"Reading" Schrodinger's equation

[Vinay080]

I am starting to learn Quantum mechanics. I can't wait for my completion of QM, as I am running behind all the concepts taught in the class; but I can't even go on studying chemistry, or I can't even analyse anything, without understanding the atoms in reality. I believe in (Russell's??) principle of reading equations by converting math-symbolic statements into english statements, and also in Feynman's principle of reading equations by comparing with reality. To be clear on what help I want in Schrodinger's equation, read these statements by Feynman (you may also enjoy reading Russell if interested):

..I have the specific, physical example of what he's trying to analyze [from math equation], and I know from instinct and experience the properties of the thing...

So, I want to understand atoms, there reality, for that I want to read this (Schrodinger's) equation by comparing with physical real atom and knowing each term with the reality, viz. converting the equation completely into plain english without any technical words...
$$i \hbar \frac{\partial}{\partial t}\psi(\mathbf{r},t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V\psi$$

I will be happy if you all can also suggest me books or papers with respect to this matter.

I have already asked this question in other physics website, I am asking this here also, to get more help.

 

[Simon Bridge]

You do not have to understand a thing to study it - if that were the case, science would be unable to make any progress.
There are plenty of works on "the quantum mechanics of atoms and molecules" - use those as search terms.

Note: "Russell's method" (putting words in place of symbols in equations) is a crutch for beginners - you should cultivate mathematics as a language in it's own right.

 

[Vinay080]

You do not have to understand a thing to study it - if that were the case, science would be unable to make any progress.

What I meant is to analyse anything, we need to understand the reality, for that I need to understand how reality is concieved using QM; Otherwise we will be able to study, but it won't be a perfect study... But, don't you think we can make a lot of progress if things are really understood than if it is not understood?

There are plenty of works on "the quantum mechanics of atoms and molecules" - use those as search terms.

Thank you. I will search for that...

But, can you guys help me by just explaining how the picture of atom is concieved mathematically? I want just that to continue my studies, for it helps me to be satisfied on analysing things in a real fashion...

Note: "Russell's method" (putting words in place of symbols in equations) is a crutch for beginners - you should cultivate mathematics as a language in it's own right.

But, mathematics is just like a shorthand, they are actually words, sounds to communicate, but written in mnemonic style. Don't you think his method should be cultivated, to understand mathematics in it's own right? By "converting into words", I mean into words they critically mean (the primitive atoms/elements of sound, viz. dt may mean infinitesimal quantity of time) and not in sloppy language...

 

[blue_leaf77]

But, can you guys help me by just explaining how the picture of atom is concieved mathematically?

The related textbooks will explain this for you, ... in tens of pages at least. So it's very likely not our capability to give a satisfying answer in textbox of a forum. My favorite textbook on atoms is "Physics of Atoms and Molecules" by Bransden and Joachain.

 

[Vinay080]

@blue_leaf77: Thank you for the reply, I accept you. I searched many places, like wiki, and works suggested by @Simon Bridge , I am getting fair idea now, but I will be happy if you all can share on the easiest path of understanding the mathematical picture, either by sharing best (mathematical) books or experience, like you did...

 

[bhobba]

What I meant is to analyse anything, we need to understand the reality, for that I need to understand how reality is concieved using QM; Otherwise we will be able to study, but it won't be a perfect study... But, don't you think we can make a lot of progress if things are really understood than if it is not understood?

There is a very deep reason for Schroedinger's equation grounded in 'reality'. Its that its implied by the principles of QM and the principle of relativity (ie the laws of physics are the same in inertial frames - which here means regardless of how fast you are travelling, where you are, or when it is, the probabilities predicted by QM are the same. If that wasn't true it would be very strange - very strange indeed - so strange most would doubt it). You will find the detail in Ballentine - Quantum Mechanics - A Modern Development - Chapter 3. Unfortunately the math is advanced and not suitable at the beginning level. Still it's will be worth your while going to a library and giving it a read. You will likely not understand the detail but hopefully will get the gist. You can return to it later when your math is more advanced.

The next question probably is - why those principles of QM? They are detailed in two axioms Ballentine gives - but why those axioms? In a certain sense that has no answer - science always accepts some things. But in this case they have a deeper reason to do with generalised probability models:
http://arxiv.org/pdf/quant-ph/0101012.pdf

At rock bottom QM is the way it is because nature is fundamentally probabilistic and physical systems are at an intuitive level continuous - by which is meant if a system goes from one state to another in a second it goes through another in half a second.

