# What are the limits of the boundaries for the Schrödigner equation

• I
Gold Member

## Main Question or Discussion Point

If one considers the quantized levels of E, for the solutions to the Schrödinger eqn,, then I am wondering: what are the lowest possible energies that can occur for the Schrödinger eqn? I take the highest possible energy is at the classical limit, but is the zero-point energy the absolute lowest? If it is, is there a model that describes lower energy levels for ANY elementary particle, including meson, muons, pions, neutrinos or even photons?

In other words, are there no solutions below ##E-0## for ANY particle type , and where does the maximum energy reach for the heaviest particles i.e electrons and protons?

Thanks

## Answers and Replies

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ZapperZ
Staff Emeritus
If one considers the quantized levels of E, for the solutions to the Schrödinger eqn,, then I am wondering: what are the lowest possible energies that can occur for the Schrödinger eqn? I take the highest possible energy is at the classical limit, but is the zero-point energy the absolute lowest? If it is, is there a model that describes lower energy levels for ANY elementary particle, including meson, muons, pions, neutrinos or even photons?

In other words, are there no solutions below ##E-0## for ANY particle type , and where does the maximum energy reach for the heaviest particles i.e electrons and protons?

Thanks
This is a very puzzling post.

In a central potential problem, such as in a hydrogen atom, you have an attractive potential (negative), and so the energy level are negative because these are bound states. It is only at n=infinity do you get E = 0.

You appear, from what I can guess, to be asking if there's any energy state lower than the ground state. This does not make any sense either because, by definition, the ground state IS the lowest energy state of the system. But even more of a puzzle here is that I don't know if you're asking for a theoretical argument, or an experimental one. The former should be easy to deal with. Just show a derivation that there is one, and we can all go home. The latter should involve published, indisputable experimental, and then, once again, we can all go home.

BTW, the title you created for this thread is a bit misleading. When I read it, I thought you were asking for the mathematical boundary conditions in solving a Schrodinger equation.

Zz.

Gold Member
This is a very puzzling post.

You appear, from what I can guess, to be asking if there's any energy state lower than the ground state. This does not make any sense either because, by definition, the ground state IS the lowest energy state of the system. But even more of a puzzle here is that I don't know if you're asking for a theoretical argument, or an experimental one. The former should be easy to deal with. Just show a derivation that there is one, and we can all go home. The latter should involve published, indisputable experimental, and then, once again, we can all go home.

Zz.

Either theoretical or empirical, I take the latter is difficult.

How can the ground state energy which involves the spacing interval L of an electron, which is substantially larger in diameter than muons or mesons as a particle, be the absolute lowest energy possible? I take it is indeed the lowest energy possible for an electron and proton, but does this account for ALL particles in the list of fermions and bosons?

The Schrödinger eqn describes fermions primarily, as far as I know, and the general wave equation describes bosons, however, having so many different particles, which several are lighter than others, and others are even mass-less, how can the lowest energy E_0 in the S.Eq be the absolute lowest AT ALL?

Should one describe a lower energy level, than I suppose the original eqn . has to be changed, and one describes a new model. Maybe I am stirring the foundations of QM, but something doesn't make sense when one puts dozens of elementary particles on one part of a scale, and ONE minimal zero point energy on the other part of the scale (figuratively speaking), representing all, light, heavy, high-velocity, spinning, zero-spining etc.

Let me ask a simple question: What is the zero-point energy of a photon?

ZapperZ
Staff Emeritus
Either theoretical or empirical, I take the latter is difficult.

How can the ground state energy which involves the spacing interval L of an electron, which is substantially larger in diameter than muons or mesons as a particle, be the absolute lowest energy possible? I take it is indeed the lowest energy possible for an electron and proton, but does this account for ALL particles in the list of fermions and bosons?

The Schrödinger eqn describes fermions primarily, as far as I know, and the general wave equation describes bosons, however, having so many different particles, which several are lighter than others, and others are even mass-less, how can the lowest energy E_0 in the S.Eq be the absolute lowest AT ALL?

Should one describe a lower energy level, than I suppose the original eqn . has to be changed, and one describes a new model.
Your post has a huge amount of error.

