On the origins of the Schrodinger equation

Apr 08, 2013 by Lisa Zyga feature
Physicists have obtained the Schrödinger equation (shown here) from a mathematical identity. Their approach shows that the linearity of quantum mechanics is intimately connected to the strong coupling between the amplitude and phase of a quantum wave.

(Phys.org) —One of the cornerstones of quantum physics is the Schrödinger equation, which describes what a system of quantum objects such as atoms and subatomic particles will do in the future based on its current state. The classical analogies are Newton's second law and Hamiltonian mechanics, which predict what a classical system will do in the future given its current configuration. Although the Schrödinger equation was published in 1926, the authors of a new study explain that the equation's origins are still not fully appreciated by many physicists.

In a new paper published in PNAS, Wolfgang P. Schleich, et al., from institutions in Germany and the US, explain that usually reach the Schrödinger equation using a mathematical recipe. In the new study, the scientists have shown that it's possible to obtain the Schrödinger equation from a simple mathematical identity, and found that the mathematics involved may help answer some of the fundamental questions regarding this important equation.

Although much of the paper involves complex , the physicists describe the question of the Schrödinger equation's origins in a poetic way:

"The birth of the time-dependent Schrödinger equation was perhaps not unlike the birth of a river. Often, it is difficult to locate uniquely its spring despite the fact that signs may officially mark its beginning. Usually, many bubbling brooks and streams merge suddenly to form a mighty river. In the case of quantum mechanics, there are so many convincing that many of the major textbooks do not really motivate the subject [of the Schrödinger equation's origins]. Instead, they often simply postulate the classical-to-quantum rules….The reason given is that 'it works.'"

Coauthor Marlan O. Scully, a physics professor at Texas A&M University, explains how physicists may use the Schrödinger equation throughout their careers, but many still lack a deeper understanding of the equation.

"Many physicists, maybe even most physicists, do not even think about the origins of the Schrödinger equation in the same sense that Schrödinger did," Scully told Phys.org. "We are often taught (see, for example, the classic book by Leonard Schiff, 'Quantum Mechanics') that energy is to be replaced by a time derivative and that momentum is to be replaced by a spatial derivative. And if you put this into a Hamiltonian for the classical dynamics of particles, you get the Schrödinger equation. It's too bad that we don't spend more time motivating and teaching a little bit of history to our students; but we don't and, as a consequence, many students don't know about the origins."

Scully added that understanding the history of both the science and the scientists involved can help in providing a deeper appreciation of the subject. In this way, the authors of the current paper are building on Schrödinger's own revolutionary discovery.

"Schrödinger was breaking new ground and did the heroic job of getting the right equation," Scully said. "How you get the right equation, is less important than getting it. He did such a wonderful job of then deriving the hydrogen atom wave function and much more. So did he understand what he had? You bet, he was really right on target. What we are trying to do is to understand more deeply the connection between classical and quantum mechanics by looking at things from different points of view, getting his result in different ways."

As the river analogy implies, there are many different ways to obtain the Schrödinger equation, with the most prominent one having been developed by Richard Feynman in 1948. But none of these approaches provides a satisfying explanation for one of the defining features of quantum mechanics: its linearity. Unlike the classical equations, which are nonlinear, the Schrödinger equation is linear. This linearity gives quantum mechanics some of its uniquely non-classical characteristics, such as the superposition of states.

In their paper, the physicists developed a new way to obtain the Schrödinger equation starting from a mathematical identity using classical statistical mechanics based on the Hamilton-Jacobi equation. To make the transition from the nonlinear classical wave equation to the linear Schrödinger equation—that is, from classical to —the physicists made a few different choices regarding the amplitude of the wave and thereby linearized the nonlinear equation. Some of the choices resulted in a stronger coupling between the wave's amplitude and phase in comparison with the coupling in the classical equation.

"We have shown in a mathematical identity—the starting point of everything—that the choice of the coupling determines the nonlinearity or the linearity of the equation," Schleich, a physics professor at the University of Ulm, said. "In some wave equations, there is coupling between the amplitude and phase so that the phase determines the amplitude, but the amplitude does not determine the phase. In quantum mechanics, both amplitude and phase depend on each other, and this makes the quantum wave equation linear."

