How is the Schrodinger Equation used in calculating forces of nature?

[mieral]

First, if you want to refer to the Schrodinger Equation only.. is it only called Quantum Mechanics? Or does the word Quantum Mechanics also include Quantum Field Theory? If true. Then what words should you refer to the non-relativistic Schrodinger Equations only?

About the forces of nature. The Hamiltonian contains a term for the forces of nature like strong force, weak force or em force contribution to the potential energy, and we use the Hamiltonian to calculate the wave function.. is this correct?

But do people use the Schrodinger Equation to compute for the strong force or weak force? Can you cite an application where the original non-relativistic Schrodinger Equation is used in the strong or weak force instead of QFT?

 

[houlahound]

I don't think there is an official glossary stating when you call it QM or QFT or anything else.

there is just a bunch of problems to solve, physical laws and invented mathematical tools to solve them - call them what you prefer.

if you get to the task of solving a specific problem and you want to be peer reviewed you will have to state what actual methods you employed precisely.

 

[mieral]

What I was specifically asking was. In atoms we commonly use schrodinger equation only because it is not relativistic and it is connected to the electromagnetic field. How about in the strong and weak force. Do physicists also use schrodinger equation a lot too or mostly QFT?

 

[bhobba]

What I was specifically asking was. In atoms we commonly use schrodinger equation only because it is not relativistic and it is connected to the electromagnetic field. How about in the strong and weak force. Do physicists also use schrodinger equation a lot too or mostly QFT?

QM is formally just an extension of probability theory:
http://www.scottaaronson.com/democritus/lec9.html
https://arxiv.org/abs/quant-ph/0101012
https://arxiv.org/abs/1402.6562

Now that can be applied to normal; classical mechanics and you get what we usually call QM. Schrodinger's equation etc believe it or not follow from symmetry considerations ie the probabilities are frame independent ie the same regardless of where you are, what time it is or how fast you are travelling, It's very intuitive, but really you are applying the Principle of Relativity. It is this that is usually applied to the hydrogen atom etc.

To go to QFT is easy - you simply divide the field into a lot of small blobs and apply the QM above to those blobs then let the blob size to go to zero. Strangely and interestingly when you do some mathematical manipulation you end up with the second quantization formulation of ordinary QM:
http://www.colorado.edu/physics/phys5260/phys5260_sp16/lectureNotes/NineFormulations.pdf

In this way particles emerge from fields. It can be applied to atoms as well and you get strange phenomena like spontaneous emission etc, but normal QM is considered good enough for most things. QFT is used for the stuff you mention because it combines fields and particles in the one framework.

Even in string theory many now think it's really QFT in disguise - in modern times it has morphed from what was originally thought:
https://www.theatlantic.com/science/archive/2016/09/the-strange-second-life-of-string-theory/500390/

Thanks
Bill

 

[vanhees71]

Usually, at least in Germany, Quantum Mechanics deals with QT formulated in the "1st-quantization formulation", i.e., it's strictly speaking only applicable to non-relativistic quantum theory since it deals with situations only, where the number of particles doesn't change in interactions. It can be formulated in the configuration-space representation, where the pure states are represented as functions $\psi(t,\vec{x}_1,\ldots,\vec{x}_N)$, where $N$ is the fixed number of particles.

Sometimes one talks about "Relativistic Quantum Mechanics" which labels the old-fashioned approach to relativistic QT that tries to formulate it in terms of the 1st-quantization formulation (as it is, e.g., treated in Bjorken&Drell vol. I). Personally for me relativistic QT should not be introduced in this way but right away as QFT ("2nd quantization formulation"). Of course, there's also non-relativsitic QFT, which is equivalent to QM if the interactions are thus that the particle number is conserved, but it's very flexible to also deal with quasiparticles, which are a very powerful concept.

 

[mikeyork]

Usually, at least in Germany, Quantum Mechanics deals with QT formulated in the "1st-quantization formulation", i.e., it's strictly speaking only applicable to non-relativistic quantum theory since it deals with situations only, where the number of particles doesn't change in interactions. It can be formulated in the configuration-space representation, where the pure states are represented as functions $\psi(t,\vec{x}_1,\ldots,\vec{x}_N)$, where $N$ is the fixed number of particles.

Sometimes one talks about "Relativistic Quantum Mechanics" which labels the old-fashioned approach to relativistic QT that tries to formulate it in terms of the 1st-quantization formulation (as it is, e.g., treated in Bjorken&Drell vol. I). Personally for me relativistic QT should not be introduced in this way but right away as QFT ("2nd quantization formulation"). Of course, there's also non-relativsitic QFT, which is equivalent to QM if the interactions are thus that the particle number is conserved, but it's very flexible to also deal with quasiparticles, which are a very powerful concept.

