Musings on Schrödinger's equation

[MauroLanzini]

Hello. I'm not sure I'm posting in the right forums, please excuse me in case. I may also be saying something very silly, my math is not that bad but it's quite below what is needed to grab quantum theory, so please excuse me again in case.

 

This is a form of Schrödinger's equation:

 

i * dΨ/dt = H * Ψ

 

In my understanding the imaginary 'i' is what allows a first-order differential equation to have periodic, 'wave' solutions, while the same equation with strictly real coefficients only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor).

 

What about writing Schrödinger's equation this way:

 

e^Ki * dΨ/dt = H * Ψ

 

Now if K = π/2 then e^Ki = i and we have the normal Schrödinger's. But if K = 0 then e^Ki = 1, there are no more imaginary coefficients and we have an equation where the solutions (after some time constants) will set to a well-definite value (of course many different functions could be used in place of e^Ki, just it looked the simplest one).

 

K could vary with time, and could be made (inversely) proportional to some measure of the interactions with the sorrounding environment, and so it would (possibly ) bridge neatly the gap from a fully-quantum to a fully-classical description of a system.

 

My math skills stop about here, so no idea on how to elaborate on that. But I thank whoever will let me know if this looks interesting, or why it's utterly silly (or, more probably yet, who already thought the same and why it did not work). Thanks for reading.

Edited January 30, 2014 by MauroLanzini

[imatfaal]

Hello. I'm not sure I'm posting in the right forums, please excuse me in case. I may also be saying something very silly, my math is not that bad but it's quite below what is needed to grab quantum theory, so please excuse me again in case.

 

This is a form of Schrödinger's equation:

 

i * dΨ/dt = H * Ψ

 

In my understanding the imaginary 'i' is what allows a first-order differential equation to have periodic, 'wave' solutions, while the same equation with strictly real coefficients only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor).

 

What about writing Schrödinger's equation this way:

 

e^Ki * dΨ/dt = H * Ψ

 

Now if K = 1 then e^Ki = i and we have the normal Schrödinger's. But if K = 0 then e^Ki = 1, there are no more imaginary coefficients and we have an equation where the solutions (after some time constants) will set to a well-definite value (of course many different functions could be used in place of e^Ki, just it looked the simplest one).

 

K could vary with time, and could be made (inversely) proportional to some measure of the interactions with the sorrounding environment, and so it would (possibly ) bridge neatly the gap from a fully-quantum to a fully-classical description of a system.

 

My math skills stop about here, so no idea on how to elaborate on that. But I thank whoever will let me know if this looks interesting, or why it's utterly silly (or, more probably yet, who already thought the same and why it did not work). Thanks for reading.

 

Sorry " Now if K = 1 then e^Ki = i and we have the normal Schrödinger's."

 

The maths here is wrong e^(ki) = cos(k) +i sin(k) setting k =1 does not make cos(k) +i sin(k) = i . To get that equation to equal i - you need to set k to equal pi/2

[MauroLanzini]

Yes imatfaal you are reason of course, thank you (I amended my post). The basic idea stays the same though (just a matter of constants), and actually many functions which give as a result i at one extreme and 1 at the other one could be used instead of e^Ki

[studiot]

Yup imatfaal pretty well summed it up, but do you also understand the significance of the RHS?

 

That is the significance of H?

[MauroLanzini]

Hello studiot, well as I said my maths doesn't go too far, so I have an idea of what an Hamiltonian (or a Lagrangian) are, I know they represent the total energy of the system and in the case of the Lagrangian I understand how it can be used to derive the laws of motion, but I don't go much further (I have a degree in electronics egineering, that's where my math comes from).

 

But of course multiplying by e^Ki on the left side is the same as multiplying the H by e^-Ki _... is this what you wanted to say?

[imatfaal]

and to be uber-picky you are missing an h-bar from the LHS. i h-bar - crops up everywhere quantum

 

You really cannot cope with Schroedinger without a good grounding in maths and physics. The Hamiltonian Operator is a description of the total energy - but you really need to understand the concept of a quantum mechanical operator for that to make any sense.