Note - this is the formalism of QM - what it means is another matter and a whole different ball game that people argue about all the time. But virtually everyone agrees on the formalism and the above is the rock bottom essence of its 'why'.

Thanks
Bill

 

[vanhees71]

I'd rather recommend to read the Prologue in

Schwinger, Quantum Mechanics, Symbolism of Atomic Measurements, Springer

This is really great to get the big view before going into the details of the formalism.

 

[Simon Bridge]

What I meant is to analyse anything, we need to understand the reality, for that I need to understand how reality is concieved using QM; Otherwise we will be able to study, but it won't be a perfect study... But, don't you think we can make a lot of progress if things are really understood than if it is not understood?

If things are "really understood", then no progress can or need be made.
There is no such thing as a perfect study.
You need to be able to analyse reality before you understand it... otherwise, how can you learn what it is?
QM is a toolkit for constructing models of Nature, these models are not the reality... but it does help to understand QM if you see an example of how it is used.

But, mathematics is just like a shorthand, they are actually words, sounds to communicate, but written in mnemonic style. Don't you think his method should be cultivated, to understand mathematics in it's own right? By "converting into words", I mean into words they critically mean (the primitive atoms/elements of sound, viz. dt may mean infinitesimal quantity of time) and not in sloppy language...

No. Mathematics is the language of physics... the symbols are not "actually words" just like "fiziko" isn't "actually" "physics". The same concept may be expressed in different languages... this does not mean that one language is actually some shorthand for another one.

Learning to treat math as a language is a major breakthrough for students. While you are starting out, just bear that in mind.

 

[Vinay080]

If things are "really understood", then no progress can or need be made.
There is no such thing as a perfect study.
You need to be able to analyse reality before you understand it... otherwise, how can you learn what it is?
QM is a toolkit for constructing models of Nature, these models are not the reality... but it does help to understand QM if you see an example of how it is used

I agree with you in the context you said.

... the symbols are not "actually words" just like "fiziko" isn't "actually" "physics". The same concept may be expressed in different languages... this does not mean that one language is actually some shorthand for another one.

Let me be clear, here are the words of Thomas (translator) in his preface to the book of Lagrange's "Lectures on Elementary mathematics":

..For the development of science all such short-mind symbols are of paramoundt importance, and seem to carry within themselves the germ of a perpetual mental motion whihc needs no outward power for its unfoldment...But it should never be forgotten that the mighty stenophrenic engine of which we here speak, like all machinery, affords rather a mastery over nature than an insight into it; and for some, unfortunately, the higher symbols of mathematics are merely brambles that hide the living springs of reality...

This is what I meant; math-symbols should not become technology as "Keith Delvin" called calculus symbols as understood now; we should be like Leibniz, in his own words (extracted from the book The Calculus Gallery), "..[ I was] ready to get along without help, for I read [mathematics] almost as one reads tales of romance.."; we should be like Lagrange who had insights on symbols. This is what I meant to say, we should understand the meaning of symbols, which can be reproduced in words...Why I said all these is, I don't want folks to learn math without understanding its meaning..i.e without understanding the meaning words of symbol.

Learning to treat math as a language is a major breakthrough for students. While you are starting out, just bear that in mind.

I agree with you, we should treat math as a language; but we should also have insight on what its symbols mean...which can actually be reproduced in words like oure folks of antiquity who wrote equations in words, or we itself used to write in our childhood "word problems"...this is what I expected from you all before to help me "read" the equation (if Schrodinger) which describes the atom..

 

[andrewkirk]

Just looking at the very specific example in the OP:

converting the equation completely into plain english without any technical words...
$$i \hbar \frac{\partial}{\partial t}\psi(\mathbf{r},t) = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V\psi$$

You might have more luck with the following more general, and concise, version of the equation:

$$\frac{d}{dt}|\psi\rangle=\frac{1}{i\hbar}H|\psi\rangle$$

Verbal version: 'the time rate of change of the wavefunction is the same as what you get when you apply the Hamiltonian operator to the wavefunction and then divide by ##i\hbar##. Scaling aside, the Hamiltonian operator maps the wavefunction to its time derivative.

That contains technical words though. I am confident the task is impossible without technical words.