I don't understand why you claim that QFT is the only way to consider interactions where particles are not conserved. First of all, there is also S-matrix theory. But in principle we can view particle transitions in the same way that we consider quantum number "addition" using vector-coupling coefficients. In fact this is how, using symmetry groups, that even in QFT we treat quantum numbers other than energy-momentum. But energy-momentum coupling has not yet been fully resolved even in QFT.

 

[vanhees71]

Well, I've no clue, how you want to describe a system with changing particle numbers by a single wave function, with a fixed number of configuration-space variables, as in the 1st-quantization formalism. Of course, you can consider abstract S-matrix theory, but usually we derive the S matrix from QFT.

 

[mikeyork]

Well, I've no clue, how you want to describe a system with changing particle numbers by a single wave function, with a fixed number of configuration-space variables, as in the 1st-quantization formalism. Of course, you can consider abstract S-matrix theory, but usually we derive the S matrix from QFT.

If you think of your wave-function as a special case of a scalar product in Hlbert space that describes a change of representation, then you can extend that to include transitions between representations of differing numbers of particles. For example, particle decay is a transition from a single-particle representation to a two- (or more) particle representation -- just like the way we add orbital and spin angular momentum for instance. If you think about it, this is really what you do to describe dynamic coupling in either S-matrix theory or Feynman diagrams. As regards deriving the S-matrix from QFT, I would say the whole point of S-matrix theory was that it was very definitely an alternative to QFT. (Just check out anything from Chew or Stapp or many others in the 60s and 70s.)

 

[vanhees71]

What you describe above, however, in fact should be equivalent to QFT, right? It's just most convenient to use QFT and not the cumbersome formalism as Dirac's sea formulation of QED. S-matrix theory has become out of fashion. I don't know whether anybody is considering it anymore today. The reason is the proof of the renormalizability of non-Abelian gauge theories by 't Hooft and Veltman in the early 70s.

 

[mikeyork]

What you describe above, however, in fact should be equivalent to QFT, right?

In most circumstance, yes. But I think it is QM itself that is the fundamental underlying theory and QFT merely one incomplete development.

It's just most convenient to use QFT and not the cumbersome formalism as Dirac's sea formulation of QED. S-matrix theory has become out of fashion. I don't know whether anybody is considering it anymore today. The reason is the proof of the renormalizability of non-Abelian gauge theories by 't Hooft and Veltman in the early 70s.

Well, S-matrix phenomenology was the starting point for string theory. I don't consider fashion to be a strong physical argument. In any case, it is the idea of representation transition by vector coupling that I consider to be the appropriate starting point for all dynamical theories. So my point is that I don't think your characterization of QM not providing interaction dynamics is correct.

 

[vanhees71]

I don't what you have in mind with this "vector-coupling" thing. Do you have a reference?

Well, I call the overall concept quantum theory (QT). QM is non-relativistic QT in the first-quantization formalism, i.e., a proper subset of QT. It's just a question of definition.

 

[mikeyork]

I don't what you have in mind with this "vector-coupling" thing. Do you have a reference?

A vector coupling is just a scalar product between representations. It is just more often used to describe couplings of pairs of eigenstates to a single eigenstate such as $<JM|l,m,s,\nu>$ tells us how to couple angular momentum. So Clebsch-Gordon coefficients are just another name for vector-coupling coefficients in the specific context of SU(2). So when I say that dynamics come from vector coupling, what I mean is extending C-G coefficients to include all quantum numbers without factorizing so that one explicitly includes the energy-momentum dependencies. My suggestion is that QFT boils down to a particular way of calculating them. Is that incorrect?

 

[mikeyork]

Oops! I just realized that most people probably think of the word "vector" in vector coupling to refer to Cartesian vectors such as classical ideas of angular momentum, whereas I have always thought of it as coupling state vectors from different representations! Could be a possible source of confusion.

 

[PeterDonis]

if you want to refer to the Schrodinger Equation only.. is it only called Quantum Mechanics? Or does the word Quantum Mechanics also include Quantum Field Theory? If true. Then what words should you refer to the non-relativistic Schrodinger Equations only?

All of these are questions about words, not physics. There is no "right" answer to them.

 

[PeterDonis]

do people use the Schrodinger Equation to compute for the strong force or weak force?

As far as I know, no. In commonly studied cases such as the structure of atoms and molecules, the effects of the weak force are negligible and the effects of the strong force are limited to the nucleus, which is just treated as a point charge at $r = 0$, with no internal structure.

When modeling the internal structure of the nucleus, in order to compute things like the expected rest masses of various nuclei (which involves the strong force) or the expected half-life of unstable nuclei (which involves both the strong and weak forces), there are various phenomenological models (such as the "liquid drop" model), but I don't think those are properly understood as using the Schrodinger Equation to compute bound states the way we do for atoms and molecules. A key part of the problem here is that we don't know the correct form of the potential $V$ for the strong or weak forces; in fact it's not even clear that there is such a potential for those interactions. So we can't even write down a Schrodinger Equation anyway.