 

I think maybe what you are getting at - and you have gone a bit too far trying to change Schroedinger - is Euler's Equation, which I hinted at above

 

[latex]e^{ix} = cosx + i sin x[/latex]

 

And in the above the addition of the imaginary number does change the shape of the curve and create the periodicity. You do realise that Schroedinger's has Psi which is the wave function of the quantum system.

 

I would recommend a good dose of Hyperphysics - on subjects like this wikipedia is too dense and written by too many people who want to show off their insight and knowledge. although the operator page on wiki seems very good

https://en.wikipedia.org/wiki/Operator_%28physics%29

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/schr.html#c1

[MauroLanzini]

Hehe imatfaal, thanks a lot.

 

Yes I knew I was missing an h-bar (a character hard to find on the keyboard) I lumped it together with the H.

 

The whole basic idea of changing 'i' to 'e^Ki' comes of course from Euler's equation (that one at least is inside my math skills), as you guessed. Schrödinger's equation stays the same but as a special case of a generalized equation where the 'quantumness' (that is to say the oscillations, that is to say the imaginary coefficient), can be tuned at will. This if I'm not badly wrong of course, but both 'i' and 'e^Ki' are unit vectors, so I think it stands up.

 

And I wholly agree with you.. no hope to understand without much more math skills than I have. That's why I posted in the first place.. to ask someone with more knowledge than me if this is just a crazy idea or if it could possible make sense (or maybe if it's simply useless/not original). Eagerly waiting for an answer .

 

Thanks for the links too. I knew the wikipedia entry on operators.. eh.. not easy at all to understand without help. I'll check out Hyperphysics. Not that I ever hope to understand enough math for quantum mechanics, but I'm sure it'll make an interesting reading.

[studiot]

 

(I have a degree in electronics egineering

 

Then you will have studied operational calculus.

 

The Hamiltonian is an operator. The star is not just a multiplication sign it means that you apply the Hamiltonian operator to the wave variable.

[MauroLanzini]

 

Then you will have studied operational calculus.

 

The Hamiltonian is an operator. The star is not just a multiplication sign it means that you apply the Hamiltonian operator to the wave variable.

 

I think I've understood what you mean, studiot: the entries in the H matrix can be real numbers, but they can also be differential operators. Thus what I said is wrong: it's not true that:

 

" the imaginary 'i' is what allows a first-order differential equation to have periodic, 'wave' solutions, while the same equation with strictly real coefficients only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor). "

 

I should have written:

 

" the imaginary 'i' is what allows the equation to exhibit the 'weird' quantum behaviour: for instance if the coefficients of H are all real numbers the Schrödinger's equations admits periodic, 'wave' solutions, while the same equation with a strictly real coefficient on the left side only admits decaying exponentials as solutions (as it happens in the 'real' world, for instance with the charging at constant voltage of an electric capacitor). "

 

So I'm not dead yet I think.... I try to resume: replacing 'i' with 'e^Ki' does not change the Schrödinger's equation if we take K = pi/2, but as K approaches zero the quantum 'behaviour' of the equation becomes smaller and smaller and totally vanishes when K = 0. So this is a mathematical trick which could allow to smoothly go from a fully-quantum to a fully-classical description of the system (I'm sure about the fully-quantum side, it's pretty obvious, I can only guess it may work also for the fully-classical side).

Edited January 31, 2014 by MauroLanzini

[studiot]

The actual form of H in the equation depends upon the situation you are modelling.

 

Several different forms are shown in my reference.

 

Are you referring to the case of a static electric dipole in an electrostaic field as described about 2/3 the way down this Wiki page?

 

http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)

 

Since everything is unmoving and unvarying and independent of time there are no space derivatives or time derivatives.

Edited January 31, 2014 by imatfaal
fixed link

[MauroLanzini]

The actual form of H in the equation depends upon the situation you are modelling.

 

Several different forms are shown in my reference.