Because this coupling between amplitude and phase ensures the linearity of the equation, it is essentially what defines a quantum wave; for classical waves, the phase determines the amplitude but not vice versa, and so the wave equation is nonlinear.

"As we show in our paper, the Hamilton-Jacobi plus continuity logic leads to an equation which is very similar to the Schrödinger equation," Scully said. "But it's different and this difference is something that we consider important to understand. From one point of view, the extra term that comes into the nonlinear wave equation corresponding to classical physics (as opposed to the linear Schrödinger equation) shows that the classical equation is not linear and we cannot have superpositions of states. For example, we can't have right and left running waves adding to get standing waves because of this nonlinear term. It's when we have standing waves (left and right running wave solutions) that we most naturally get the eigenvalue solutions which we must, like the hydrogen atom eigenstates. So emphasizing linearity is very important."

The analysis also sheds some light on another old question regarding the Schrödinger equation, which is why does it involve an imaginary unit? In the past, physicists have debated whether the imaginary unit—which does not appear in classical equations—is a characteristic feature of quantum mechanics or whether it serves another purpose.

The results here suggest that the imaginary unit is not a characteristic quantum feature but is just a useful tool for combining two real equations into a single complex .

In the future, the physicists plan to extend their approach—which currently addresses single particles—to the phenomenon of entanglement, which involves multiple particles. They note that Schrödinger called entanglement the trait of quantum mechanics, and a better understanding of its origins could also reveal some interesting insight into the workings of the tiniest components of our world.

"We are presently looking at the problems from the point of view of current—how and to what extent can we regain quantum mechanics by relaxing the classical current idea and focus instead on a quantum-type current," Scully said. "From this perspective, we get into gauge invariance. There are lots of fun things that one can consider and we are trying to fit these together and see where each of these perspectives takes us. It is also fun to find out who has had ideas like this in the past and how all the ideas fit together to give us a deeper understanding of . If our paper stimulates interest in this problem, it will have served its purpose."

Explore further: When parallel worlds collide, quantum mechanics is born

More information: Wolfgang P. Schleich, et al. "Schrödinger equation revisited." PNAS Early Edition. DOI: 10.1073/pnas.1302475110

Related Stories

Quantum simulation of a relativistic particle

Jan 06, 2010

(PhysOrg.com) -- Researchers of the Institute for Quantum Optics and Quantum Information (IQOQI) in Innsbruck, Austria used a calcium ion to simulate a relativistic quantum particle, demonstrating a phenomenon ...

Pure mathematics behind the mechanics

Feb 07, 2008

Dutch researcher Peter Hochs has discovered that the same effects can be observed in quantum and classical mechanics, if quantisation is used.

Recommended for you

Quantum holograms as atomic scale memory keepsake

Oct 21, 2014

Russian scientists have developed a theoretical model of quantum memory for light, adapting the concept of a hologram to a quantum system. These findings from Anton Vetlugin and Ivan Sokolov from St. Petersburg ...

1980s aircraft helps quantum technology take flight

Oct 20, 2014

What does a 1980s experimental aircraft have to do with state-of-the art quantum technology? Lots, as shown by new research from the Quantum Control Laboratory at the University of Sydney, and published in Nature Physics today. ...

User comments : 27

Adjust slider to filter visible comments by rank

Display comments: newest first

Lurker2358
1 / 5 (12) Apr 08, 2013
The results here suggest that the imaginary unit is not a characteristic quantum feature but is just a useful tool for combining two real equations into a single complex equation. Unlike in the classical version, the quantum formulation involves two equations instead of one.


The problem with this conclusion is that all the terms in any physics equation are supposed to be indicative of real world relationships of all the entities which make up reality, or all the entities which make up the system you are studying.