 

[mieral]

As far as I know, no. In commonly studied cases such as the structure of atoms and molecules, the effects of the weak force are negligible and the effects of the strong force are limited to the nucleus, which is just treated as a point charge at $r = 0$, with no internal structure.

When modeling the internal structure of the nucleus, in order to compute things like the expected rest masses of various nuclei (which involves the strong force) or the expected half-life of unstable nuclei (which involves both the strong and weak forces), there are various phenomenological models (such as the "liquid drop" model), but I don't think those are properly understood as using the Schrodinger Equation to compute bound states the way we do for atoms and molecules. A key part of the problem here is that we don't know the correct form of the potential $V$ for the strong or weak forces; in fact it's not even clear that there is such a potential for those interactions. So we can't even write down a Schrodinger Equation anyway.

But even when solving for QFT.. we also need to know the potential.. is it not?
Or is there alternative form of computation that doesn't use the potential in QFT for the strong or weak force.. and what method is that? I heard about S-matrix.. doesn't it use the potential?

 

[PeterDonis]

even when solving for QFT.. we also need to know the potential

No, we don't. We need to know the Lagrangian. They're not the same.

is there alternative form of computation that doesn't use the potential in QFT

QFT never uses the potential. It uses the Lagrangian. The various formulations of QFT (such as S-matrix) just do different things with the Lagrangian. If you want to find out more, consult a QFT textbook; a discussion of this would be far too much for a PF thread.

 

[mikeyork]

The various formulations of QFT (such as S-matrix) just do different things with the Lagrangian.

Only a hard-bitten field theorist or someone who graduated after about 1975 would imagine that the S-matrix was a formulation of QFT or had anything to do with a Lagrangian. Leaving aside the issue of taking the unphysical asymptotic limit of a time-dependent operator -- actual S-matrix theory itself is a fundamentally time-independent theory -- instead of Lagrangians, it is founded on unitarity and analyticity and, in most versions, the "particle bootstrap" (AKA "particle democracy"). And it was the foundational basis on which string theory was built. The wikipedia entry gives a simple and reasonably accurate summary.

 

[PeterDonis]

Only a hard-bitten field theorist or someone who graduated after about 1975 would imagine that the S-matrix was a formulation of QFT or had anything to do with a Lagrangian.

Well, my degrees are from 1987 and 1988, so I guess I qualify.

I think it would still be valid to state that S matrix theory does not use a potential in the sense of the Schrodinger Equation, though.

 

[mikeyork]

I think it would still be valid to state that S matrix theory does not use a potential in the sense of the Schrodinger Equation, though.

Yes, that is certainly correct.

Edit: On reflection, I think dispersion relations for elastic scattering might have originally been treated using potential theory. But it is obviously essential to go beyond potential theory in discussing inelastic interactions.

 

[Denis]

Well, I've no clue, how you want to describe a system with changing particle numbers by a single wave function, with a fixed number of configuration-space variables, as in the 1st-quantization formalism.

The configuration space of a field theory is a space of functions $\varphi(x)$ on usual space. That this space in infinite-dimensional causes problems, but one can regularize these problems away with, say, a lattice regularization with a finite number of lattice points. This regularized theory is, yet, a field theory, in the sense that the configuration space is a set of functions, only the space on which these functions are defined is no longer $\mathbb{R}^3$ but some discrete space of lattice nodes.
Particles are in above cases nothing but a strange name for discrete energy levels of that field theory. The configuration space remains the same over time, because these discrete energy levels play no role in its definition.

 

[vanhees71]

There's a lot of confusion in this thread.

First of all, I still don't get the point about "vector couplings". Of course, e.g., in effective field theories of hadrons we have tons of Clebsch-Gordan matrix elements to couple different isospin degrees of freedom (or analogous extensions to higher SU(N) groups).

Then the S-matrix theory of the 60ies was considered because of a deep crisis of QFT, particularly due to the phenomenon of Landau poles and the lack of a theory to describe the strong interaction. This has changed in 1971, when 't Hooft in his PhD thesis and his Veltman provided the proof for the renormalizability of non-Abelian gauge theories, providing also the computational tools like dimensional regularization to systematically study these theories, which lead to the discovery of asymptotic freedom by Wilczek&Gross, and Politzer (also by 't Hooft and Weinberg). This opened the door for concrete QFT models of the strong interaction, leading finally to QCD. In the electroweak sector another crucial ingredient was the discovery of the Anderson-Higgs-Kibble-Guralnik-Englert-Brout-et-al mechanism to describe also massive gauge bosons without breaking the underlying local non-Abelian gauge symmetry. Abstract S-matrix theory is nowadays not used anymore, as far as I know.

