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.

Although much of the paper involves complex mathematical equations, 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 experimental results 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 quantum physics—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 equation.

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 quantum mechanics. If our paper stimulates interest in this problem, it will have served its purpose."

**Explore further:**
Beam me up ... Quantum teleporter breakthrough

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

## Lurker2358

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

## antialias_physorg

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

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

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

## Gawad

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

## GuruShabu

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

## GuruShabu

To describe mathematics as a divine topic of study is not, I think, to overstate the case.

## GuruShabu

"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

Guru please elaborate, we need further details

## vacuum-mechanics

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

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

## christophe_galland1

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

## angelhkrillin

## Mike_Massen

Lurker2358 went on with 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

## Higgsbengaliboson

## jdbertron

## ValeriaT

## flashgordon