Schrodinger's equation

[thigle]

thank you jinidu, i gained some micro-insights.

not due to rigor and precision of physics theory, but to certain passages/modes in your explanation where our 'languague-clusters' partly overlap, or structurally couple.

anyway, thanx again.

[jinydu]

Actually, there is one point in my previous post that is not totally accurate. I (essentially) said that a wavefunction can be uniquely determined using a statistical measurement on a large number of identically prepared systems.

In fact, such a measurement can only determine a wavefunction up to a factor of e^(i theta), where theta is any real number.

The reason for this is that computing probability density requires taking the absolute value of the wavefunction, so of course, any two wavefunctions whose absolute values are always equal will have identical physical characteristics. Furthermore, the absolute value of e^(i theta) is always equal to one for any real value of theta. Thus, multiplying a wavefunction by e^(i theta) doesn't change any of its physical properties, and is therefore an undetectable change.

By the way, do you understand the mathematics in my previous post?

You can't truly understand quantum mechanics without understanding the mathematics behind it.

[thigle]

well jin, there are many things i don't understand and many I do. also, the point of inaccuracy that you mention is not single. to mention its mate, you wrote:

"First though, you have understand what a system is. Simply put, in quantum mechanics, a system is a collection of physical objects."

system is not a simple collection of (physical) entities, even in a simply put statement of what a system is. system is (irrelevant of discipline, as systems are just schemas, or ways of distinction, applicable from outside of disciplinary boundaries) not a simple collection. than it would not differ from a simple sum.
according to most system theorists, system is :
[(sum(or set) of whatever) + (its (global) emergent properties)]
(btw, meta-system is taken to mean the system's niche, context, ecology, proto-field, openscape, infra-system...whathaveyou)

but back to your question: did I understand the math in your previous post ? I understand all the WORDS(as concepts) and NOTATION(as formalism). but I surely don't understand the whole MEANING. because i have no firm knowledge of math behind, just understanding if the information you present.
actually, it's been long time since I applied any rigorous math. the last time I solved any integrals was 8 years ago.! . i don't DO any math much these days, i mostly swim in geometrical oceans of VISION. IMAGINATION is the nectar I drink.
perhaps I should start playing with/studying the Geometric Algebras, to overcome this duality ?

but most essentialy (or meta-essentialy if you want(or not)), this boils down to epistemological considerations like...
what makes an understanding of information? what makes knowledge ? what makes wisdom ? these surely differ in intensity and viscosity of experiential flow, with the amount of friction decreasing with the dissolution of subject/object structure in the operation of what we call individual. you wrote:
"You can't truly understand quantum mechanics without understanding the mathematics behind it."
seems that 'truly' is redundant. theories (including quantum mechanics) can be truly grasped (as theories/seeings) without following the specific execution (computation in this case) of the given theory. don't you overvalue specific/local (within conceptual terrain of a given theory) over global meaning of the theory ?
the essence of understanding you're getting at via math, is non-math wisdom, beyond the math-specific knowledge (that is itself gained through authentic experience of information.)

well i went a bit off topic as usually

[jinydu]

system is not a simple collection of (physical) entities, even in a simply put statement of what a system is. system is (irrelevant of discipline, as systems are just schemas, or ways of distinction, applicable from outside of disciplinary boundaries) not a simple collection. than it would not differ from a simple sum.
according to most system theorists, system is :
[(sum(or set) of whatever) + (its (global) emergent properties)]
(btw, meta-system is taken to mean the system's niche, context, ecology, proto-field, openscape, infra-system...whathaveyou)

There is no inaccuracy in what I originally posted. The professor and the textbook were very clear about this point. In physics, a system is defined as a collection of physical entities. That's all there is to it.

actually, it's been long time since I applied any rigorous math. the last time I solved any integrals was 8 years ago.!