What is this spacing "L"? Can you look up the Schrodinger equation for a hydrogen atom and tell me?

The schrodinger equation is NOT only for "fermions" and the wave function is a SOLUTION to a schrodinger equation, so how come that is only for boson, as you stated? This is very confusing.

There is a fundamental lack of understanding of the Schrodinger equation here. I think you need to go back and re-learn this.

Zz.

Gold Member
Your post has a huge amount of error.

What is this spacing "L"? Can you look up the Schrodinger equation for a hydrogen atom and tell me?

I never talked about the hydrogen atom. You did. I talk about free electrons. They still have a minimum energy.

The schrodinger equation is NOT only for "fermions" and the wave function is a SOLUTION to a schrodinger equation, so how come that is only for boson, as you stated? This is very confusing.
I never talked about a wave function, I talked about the general wave equation. Please look it up on Wiki.

There is a fundamental lack of understanding of the Schrodinger equation here. I think you need to go back and re-learn this.
Forget this eqn for a second. Take it under consideration that it has solutions only within a boundary of E. If you cross it over the classical limits, it has no solutions. If you go at the minimum, you reach the zero point energy.

I am asking, can one develop a model that has solutions BELOW the zero-point energy, and which elementary particles or wave-phenomenae could reflect that?

PeterDonis
Mentor
2019 Award
How can the ground state energy which involves the spacing interval L of an electron, which is substantially larger in diameter than muons or mesons as a particle, be the absolute lowest energy possible?
I think I see what you are asking, but you are phrasing it very clumsily. Let me try to rephrase it:

Suppose we have two quantum systems: one consisting of an electron and a proton bound together (i.e., a hydrogen atom), and one consisting of a muon and a proton bound together (i.e., a "muonic hydrogen" atom). The binding energy of these two systems--i.e., the energy it would take to separate the two components and make them free particles--will not be the same. I don't know if anyone has actually tried to calculate or measure what the binding energy of "muonic hydrogen" would be, but it seems evident that it won't be the same as that of ordinary hydrogen, and intuitively I would expect it to be larger--i.e., for the muon to be more tightly bound to the nucleus than the electron in ordinary hydrogen is. That would mean that, if we consider the energy of the unbound free particles at rest to be zero, the energy of muonic hydrogen is lower--more negative--than the energy of ordinary hydrogen.

Your question, then, appears to be: is there a lowest possible energy for any bound state, considering all possible ones that could exist? Or can the energy of a bound state be as negative as we like?

Is this a fair restatement of your question?

SemM
Gold Member
I think I see what you are asking, but you are phrasing it very clumsily. Let me try to rephrase it:

Suppose we have two quantum systems: one consisting of an electron and a proton bound together (i.e., a hydrogen atom), and one consisting of a muon and a proton bound together (i.e., a "muonic hydrogen" atom). The binding energy of these two systems--i.e., the energy it would take to separate the two components and make them free particles--will not be the same. I don't know if anyone has actually tried to calculate or measure what the binding energy of "muonic hydrogen" would be, but it seems evident that it won't be the same as that of ordinary hydrogen, and intuitively I would expect it to be larger--i.e., for the muon to be more tightly bound to the nucleus than the electron in ordinary hydrogen is. That would mean that, if we consider the energy of the unbound free particles at rest to be zero, the energy of muonic hydrogen is lower--more negative--than the energy of ordinary hydrogen.

Your question, then, appears to be: is there a lowest possible energy for any bound state, considering all possible ones that could exist? Or can the energy of a bound state be as negative as we like?

Is this a fair restatement of your question?

Thank you Peter!! You described my question exactly. And thanks for the answer. Can a different ODE describe such an energy system?

PeterDonis
Mentor
2019 Award
I never talked about a wave function, I talked about the general wave equation. Please look it up on Wiki.
@ZapperZ is quite familiar with the relevant equations, and his criticism of your posts is valid. Instead of asking him to look up the equations, you should be the one looking up things in textbooks and learning more about the relevant equations and concepts. You labeled this thread as "A", indicating that you have a graduate level understanding of the subject matter. Your posts so far do not at all indicate that you have such an understanding. Please take note.