 

[bhobba]

This is what I meant; math-symbols should not become technology as "Keith Delvin" called calculus symbols as understood now; we should be like Leibniz, in his own words (extracted from the book The Calculus Gallery), "..[ I was] ready to get along without help, for I read [mathematics] almost as one reads tales of romance.."; we should be like Lagrange who had insights on symbols. This is what I meant to say, we should understand the meaning of symbols, which can be reproduced in words...Why I said all these is, I don't want folks to learn math without understanding its meaning..i.e without understanding the meaning words of symbol..

A physicist acquaintance said to the great mathematician and polymath, Von Neumann - 'I'm afraid I don't understand the method of characteristics.' His reply was 'Young man, in mathematics you don't understand things. You just get used to them.'

The view of mathematics you are elucidating is rather old fashioned because of the rise of pure mathematics as a discipline distinct from applied math. What that has taught us is that if you want exactitude then mathematics becomes very formal and abstract and not concrete. If you want applied math then one starts out with terms that are vague and via experience what they mean becomes concerete. For example in probability you have the Kolmogorov axioms where event is used. We all have a bit of an intuitive idea what event is, but its true meaning is only grasped when it is seen how it is applied. In QM observation is often used. We all have an intuitive idea what an observation is, but what it means in QM is only grasped with experience. Mathematics is a language, just like English is a language. One does not fully grasp words by looking them up in a dictionary - you have to see how they are used in practice.

As you probably have guessed I fully agree with what Simon said.

Thanks
Bill

 

[Vinay080]

@bhobba:

A physicist acquaintance said to the great mathematician and polymath, Von Neumann - 'I'm afraid I don't understand the method of characteristics.' His reply was 'Young man, in mathematics you don't understand things. You just get used to them.'

I have got used to differentiate and integrate, without still understanding infinitesimals. Now, I have learned to see the log table and find answer, I have got used with it, but I don't understand how it works. It is something like technology to me, which I have got used to, but don't understand it... This is a irritating thing which I don't want to happen ahead, now I am trying to understand.

Getting used with anything after understanding it, is okay; but getting used to it without understanding it, is not okay; I don't agree with Neuman statements, as there is no reasoning as I can understand, I follow Feynman (extracted from wikiquote):

Doubting the great Descartes … was a reaction I learned from my father: Have no respect whatsoever for authority; forget who said it and instead look what he starts with, where he ends up, and ask yourself, "Is it reasonable?"

I haven't still experienced the complete math, the pure and applied, and the probability concept axioms, I look forward to see them...

 

[vanhees71]

This is also a wrong approach, particularly as a physicist. You must understand the meaning of the operations you use. Differentiation and integration have very clear meanings, and that's important for the use in the natural sciences, where you apply them within models that describe (certain well defined aspects of) nature.

For mathematicians and often also scientists this is, however, still not enough of understanding math. To a certain extent you should also be aware of possible problems of naive or intuitive definitions of mathematics. For quantum theory you need quite a bit of abstract math, i.e., (rigged) Hilbert spaces, vector calculus, and partial differential equations.

From the physics side, you should have a very solid understanding of analytical mechanics. Before that, I cannot recommend to start learning quantum theory.

 

[bhobba]

For mathematicians and often also scientists this is, however, still not enough of understanding math.

Just on that point Feller gives a more detailed discussion in his classic book on probability:
https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20

Thanks
Bill

 

[ReiBaretti]

The time independent Schrodinger equation is basically a statement of energy conservation.
In classical Mechanics E = p^2/(2m ) + V(x) . From here you go to the hamiltonian operator letting p = - i h'd/dx
and we transform the classical statement into
H = - h'^2/(2m) d^2/dx^2 + V(x) .
Given that V(x) is known and the particle is constrained between 0 and L the DE
H Psi(x) = E Psi(x) has Psi(x) as unknown and also E . It is to be solved obeying boundary conditions which may be psi(0) = psi(L) =0.
The most elementary visualization is afforded by a vibrating spring with both ends fixed.

What results is a set of E(n) and Psi(n,x) , the eigenvalues and eigenvectors.( functions)

The rhs of the equation has two terms ~ kinetic energy * (probabilty/length)^(1/2) + potential energy * (probabilty/length)^(1/2) at point x.
The lhs has dimensions Total Energy * probabilty/length)^(1/2) .at point x .
So the energies are not localized but spread out over the x-axis . This is consistent with the quantum fact that we can not ascribe simultaneously an exact position and velocity to a particle.

