primary interest is not maths for its own sake, when the textbooks of physical chemistry drag you through the solution of the Schrödinger equation or something, jump straight to the solution at the end and by differentiation convince yourself that it satisfies the equation. Do a bit of that and you may get the hang of how it's done starting from the other end, although that may not be of too much interest for understanding the physics. And I also wonder how long teaching of symbolic integration in schools is going to last when you can have a calculator do it for you, whether it will go the way of forbidding calculators or learning special tricks for arithmetic.

It's anyway a skill in my experience rapidly lost. I remember that having got quite expert on it at school, less than a year afterwards a chemist asked me if I could integrate 1/√(a - x²) I think it was that can come up in chemical kinetics. I remembered I had done something like that at school, but no longer had a clue.

I have now and then uttered these subversive thoughts on this forum. I also only a few weeks ago here began to realize how we had been taught wrong, or rather that some things can be made a lot more obvious by a suitable diagram. I was beginning to induce a student to see this https://www.physicsforums.com/threads/help-me-integrate-please.886124/#post-5573560, but he let the thread drop like so many do before we got to the point.

 

[f95toli]

Maybe "many physicists" is too strong, but if say 10% of the physicists couldn't solve a partial differential equation in spherical coordinates using separation of variables, (and I suspect the 10% estimate is conservative), wouldn't that qualify as many.

But those 10% would have been able to do so at some point when they were undergraduates, or they wouldn't not have been able to pass their exams.
People forget things they don't use on a regular basis and most physicists never have to solve PDEs using analytical methods in their day-to-day work. There are plenty of experimental physicists whoNEVER have to do any calculations at all (analytical or numerical, not counting "everyday math") in their daily work.

The important thing to remember is that most of these physicists are perfectly capable of quickly picking up the necessary skills again if they encounter a situation where they need to do so.

And, as has already been pointed out. most problems are simply not possible not solve analytically which is why numerical methods are so important. This does not mean that you don't need to understand the math; QM is a good example of a very "math heavy" field which uses a LOT of algebra to formulate or understand a problem (say figure out the correct Hamiltonian) but where we very rarely use analytical methods to actually solve equations (because it is usually impossible).

 

[mpresic]

The complete solution to a partial differential equations would probably not be on most exams. Professors generally like to put more than 1 or two problems on an exam and a full solution .including the power series solution to Legendre's DE, or a solution with spherical harmonics, and the radial equation, would take too long.

As far as most students could solve them at one time. I'm not so sure. My physics classes had approximately 20 -30 students in them. I think most professors would consider their course a success if 18 - 27 students out of the 20 - 30 could fully solve a separation of variable problem in spherical coordinates , especially in a test situation.

Many times, long problems are given in homework. Some instructors count homework as a significant percentage of the grade, but not all instructors do.

Some teaching faculty (a very selective and talented group) have confided to me that they did not understand how to get the full solution to the quantum mechanical harmonic oscillator involving Hermite polynomials until they had to teach it to their students.

 

[dipole]

Being able to program well is a far more important skill for a physicist than being able to solve differential equations by hand.

 

[mpresic]

I do more programming than hand solving differential equations, but it is not an either / or thing. Knowing the characteristics of differential equations can help decide which algorithms, Ifor example in Numerical Recipes, IMSLLIB, MATLIB, etc), can be used efficiently for a solution. Physicists should study both programming and differential equations.

 

[f95toli]

As far as most students could solve them at one time. I'm not so sure. My physics classes had approximately 20 -30 students in them. I think most professors would consider their course a success if 18 - 27 students out of the 20 - 30 could fully solve a separation of variable problem in spherical coordinates , especially in a test situation.

I guess it depends on where you study then.
Separation of variables is a standard technique and doing it in spherical coordinates is not conceptually very difficult and was used in several of my math courses (PDEs, mathematical physics, Fourier analysis and probably a couple more). If you did not know how to do this you would almost certainly not pass the exam for the course (meaning you would not get your undergraduate degree).
Also, this particular problem pops up a LOT in say atomic physics; solving the SE for a hydrogen atom tends to be a standard part of any course in QM.

 

[mpresic]

Separation of Variables is not conceptually very difficult to most physics majors, but the solution of for example the Hydrogen atom, e.g the Laguerre polynomials, the spherical harmonics all that would be very tedious and long for an exam. I would be shocked if that was on quantum exam. Even if it was, there is also scattering, symmetries, linear oscillator in 1d, and 3d, perturbation methods, path integrals etc. The average QM course has a final, say 3 hr, a midterm, maybe 2 hr, homwework although sometimes the professor counts this less. etc. All told, it is probably not possible to test the students on all the course material they are supposed to know in quantum or many other fields.

I think we all have been on the lucky end and the unlucky end of exams. Sometimes everything you studied is on the exam, sometimes, almost nothing that you studied is on the exam. Most times students (and their instructors) get a feel of what problems are challenging but solvable in the time allotted for the tests.

In addition, I am quite sure you could pass a QM course without solving the H atom in depth. A QM midterm might have 6 problems on it. Most students can leave out 1 or 2. I have seen qualifying exams for graduate schools available on the internet, where the student is even instructed to clearly indicate which questions are to be graded. The questions not selected do not get credit even if the student were to complete it successfully. I know if I took the test and had any qualms on a tedious problem, I would select a problem which was shorter and I could get the same or more points with. An exam situation is not a venue for "showing off".

I do have to say between classroom tests, qualifying exams, and candidacy exams, physicists are probably the most tested professionals around. The ones that make it through this obstacle course are exceptional.

I understand these concepts are part of the undergraduate curriculum in most if not all undergraduate institutions. I often hear graduates say, Yeah we had it.
Often, this does not mean, they can do it, presently. It does not mean they were tested on it. It just means it was shown to them.