Integrals are absolutely necessary to understand quantum mechanics; without them, you cannot calculate expectation values of observables, for instance.

but most essentialy (or meta-essentialy if you want(or not)), this boils down to epistemological considerations like...
what makes an understanding of information? what makes knowledge ? what makes wisdom ? these surely differ in intensity and viscosity of experiential flow, with the amount of friction decreasing with the dissolution of subject/object structure in the operation of what we call individual. you wrote:

You used a lot of words that quantum mechanics never uses: "epistemology" (I'm quite certain that's never used by actual scientists at all), "viscosity of experimental flow", etc.

"You can't truly understand quantum mechanics without understanding the mathematics behind it."
seems that 'truly' is redundant. theories (including quantum mechanics) can be truly grasped (as theories/seeings) without following the specific execution (computation in this case) of the given theory. don't you overvalue specific/local (within conceptual terrain of a given theory) over global meaning of the theory ?
the essence of understanding you're getting at via math, is non-math wisdom, beyond the math-specific knowledge (that is itself gained through authentic experience of information.)

I disagree. Mathematics is the foundation for quantum mechanics (and any serious physics theory). Before you can even begin to talk about wavefunctions, you have to understand the notion of a "function" and a "complex number". And there's no way you can understand Schrodinger's equation without knowing what a derivative is.

In fact, what I described is only part of what I learned in a first course on quantum mechanics. More advanced quantum mechanics requires even more advanced mathematics, such as Hilbert spaces.

[bo198214]

Oh come one people. Pleaso no discussions about what is real understanding in this thread.

The point is, if Jinydu (I exaggerate) calls everyone an idiot that cannot or will not follow his mathematical derivations, then he will make no friends and probably will miss something.
And as a master of mathematics I know what I speak of.
So please any further discussions about it in a new thread.

[thigle]

so i forget ethics and don't care anymore

anyway, i will start a new thread on that issue.

and Jin, I have no problem with "function", "complex number" or "hypercomplex number" or "derivation". and no fear of Hilbert spaces. :twisted:

just that much.

[PWrong]

anyway, i will start a new thread on that issue.

Ok, I'm glad that's over. By the way, epistimology is a branch of philosophy, if anyone's interested.

I've had a go at 5D hydrogen, but it's not looking too good. I've reduced it to this:

xy'' + (2x + 2l + 4) y' + (a / x^2 + 2l + 4) y = 0
where l is a constant (that hopefully must be an integer), and
'a' is some function of E and k, (I can't remember what, but it doesn't matter yet.)

Anyway, this hideous thing has an x^3 hidden in it. I can't find it on Mathworld or EqWorld, and Mathematica can't solve it. And the Frobenius method gives an ugly reccurance relation. I've tried solving it numerically, but I haven't quite got the hang of that sort of thing.

Jinydu, do you know of any way to calculate energy levels without actually solving the system? That would be extremely useful here.

If it turns out that 5D has stable atoms but not 4D, maybe we could introduce some kind of Kaluza-Klein thing. We have 4 extended dimensions, then the 5th one is wrapped around a circle, about the size of a hydrogen atom. That way the inhabitants think they're 4D, but their atoms are 5D (Just like we're actually 11D ). It's probably too complicated for us just yet, but I'll start a thread on it anyway.

[bo198214]

I've had a go at 5D hydrogen, but it's not looking too good.

poor PWrong :wink:

(Just like we're actually 11D ). It's probably too complicated for us just yet, but I'll start a thread on it anyway.

Yes, please do it, and tell me whether the 11 dimensions have equal rights, or if some behave differently from the others, i.e. whether it are spatial dimensions similar to our 3 dimensions (answer probably in this thread).

[moonlord]

The point is, if Jinydu (I exaggerate) calls everyone an idiot that cannot or will not follow his mathematical derivations, then he will make no friends and probably will miss something.

That's not necessary something bad. There are some people reading this thread which (like me) were probably helped by Jinydu's help-a-noob post. In a year or two I will most probably be able to follow this better, but till then those posts help me understand the point. However, if someone finds that offending or disturbing or whatsoever, then... I'll use the personal message button .

[jinydu]

[Jinydu, do you know of any way to calculate energy levels without actually solving the system? That would be extremely useful here.