Greg Bernhardt
PeterDonis
Mentor
2019 Award
Moderator's note: I have changed the level of this thread to "I".

Gold Member
@ZapperZ is quite familiar with the relevant equations, and his criticism of your posts is valid. Instead of asking him to look up the equations, you should be the one looking up things in textbooks and learning more about the relevant equations and concepts. You labeled this thread as "A", indicating that you have a graduate level understanding of the subject matter. Your posts so far do not at all indicate that you have such an understanding. Please take note.

Sorry for insulting, it was not the intention. I never wrote the "wave function", that created confusion along the thread and moved the focus towards a completely different topic. I think I am fine with what you wrote, and I understand this rather well.

PeterDonis
Mentor
2019 Award
You described my question exactly.
Ok, good.

And thanks for the answer.
I didn't give you an answer. I only restated your question.

Now that I've restated it, what do you think the answer is?

Gold Member
Ok, good.

I didn't give you an answer. I only restated your question.

Now that I've restated it, what do you think the answer is?

I think one needs to develop an own ODE/PDE to describe that energy system. I have an idea of how one could look like.

ZapperZ
Staff Emeritus
@ZapperZ is quite familiar with the relevant equations, and his criticism of your posts is valid. Instead of asking him to look up the equations, you should be the one looking up things in textbooks and learning more about the relevant equations and concepts. You labeled this thread as "A", indicating that you have a graduate level understanding of the subject matter. Your posts so far do not at all indicate that you have such an understanding. Please take note.
You also forgot to advice the OP with one more thing: Never, EVER tell ZapperZ to go look up Wikipedia!

Zz.

Greg Bernhardt and SemM
Gold Member
You also forgot to advice the OP with one more thing: Never, EVER tell ZapperZ to go look up Wikipedia!

Zz.

Sorry Zapp! I was surprised when you interpreted the general wave equation for a wavefunction and didnt look at your history/merits!

PeterDonis
Mentor
2019 Award
I think one needs to develop an own ODE/PDE to describe that energy system.
You are incorrect. If we are talking about non-relativistic systems, the Schrodinger Equation works fine for all of them. No other equation is needed.

If you want to talk about relativistic systems, then you need to be doing quantum field theory, which doesn't use the Schrodinger Equation. But the equations for that are well known as well.

Finally, you did not actually answer my question: do you think the energy of a bound quantum system can be as negative as you like, or is there a limit to how negative it can be?

I have found one.
First, please review PF's rules on personal speculation. Your statement here looks like it is a personal speculation, which is not allowed here.

Second, you don't need to find any such equation anyway. See above.

Gold Member
First, please review PF's rules on personal speculation. Your statement here looks like it is a personal speculation, which is not allowed here.

Sorry for reinsulting, now not just one, but thousands of members. I have rephrased.

Gold Member
Second, you don't need to find any such equation anyway. See above.
Should we stop looking?

PeterDonis
Mentor
2019 Award
Sorry for reinsulting, now not just one, but thousands of members. I have rephrased.
What you rephrased wasn't "insulting" to anyone; it just indicated a possible violation of PF rules. Your rephrasing does not fix that.

In any case, you are focusing on the wrong thing. Please go back and read my post #15, the third paragraph I wrote there, starting with "Finally". That's what you should be focusing on.

Should we stop looking?
See above.

Gold Member
What you rephrased wasn't "insulting" to anyone; it just indicated a possible violation of PF rules. Your rephrasing does not fix that.

In any case, you are focusing on the wrong thing. Please go back and read my post #15, the third paragraph I wrote there, starting with "Finally". That's what you should be focusing on.

See above.

Take this for example:

Say that the S.Eq describes quantized waves in the range of zero point energy to the classical limit. Newtonian equations describe from the classical limit and above to the relativistic level.

What if an equation described what is below the zero-point energy? You mention QFT in another context or a related context. I have no knowledge of QFT, but should thre exist a PDE/ODE for the spectrum of energy below the zero point energy, wouldn't it be interesting?

PeterDonis
Mentor
2019 Award
@SemM you are continuing to speculate instead of answering the question I asked you. There is no point in continuing the discussion if you are going to do that. Thread closed.

SemM