Imaginary terms ought to signify something "real". It may be that we don't understand the "interpretation" of the meaning of those terms; as dimensions or cycles or some other entity, but they must nevertheless be "real," else they don't belong in the fundamental equations of reality. They need not be a "material" entity, but the law or relationship they represent must be "real".
ValeriaT
1.3 / 5 (12) Apr 08, 2013
IMO it's important to understand the physical meaning of quantum wave equation. The Schrodinger equation is actually the equation of quite normal transverse wave, but the wave of the strange material, which gets more dense reversibly under introduction of energy in similar way, like the soap foam gets thick under shaking. It's an equation of elastic material, the material density (blue line at this graph) of which in each time and space interval is proportional to its energy density (red line) in accordance to E=mc^2 equation. In this way the world of quantum mechanics remains connected with general relativity.
antialias_physorg
4.3 / 5 (16) Apr 08, 2013
Imaginary terms ought to signify something "real"

They do. As an example look at quaternions vs. matrix mechanics in doing geometrical manipulations or writing down electromagnetic wave components in engineering. Imaginary units are 'just' a concise way of writing something multidimensional down into a single equation. It's pretty handy for any kind of problem with more than one variable once you get the hang of it. .

In the phrase with 'imaginary unit' the author does not refer to a physics unit, here, but to a unit (i.e. a part) of the equation (which has a real part and an imaginary part).
Though that part of the equation is related to a physical unit (the phase). The wording is a bit unfortunate.
Higgsbengaliboson
1 / 5 (6) Apr 08, 2013
Schrodinger equation can be shown through 1-D and 3-D representation.Scrodinger wave equation properly depicts it's phenomena but using grad we can represent it in vector form.

Schrodinger's cat is certainly one of the most interesting characteristics in quantum physics(2012 Nobel was awarded based on this phenomena)but I think these equations represent mostly mathematical operation than physical phenomena(it's origin can be understood from mathematical representation).
tadchem
4.7 / 5 (13) Apr 08, 2013
In studying chemical physics I came to recognize the fact that the equation is simply the statement of the Law of Conservation of Energy expressed in the mathematics of wave mechanics: Kinetic energy plus Potential energy = Total energy.
I learned much when I realized that the equation admits both steady-state and periodic (time-dependent) solutions.
I learned even more when I re-expressed it in the mathematics of four-tensors in Minkowski Space. The photon becomes an elementary four-tensor in motion which collapses into either a wave or a particle representation depending on the operator required to tease an observable parameter from the solution.
GuruShabu
1 / 5 (13) Apr 08, 2013
The next revolution in theoretical physics is unstoppable; its time has come, and this paper is indicative of it. Thank you for the link. Having said that, people are ultimately going to find the revolution a somewhat disappointing one: At the moment we take one of a few equations (with empirical parameters such as h,G etc), add initial conditions and boundary conditions, make a prediction and compare with experiment.
Gawad
4.4 / 5 (15) Apr 08, 2013
The wording is a bit unfortunate.


That's putting it mildly.

To QC and anybody else who has this misconception: there isn't anything more innately unphysical about that "i" than mathematical operations on a plane vs. those on a number line. The original naming oddity came from those who came up with the appellation "imaginary" (in the 16th century IIRC) because in their minds these operations stood in contrast with those performed on the Reals (i.e., operations performed on the "Real numbers" on the Real number line). "Well, if they're not 'real' then they must be..." Get it? Nothing more. And no, I'm NOT kidding. And there's nothing more "physical" about 1/0 (Reals) than there is about the square root of -1 (imaginary). So yes, as A_A says, the term is "unfortunate" as it causes unimaginable, ahem, confusion with the idea of imaginary equaling "unphysical". But in the end, it all only depends on why and how you're using complex numbers, really (sorry).
GuruShabu
1 / 5 (13) Apr 08, 2013
In the future we will take a single equation without parameters and no boundary conditions (since the equation has no boundary!), add initial conditions, make predictions and compare to experiment. Most people will neither care about nor realise the importance of this change, and will still be left wondering the imponderable why? of it. The best answer I can give is quite unsatisfactory: "it is what it is, and always has been, because it is the only way it can be" which is a subtly different answer to the equally unsatisfactory multi-verse interpretation "all universes are possible and we live in the one we do, because we do", or the subtly different religious one ('it was called into existence by the one"). The question in fact might well never be empirically answered, even in principle (assuming no mistake has been made in Goedel's incompleteness theorem) so I have decided to ponder other things instead and remove "why" from my vocabulary.
GuruShabu
1 / 5 (13) Apr 08, 2013
I do know that the practical consequences of the unified physics equation are deeply rooted in the mathematics of number theory and transfinite numbers (Cantor) . In fact the reason mass quanta are free to move at all and not completely "fill" space is because mass is quantised ; (infinite with cardinality on the set of natural numbers), while space is a continuum (infinite cardinal on the complex number set), and the prior infinity is smaller than the latter.