 

[rrogers]

You do not have to understand a thing to study it - if that were the case, science would be unable to make any progress.
There are plenty of works on "the quantum mechanics of atoms and molecules" - use those as search terms.

Note: "Russell's method" (putting words in place of symbols in equations) is a crutch for beginners - you should cultivate mathematics as a language in it's own right.

I must disagree a little here; although the opinion is popular. Mathematical formulas and procedures are also just symbols like words. The difference is that the mathematical symbols have a rigid structure. To presume that our present mathematical language is the final word (or are symbols for "reality") is presumptuous. The advice is good though; our present mathematical systems vastly surpass what our normal intuitions. OTOH: learning the mathematics can lead to a more sophisticated (realistic) intuition. Look at people like Feynman, Poincare, and others. My point is that the OP must learn to read the equation (using crutches along the way) and use that exercise to expand (and limit) his intuition; but not take a particular expression (in mathematical symbols) as the end. One time I took a long and tortuous derivation of Schrodinger's equation from Dirac's equation in Maple. And then realized that it had already been done by establishing the correspondence in the book. But we are where we are in understanding and students must walk the road from baseball to QM steadily one step at a time.

 

[Xu Shuang]

I must disagree a little here; although the opinion is popular. Mathematical formulas and procedures are also just symbols like words. The difference is that the mathematical symbols have a rigid structure. To presume that our present mathematical language is the final word (or are symbols for "reality") is presumptuous. The advice is good though; our present mathematical systems vastly surpass what our normal intuitions. OTOH: learning the mathematics can lead to a more sophisticated (realistic) intuition. Look at people like Feynman, Poincare, and others. My point is that the OP must learn to read the equation (using crutches along the way) and use that exercise to expand (and limit) his intuition; but not take a particular expression (in mathematical symbols) as the end. One time I took a long and tortuous derivation of Schrodinger's equation from Dirac's equation in Maple. And then realized that it had already been done by establishing the correspondence in the book. But we are where we are in understanding and students must walk the road from baseball to QM steadily one step at a time.

I think mathematics is more fundamental than understanding, since our thinking and consciousness exist only as mathematical structures in our brain. Basically, anything we think and feels are no more physical than mathematics. We really don't know if the universe is actually anything more than mathematics.

 

[andrewkirk]

We really don't know if the universe is actually anything more than mathematics.

Max Tegmark's Mathematical Universe

 

[rrogers]

I think mathematics is more fundamental than understanding, since our thinking and consciousness exist only as mathematical structures in our brain. Basically, anything we think and feels are no more physical than mathematics. We really don't know if the universe is actually anything more than mathematics.

Basically I agree with you and have a Platonic view about the objects we study in the field we call mathematics. But I disagree in the sense that I don't think that what we call mathematics is the final word. I view it as a exploratory and constructive tool; limited by the ideas of particular times. Take calculus: Archimedes could calculate, in theory, areas by geometrical limits without formalizing limits or the ideas non-standard analysis. As did Newton but in a much more useful manner. So I view (intuitively) mathematics as an explorer does a map or a builder his tools. In mathematics we emphasize logical correctness; but know the standard for that changes over time and culture. An explorer really wants an accurate map but may discover things of interest while following it. A builder wants reliable tools even though the plans might change and require different tools. So I view it both as a description and a tool; but the internal object of study (versus application) is separate from the map or tool used.
Back to the question: an explorer must learn to read a map to follow a path and a builder how to use the tools available. The Schrodinger equation can be considered as a formula involving complex variables but also has an interpretation in terms of operators. Reading it as a string of symbols like one could for the language APL tells nothing without the internal vision to "see" the relationship it is referring to; that is called intuition.
As far as my philosophy goes: it's just an opinion or observation. I have no "answers" and try to take a lesson from Pythagoras who had a concrete idea of perfection/reality; that was wrong. Yes all that we perceive is a reconstruction or construction of our nervous system but I personally don't have the intuition necessary to "understand" that.

 

[bjbbshaw]

I strongly recommend the book "The Meaning of Quantum Theory" by Jim Baggott, Oxford University Press, 1992.
I think you will find something pretty close to what you're looking for on pages 21, 22 and the following discussion. Baggott shows how Schrodinger probably developed the equation and what he did with it afterward. It's the right level for you and it's specifically written for students of chemistry and physics.