 

Are you referring to the case of a static electric dipole in an electrostaic field as described about 2/3 the way down this Wiki page?

 

http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)

 

Since everything is unmoving and unvarying and independent of time there are no space derivatives or time derivatives.

 

Hmm interesting question, but I need some time to prepare an answer. Well, nice discussion anyway.. all your remarks have been thought-provoking, I'm grateful.

Let me try to explain what the idea is in a more informal way. Sorry if I cannot do better, always big thanks for your patience.

 

Consider a plain 1st-grade, linear, one variable differential equation:

 

A * d y(t)/dt + y(t) = 0

 

If A is real the only solutions are decaying exponentials with the general form e^-t/tau (add some constants depending on initial conditions and on the right-side part being zero or something else). This for instance is the case of a charged capacitor which is discharged through a resistor: V_capacitor(t) = V_initial e^-t/tau . After some time V_capacitor will settle to zero (rigorously after an infinite time, but in practice after some tau's).

 

 

Now what takes Schrödinger's equation apart from every other physical equation I know of is that it has an explicitly imaginary coefficient (I got my first glimpse of this browsing http://superstringtheory.com/index.html, by the way).

 

By adding an 'i' to the same equation as above it becomes:

 

i * A d y(t)/dt + y(t) = 0 (where A is real)

 

But now the equation is deeply changed because it admits periodic 'wave' solutions which were previosly impossible, and this turns the equation into something weird: with the same analogy as before you cannot say any more what the final voltage on the capacitor is because it can be always oscillating, and every time you measure it (provided you can measure it more than once) you'll get a different result because you're sampling a sinewave on a random point (allow me for the sake of the explanation that there is no 'energy dissipation' and that the wave does not decay, please).

 

So the background idea is that what turns quantum mechanics 'weird' is, mathematically, the imaginary 'i' in the Schrödinger equation. Can anyone tell me if I'm right upon this at least?

 

 

 

Now, what if the 'i' is not actually just 'i', but rather 'e^Ki' ?

 

e^Ki * A d y(t)/dt + y(t) = 0 (where A is real)

 

If K = π/2 we have (obviously) the 'weird' imaginary version of the equation, with oscillations and no steady final state. But if K = 0 we have the plain vanilla equation, where things settle to a definite 'real-world' value (after some tau's). So by varying K from π/2 to 0 we go 'smoothly' from a 'quantum' to a 'classic' description of the system.

 

As a corollary K could be (inversely) proportional to the 'strength' of the interactions with the sorrounding environment. So a system without any interactions with the outside world would have K = π/2 and it would be described in a full 'quantum' way. But when it starts to interacts with the outside world (by way of a measurement, or just because it becomes embedded say in a piece of rock) K will go towards zero and the system will 'collapse' to a well-definite state. And at least in principle, the trajectory taken by the system going from the 'quantum' to the 'classical' state could be solved (knowing the function K = K(i_don't_know_what), of course).

 

 

I need not to forget the disclaimer..: I have no idea if this theory would actually work or even if it is sensible, and I'm aware it's with 99.9999% probability NOT sensible at all hehe. It just looked interesting to me, and the worse which can happen is that I'll learn something from your answers, so it's all pretty cool anyway.

Edited January 31, 2014 by MauroLanzini

[studiot]

Why does the first order equation have to be linear, and what is A please?

 

For instance how about

 

[math]\frac{{dy}}{{dx}} = \frac{1}{{x\tan y}}[/math]

[MauroLanzini]

Why does the first order equation have to be linear, and what is A please?

 

For instance how about

 

[math]\frac{{dy}}{{dx}} = \frac{1}{{x\tan y}}[/math]

 

 

Well (blushing) it's linear because I feel a comfortable (a little) with linear equations, but not at all with non-linear ones. Sorry.

 

A is a real number.

 

But it's not the equation in itself which is important, that's just an example. The point is that (sorry for having to use a non-technical language) it looks to me that the 'weird' behaviour of the Schrödinger's equation stems, mathematically, from having one (or more) coefficients with an imaginary part, while in classical physics (for what I know) this never happens elsewhere (ie. in Maxwell's equations, Navier-Stokes' etc. etc.).