The existence of an effectively random variable expressing quantum probability is also in fact an emergent property of number theory and the unified equation, emerging directly from uncertainty about future states due to chaos theory (the transfinite relationships between rational, irrational and transcendental numbers.)
GuruShabu
1 / 5 (12) Apr 08, 2013
This fundamentally arises because an ellipse is only an approximation to the state transitions of position that can occur with time (for example in the position of one planet orbiting another): the assumption of an elliptical solution explicitly forces the end state and starting state to be identical, but in fact tracking the infinity of state transitions along the orbit simply does not allow this assumption to be made in practice. The practical consequence is that no two orbits are identical, and over time, all possible equal energy orbits will be enacted according to chaos theory ; the system energy being the attractor point (because it is the "conserved" value in the chaotic system). In an atom, averaged over time this results in a "probability of position" (of the randomised constant energy orbit of the constituent quanta over time) - which we observe empirically as the quantum wave-function.
GuruShabu
1 / 5 (12) Apr 08, 2013
In fact if we left the mass in a planet to its own devices (that is switched off the electromagnetic force holding it together), the position of the mass quanta over time would be indistinguishable from such a quantum probability wave-function (at least according to Quirk theory) - the mass quanta of the planet want to follow all possible constant energy trajectories, but in fact cannot because they are bound by the electromagnetic force which constrains them to follow only one at a time (the classical orbit); But note well that even this classical orbit remains uncertain by exactly the same underlying number theory (the chaos of sensitivity to initial conditions which derives from the state transitions allowed by transfinite number theory).
To describe mathematics as a divine topic of study is not, I think, to overstate the case.
GuruShabu
1 / 5 (12) Apr 08, 2013
As to why, I find comfort in the art of prose:
"Such a sweet and grand illusion playing itself out in front of our eyes, not intended to deceive but rather to simply reflect a reality that cannot help but be what it is" (AJ)
Chasing Odysseus: "Was it not the immortal gods who spun catastrophe into a thread of events to make a song for generations not yet born?" (Sulari Gentill) - and my personal favourite
"The Multiverse Theory: Somewhere within the quantum foam of existence, amongst the very building blocks of reality there is a universe where you… are Batman." (Anon)
"Its a scene repeated a million times a day in the multiverse. Both would be fighters growled & grimaced at each other and fought to escape the restraint of their friends, only not too hard, because there is nothing worse than actually succeeding in breaking free and finding yourself all alone in the middle of the ring with a madman who is about to hit you between the eyes with a rock." (Terry Pratchett)
TopherTO
4 / 5 (9) Apr 08, 2013
Jeebus...

Guru please elaborate, we need further details
vacuum-mechanics
1 / 5 (14) Apr 08, 2013
In a new paper published in PNAS, Wolfgang P. Schleich, et al., from institutions in Germany and the US, explain that physicists usually reach the Schrödinger equation using a mathematical recipe. In the new study, the scientists have shown that it's possible to obtain the Schrödinger equation from a simple mathematical identity, and found that the mathematics involved may help answer some of the fundamental questions regarding this important equation.

We know that Schrödinger equation is a mathematical wave equation, but the problem is wave of what or what is wavy? Maybe this simple physical mechanism could help us to visualize it.
http://www.vacuum...19〈=en
robeph
4 / 5 (4) Apr 08, 2013
Jeebus...

Guru please elaborate, we need further details


Please don't ask him to do that.

...And why is it that the screw balls can be identified with almost 100% accuracy simply by the fact they post multiple volume comments rather than the normally singular of the regular folk.
rwinners
3 / 5 (6) Apr 08, 2013
I'm dead or I ain't. Please DON'T open that box!
christophe_galland1
4 / 5 (4) Apr 08, 2013
The problem with this conclusion is that all the terms in any physics equation are supposed to be indicative of real world relationships of all the entities which make up reality, or all the entities which make up the system you are studying.