I'm afraid not. I once asked my chemistry professor exactly this question; he said that such methods do exist, but that I wouldn't be taught them at such a low level quantum mechanics class. Remember, what I took last quarter was a first class in quantum mechanics; I still have a long way to go.

[PWrong]

Sorry to revive an old post, but my quantum lecturer covered the hydrogen atom about a week ago, and I now understand why we can't have stable atoms in 4D. He explained that the 1/r potential is the only radial potential with bound states. In any other potential, an electron will escape and become a free particle. That's why energy isn't quantised.

I can show you this graphically.
For any radial potential, we can reduce the equation to a 1D schrodinger equation with an "effective" potential:
V_eff = L(l)/r^2 - V(r)

Try graphing this for a 3D potential, i.e. V(r) = k / r
Depending on the values of L(l) and k, you can usually find a shallow minimum in the graph (it might be hard to find). Electrons will tend to hang around the minimum value.
Now try graphing it for V(r) = k/r^2. This time there's never a minimum, so the electron will always escape to infinity .

For a simpler example, look at the finite square potential well. When E<V, we get bound states, and E is quantised. but when E>V, we just get something like the equation for a free particle, and E can take any value. In 4D hydrogen, any amount of energy is enough to escape the well.

[jinydu]

Assuming your claim is correct:

In 5 dimensions and higher, the effective potential is dominated by the negative term as r approaches 0; thus (if I am not mistaken), the electron has a tendency to "spiral into the nucleus".

In 2 dimensions, the effective potential is of the form:

L/r^2 + ln(r) (I'm ignoring a whole lot of constants there)

The situation there is even worse here because ln(r) goes to infinity as r goes to infinity (so it takes an infinite amount of energy to ionize an electron). However, a minimum potential, as I can see from a computer graph.

More importantly though, how do you know that the effective potential really is of that form in any dimension? Looking through my old quantum chemistry folder, I can see a derivation for the hydrogen atom. It takes about two pages of derivations to get from Schrodinger's Equation for the Hydrogen atom (expressed in spherical coordinates) to the "equivalent" 1D Schrodinger Equation featuring the effective potential; and I don't see how the argument can be generalized to arbitrary dimenisons.

[PWrong]

It takes about two pages of derivations to get from Schrodinger's Equation for the Hydrogen atom (expressed in spherical coordinates) to the "equivalent" 1D Schrodinger Equation featuring the effective potential; and I don't see how the argument can be generalized to arbitrary dimenisons.

You can skip most of the derivation. Just use separation of variables to find the radial equation, and ignore the angular equations. You should find that the radial equation is similar in any dimension (there might be some different constants). By the way, the L/r^2 is unrelated to the dimension; the two comes from the "Del^2" in the laplacian.

I did this once last year, but I'll try it again in a few weeks. (I've got exams soon, so I shouldn't really be on here at all).

[jinydu]

By the way, the L/r^2 is unrelated to the dimension; the two comes from the "Del^2" in the laplacian.

It would seem that that is the crux of the derivation. So the difficult part is: What is the Laplacian in (hyper)spherical coordinates in n dimensions (sorry if this question was already answered earlier in this thread or some other thread; it has been a while since this topic was discussed)?

[PWrong]

Well, in an earlier post I found that you can separate Schrodinger into these four equations:
C''(c) + m²C(c) = 0
B''(b) + cot b B'(b) + ( P(p) -m²/sin²b) B(b)=0
A''(a) + 2cot a A'(a) + (L(l) - P(p)/sin²a) A(a) = 0
R''(r) + 3/r R'(r) - ( 2m/h² (V(r) -E) +L(l)/r²)R=0

I wrote a mathematica notebook to find the laplacian in any coordinate system. It's based on this definition. Just copy it into mathematica and it should work.