 

[Simon Bridge]

It is generally good advice for students of physics, at this level (quantum mechanics is usually year 2 college), to treat mathematics as a language.
We'll often see, "mathematics is the language of physics".
http://wordpress.mrreid.org/2013/07/22/mathematics-as-the-language-of-physics/
http://www.americanscientist.org/bookshelf/pub/the-language-of-physics
... just 2 examples.

When students are just starting out, it is often useful to get them to translate the math symbols into words - to help them grasp what the simpler equations and formulae are telling them. This is a crutch which leads to the kinds of questions here where we are asked for non-maths answers about phenomena best described by maths.

If students want to ponder platonic forms or "will the real realty please stand up", they are welcome to pursue a course in philosophy.

 

[pellis]

The OP, https://www.physicsforums.com/members/vinay080.549419/, might also find it useful, for additional insight into the Schrodinger equation, to read parts of Peter Atkins' Galileo's Finger, especially Ch 7, Quanta.
[ http://www.amazon.com/dp/0198609418/?tag=pfamazon01-20 ] .

While I agree with most of the expert advice to treat the subject as primarily mathematical, my own experience as an ex-quantum-chemist (though not an outstanding one) is that the balance between thinking about maths (and therefore also modern physics) in images, words and algebraic abstractions, varies between individuals, as investigated originally, in the 1930s,* by the mathematician Jacques Hadamard. [ Hadamard, J., The Psychology of Invention in the Mathematical Field, Dover Edition (1954), pp 142-143. ]

It could be a mistake to dismiss alternative modes of apprehension as even Einstein, referring to "elements of thought" in his answer to Hadamard's survey of the top mathematicians and physicists of the day, replied:

“The above mentioned elements are, in my case, of visual and some of muscular type”.

Perhaps such a kinesthetic approach to (the “modelling” of?) thought is natural when trying to conceive the curved space of General Relativity.

However, Hadamard reported that the majority of his survey group thought in images (which I interpret/paraphrase as 'geometrically') - often dots and lines - while many also thought mainly in abstract symbols (...'algebraically'). It is notable, especially for Vinay080 however, that Hadamard reported only one respondent replying that he thought primarily in words! So, I would agree also with those who suggest that the verbal/textual approach should be treated as a pedagogical aid rather than an endpoint.

[Above: including extracts of my letter "Einstein's Muscular Thinking" published in New Scientist, April or May 2010]

* and by others since then, e.g at U. Birminham, UK in the 1990s, if memory serves.

 

[Mark Harder]

While I wouldn't want to discourage you from trying to find a physical picture of quantum reality, esp. one described by the SE, outside rather than instead of your coursework, I think it would be a good idea for a beginner, esp. in a quantum CHEMISTRY course to accept the SE as a description of reality in terms of waves. There's some concrete evidence for wave-particle duality and other quantum notions. One course of study I undertook explained the equation by analogy with the classical wave equation. Also, there is physical evidence that electrons (& other particles?) behave as waves - the observation that electrons diffract from crystals just like EM waves do, for instance. Other propositions are: The wave function solution of an appropriate SE equation is a description of the system under study and multiplication of a wave function and its complex conjugate yields the probability that the quantum system exists in the state described by that particular wavefunction. BTW, be grateful that you are not required to understand QM in its linear algebraic (matrix) formulation. My own attempts to understand the matrix formulation lead to much more head scratching.

The problem that we, and that includes physicists, have in making an intuitive picture of quantum physics is that our intuition is based on our experience and our experience of reality is limited to macroscopic, i.e. classical, systems. Attempts to arrive at physically intuitive pictures of quantum reality have given rise to a bunch of "schools" of quantum philosophy. My personal inclination is to ignore these claims until I attain fluency with the mathematical language of QM, which I haven't done to my own satisfaction yet. The popular literature likes to describe these interpretations of quantum reality because, in my judgement, those descriptions attract readership precisely because they are so weird.

 

[Mark Harder]

PS. to my previous post: Some years ago, I read "The Strange Story of the Quantum". I remember liking it. Perhaps it will help you understand QM at elementary level. I don't know what to think of Leonard Sussman's Theoretical Minimum series yet, but it includes a volume on QM. He begins by describing quantum "systems" , which are both abstract and sort of concrete (at least that's how I see it) at the same time. His classical mechanics book also introduces this conception of physical reality in terms of "systems". While I haven't worked my way to the end of either of these books, I like his approach thus far.

Edit: "The strange ..." can be found on http://www.bookyards.com/ in .pdf format free of charge. The book was published a long time ago and site doesn't behave like a pirate site, but caveat emptor.