Imaginary terms ought to signify something "real". It may be that we don't understand the "interpretation" of the meaning of those terms; as dimensions or cycles or some other entity, but they must nevertheless be "real," else they don't belong in the fundamental equations of reality. They need not be a "material" entity, but the law or relationship they represent must be "real".


In this case "imaginary" as nothing to do with "unreal". Multiplying by the imaginary unit is nothing but a 90 degree change in the phase of a wave. When "i" is in an exponential function it is just a shortcut for a sum of cosine and sine (the quadratures of the wave). Nothing unreal in this.
Phil DePayne
1 / 5 (8) Apr 08, 2013
The imaginary unit has always been used to express two scalar dimensions. Why does there have to be a brain trust working on this?
angelhkrillin
1 / 5 (4) Apr 08, 2013
I also agree her opinion is not in par with the "real world" instead of wanting to turn away from classical mechanics to allow more of a quantum feel, we should be looking at the bigger picture of both quantum and classical mechanics. I have some theories as well hopefully someday I can understand them fully enough to represent them on paper as Schrodinger did.
Mike_Massen
1.7 / 5 (6) Apr 09, 2013
Lurker2358 postulated
Imaginary terms ought to signify something "real".
This is a nice assumption but its *only* that. The closest analogy I can offer Eg. (so far) is bipedal motion, ie. We pass through an unstable state whilst lifting one leg with a probabilistic aspect we *will* achieve stability when the foot later on reaches the ground. The unstable state may be inferred as 'unreal/imaginary' & the stable as 'real' but both are obviously necessary in *that* sequence.

Lurker2358 went on with
They need not be a "material" entity, but the law or relationship they represent must be "real".
Again, nice idea but based on an assumption probably ingrained from a close observation of causality where we are so uncomfortable with so called 'unreal' un-testable causal factors. Bear in mind:-
Maths describes the world, its does *not* explain it.
Hence any linguistic aspect (tied up with math) need not (at all) have any 'real' property as we are only ever at the descriptive stage !
Disproselyte
1.3 / 5 (7) Apr 09, 2013
Schrödinger's equation is Newton's equation of dynamics, but expressed in a non-differentiable space-time: under specific conditions the one can be transformed in the other, see http://arena.obsp...EJTP.pdf §5.7.
Higgsbengaliboson
1 / 5 (4) Apr 09, 2013
One final comment:Schrodinger actually derived the equation based on quantum mechanical nature of wave.Now if anybody know the characteristic equation of classical wave(i.e both transverse and longitudinal) then it's very easy to convert that equation in quantum mechanical form.For example amplitude of classical wave equation A can be represented using quantum form of ih- and so on the other terms...
swordsman
1.2 / 5 (6) Apr 09, 2013
If you rewrite the wave equation as a vector equation, it appears similar to the Schrodinger equation in form with the imaginary term.
jdbertron
2.1 / 5 (7) Apr 09, 2013
Of course, it's behind a paywall, because we, citizens who pay for this research don't deserve to get the results.
ValeriaT
1.7 / 5 (12) Apr 09, 2013
With respect to the fact, that A) the double slit experiment is completely described with Schrodinger equation B) the same experiment can be faithfully modeled with droplets jumping above water surface, then it's evident, that Schrodinger equation models the undulating water surface too (just with different constants). It's because the speed of ripples at the water surface is similar function of its frequency and amplitude, like at the case of vacuum. The undulating water surface is deformed and due its higher specific area it slows down the spreading of another water ripples accordingly. Actually the Schrodinger equation is often used for modeling of ocean freaky waves, which have solitoni character too. Therefore the Schrodinger equation is the equation describing quite normal elastic environment with the only exception: this environment enables the spreading of transverse waves only in similar way, like the soap foam.
beleg
1 / 5 (6) Apr 12, 2013
Perhaps the origin is not so much at stake. The incentives for motivation - how this can aid our progress and thinking - is just as important.
The wheel has no known origin. Universal application is important.
flashgordon
1 / 5 (2) Apr 13, 2013
Crease and Mann stress that Shroedinger and those who did the original quantum mechanics considered quantum mechanics coming out of classical mechanics as essential in their "The Second Creation." Although, they don't get into technical details.