Code: Select all

JacobianMatrix[xs_, us_] := Outer[D, xs, us]

ScaleFactor[xs_, us_, i_] := Sqrt[Transpose[
JacobianMatrix[xs, us]][[i]].Transpose[JacobianMatrix[xs, us]][[i]]]

ScaleFactorProduct[xs_, us_] := Product[ScaleFactor[xs, us, i], {i, 1,
Length[xs]}]

Laplacian[ψ_, xs_, us_] := Sum[D[(ScaleFactorProduct[xs, us] D[ψ , us[[k]] ] /
ScaleFactor[xs, us, k]^2), us[[k]]], {k, 1, Length[xs]}] / ScaleFactorProduct[xs, us]


Anyway, with separation of variables, you can reduce Schrodingers equation to the following:
r² R''/R + 3 r R'/R + r²(V(r) - E) = a bunch of stuff in theta, phi and gamma = a constant L(l)

In n dimensions, you get something similar:
r² R''/R + (n-1)r R'/R + r²(V(r) - E) = L(l)

You can turn this into a 1D schrodinger equation using R(r) = u(r) r^p
The coefficient of R' vanishes when 2p+n-1 = 0.
So in 3D, p = -1, and in 4D, p = -3/2

We end up with this effective potential:
V_eff = V(r) - (L(l) + 3/4)/r^2

So the only difference is we've added something onto L(l).

[quickfur]

Sorry to revive an old post, but my quantum lecturer covered the hydrogen atom about a week ago, and I now understand why we can't have stable atoms in 4D. He explained that the 1/r potential is the only radial potential with bound states. In any other potential, an electron will escape and become a free particle. That's why energy isn't quantised.

Hmm. This is very interesting. Even though it is disappointing that we can't generalize our familiar atoms to 4D, the upshot of all this is that maybe the reason the universe has only 3 macroscopic spatial dimensions is related to the fact that only 3D gives you the right potential function to form stable atoms.

[...]In 4D hydrogen, any amount of energy is enough to escape the well.

Ahh, now I understand. I've had trouble understanding why electrons won't form standing waves around a positive charge, but now I get it. The well is just too shallow to keep anything in there for a long time.

[PWrong]

Hmm. This is very interesting. Even though it is disappointing that we can't generalize our familiar atoms to 4D, the upshot of all this is that maybe the reason the universe has only 3 macroscopic spatial dimensions is related to the fact that only 3D gives you the right potential function to form stable atoms.

That's possible. It's kind of like the anthropic principle.

Ahh, now I understand. I've had trouble understanding why electrons won't form standing waves around a positive charge, but now I get it. The well is just too shallow to keep anything in there for a long time

That's pretty much it. I think if you solved the system properly, including the time dependent part of the wavefunction, all the solutions would be unstable.

[quickfur]

Hmm. This is very interesting. Even though it is disappointing that we can't generalize our familiar atoms to 4D, the upshot of all this is that maybe the reason the universe has only 3 macroscopic spatial dimensions is related to the fact that only 3D gives you the right potential function to form stable atoms.

That's possible. It's kind of like the anthropic principle.

Personally, I never liked the idea behind the anthropic principles... sounds like circular reasoning to me. But anyways.

Ahh, now I understand. I've had trouble understanding why electrons won't form standing waves around a positive charge, but now I get it. The well is just too shallow to keep anything in there for a long time

That's pretty much it. I think if you solved the system properly, including the time dependent part of the wavefunction, all the solutions would be unstable.

And clearly, such a state of things is just begging for a new physics to be made, a different kind from the familiar 3D one, that does produce interesting structures in 4D! Maybe a force that instead of having a positive/negative charge, has a 3-fold charge. Something like QCD. Maybe the strong nuclear force will play a much more extensive role in 4D?

[bo198214]

Hey, that was my thought! Lets create a universe that makes really sense not such 3d crap. *giggles his ass off*

[PWrong]

Maybe a force that instead of having a positive/negative charge, has a 3-fold charge. Something like QCD. Maybe the strong nuclear force will play a much more extensive role in 4D?

We might as well look at the strong force anyway. It'll be interesting to see whether protons can form in 4D.

I guess we have to derive the Yukawa potential for 4D. The best way to do this would be to solve the integral down the page, except with k as a 4-vector. Unfortunately I don't know how to solve it, even with mathematica. I suppose we could just divide by r^2 instead of r to get the potential. Then the force has a 1/r^2 term and a 1/r^3 term. That should hopefully be enough to hold quarks together.