 

Then replacing 'i' with 'e^Ki' is just a trick (which may or may not work) which allows the same equation to switch from 'quantum' to 'classical' (and in-between).

[studiot]

I'm not sure what your key point is.

 

The use of a first order equation, the use of the Hamiltonian or what?

 

Most uses of H lead to a second order equation and there are many of these in classical physics that have periodic solutions.

 

I pointed to a situiation where H leads to a first order solution in classical physics that is non periodic.

[MauroLanzini]

I'm not sure what your key point is.

 

The use of a first order equation, the use of the Hamiltonian or what?

 

Most uses of H lead to a second order equation and there are many of these in classical physics that have periodic solutions.

 

I pointed to a situiation where H leads to a first order solution in classical physics that is non periodic.

 

Hello again studiot. I try to explain myself with a somewhat different line of reasoning. Then I'll admit that my 'idea' is so poor that I can't even explain it and I stop, I promise.

 

 

- The equations of classic physics describe a world which can be very complicated, with solutions which can be stable, unstable, static or oscillating in innumerable different ways, deterministic or not, even chaotic.. any kind of things. But they are never 'strange' in the same sense as the equations of quantum mechanics, where things can be both waves and particles, pass through two different slits at once and so on: the behaviour of the quantum equations is qualitatively different from the classical ones. I hope we can agree on this point.

 

- The difference in behaviour must lie in the equations themselves: quantum mechanics equations should have something peculiar to them (with respect to classic ones) which allows them to behave the way they do.

 

- Can this difference be that in classic equations there are no complex numbers in their coefficients, while Schrödinger's equation sports a nice 'i' (usually written on the left side, but it could be anywhere)? This is the key point, in effect. Did I get this right, or am I wrong? This is what I would like to know.

 

 

The important things end here, all the rest (replacing the 'i' with 'e^Ki' and etc. etc.) is just speculation, but, well, it looked nice. And it's true that replacing the 'i' in Schrödinger's with 'e^Ki', with K = π/2, leaves it exactly the same, doesn't it ? Trivial for sure, but maybe can be useful in some way.

[Endercreeper01]

 

 

- Can this difference be that in classic equations there are no complex numbers in their coefficients, while Schrödinger's equation sports a nice 'i' (usually written on the left side, but it could be anywhere)? This is the key point, in effect. Did I get this right, or am I wrong? This is what I would like to know.

 

Not really. The main difference between quantum and classical mechanics arises from the fact that the position, time, and momentum have probabilities assigned to them.

Replacing [latex]i[/latex] with [latex]e^{\frac{\pi i}{2}}[/latex] would work, but it wouldn't be in it's simplest form. It is similar to replacing [latex]-1[/latex] with [latex]e^{\pi i}[/latex]. The replacement would work, but it wouldn't be in simplest form.

It can be useful if you have a wave function that includes an exponent of [latex]e[/latex], as you can add [latex]\frac{\pi i}{2}[/latex] to the exponent and possibly simplify [latex]i \psi[/latex].

Edited February 1, 2014 by Endercreeper01

[studiot]

You may like to look into 'the LAX heirachy'

 

A good start would be pages 95 to 102 of

 

Cambridge Texts in Applied MAthematics

 

Solitons an Introduction

 

By

 

Drazin and Johnson

[MauroLanzini]

Thank you for the tips. Browsing around I have also found http://www.scottaaronson.com/democritus/lec9.html, very interesting and mathematically very accessible, and from there http://arxiv.org/abs/quant-ph/0101012 by Lucien Hardy (more complicated, but yet accessible). Nice explanations (for what I can judge) of what (mathematically) distinguishes classic from quantum, and why complex numbers (and not reals, nor quaternions, as I learned) are so important. I'll seek out also the 'LAX hierarchy'. Thanks.

Edited February 2, 2014 by MauroLanzini