Last but not least, of course lattice QFT is another technique to regularize the QFT and provides calculational tools on the computer, called lattice QCD. This is, nevertheless, still QFT and not 1st-quantized QM with a fixed number of particles. The path integral, evaluated in its discretized form and carefully continued to the continuum limit, is over field configurations and thus includes the full many-body space, not only the subspace of fixed particle number.

 

[Denis]

There's a lot of confusion in this thread.
...
Last but not least, of course lattice QFT is another technique to regularize the QFT and provides calculational tools on the computer, called lattice QCD. This is, nevertheless, still QFT and not 1st-quantized QM with a fixed number of particles. The path integral, evaluated in its discretized form and carefully continued to the continuum limit, is over field configurations and thus includes the full many-body space, not only the subspace of fixed particle number.

The confusion seems on your side, because I have not claimed that lattice QFT is not a QFT, or that it is some theory with a fixed number of particles. It is a theory with a fixed (finite) number of lattice nodes, and, therefore, a theory with a finite-dimensional configuration space. And such a theory with a finite-dimensional configuration space fits into the standard general scheme of QT, without any problems. Even with a canonical, quadratic in momentum variables, Hamilton operator.

A variable number of "particles" you can have already in a on-dimensional simple harmonic oscillator, so that you do not even need something "carefully continued" to the ill-defined continuum limit.

 

[vanhees71]

Yes, the harmonic oscillator is the paradigmatic example for the quasi-particle technique, which comes into full glory also in non-relativistic many-body theory, and then, since the quasiparticle numbers are not conserved, again the QFT formulation is most convenient.

In lQCD getting the continuum limit right, is crucial! An example for that from recent years is the correct determination of the pseudocritical deconfinement-confinement and chiral transition temperature at vanishing baryo-chemical potential (which is now settled to be around 160 MeV).

 

[mikeyork]

Of course, e.g., in effective field theories of hadrons we have tons of Clebsch-Gordan matrix elements to couple different isospin degrees of freedom (or analogous extensions to higher SU(N) groups).

Yes. But is there any reason that they can't be extended to also include energy-momentum coupling but can't be factorized (reflecting symmetry-breaking due to electromagnetic and weak forces and perhaps even gravity)? Then the couplings to irreducible representations would represent interactive vertices. QFT, in effect, is a developed theory that enables the calculation of such vertices. S-matrix theory was another attempt that has an interesting legacy in string theory. But one doesn't have to have a developed theory to understand that, in principle at least, this provides an alternative to QFT that is founded on QM scalar products only. (C-G coefficients are scalar products.) That is really the only point I am trying to make.

 

[Denis]

In lQCD getting the continuum limit right, is crucial! An example for that from recent years is the correct determination of the pseudocritical deconfinement-confinement and chiral transition temperature at vanishing baryo-chemical potential (which is now settled to be around 160 MeV).

What means this "is crucial!"? Is there some conceptual reason why LQCD with some sufficiently small h fails so that only the limit defines a valid theory?

(I know you like relativistic symmetry, and, given that only the limit recovers Lorentz symmetry, you think the lattice theory is somehow inferior. But I'm asking about real conceptual problems, which would endanger the observable results, or make the theory invalid as a quantum theory, not that relativistic symmetry will be only approximate.)

 

[vanhees71]

I'm not an expert in lQCD, but what's taken as the result valid for the real world, is the continuum extrapolation. An example is the question, where the pseudocritical crossover temperature for the confinement-deconfinement or chiral-symmetry transition really lies. For years there were two lattice groups fighting about this value, using different discretization schemes to "measure" this quantity on the lattice. Finally it turned out that one of the groups where not in the appropriate scaling limit in terms of the number of lattice points, finally leading to an agreement between the two values, i.e., a transition temperature around 155-160 MeV at $\mu_{\text{B}}=0$, which is very close to the chemical freezeout temperature found by fits of the particle abundances in heavy-ion collisions at the highest collision energies at RHIC and LHC (where $\mu_{\text{B}}$ is indeed very small) to a thermal model.

 

[Denis]

You should not confuse two very different questions.

The first one is about conceptual approaches. Their aim is not at all computations, but to define a reasonable theory, which is not full of infinities and singularities.

The continuum limit should give what we observe, but this continuum limit is also a conceptual thing. So, if the lattice theory gives some fermion doubling, with four "doublers" instead of two which could be interpreted as electroweak pairs, this type of lattice theory has a problem.

The other aim is computation. Here, the only aim are numbers. Here, lattice theory may be inferior to otherwise completely nonsensical things like dimensional regularization. And what matters is if the lattice approximation is first order or second order accuracy. If there appear doublers, is irrelevant if one can add some terms to suppress their contributions. Once the problem you mentioned is about some number, it is of this, second type.

The phrase "continuum extrapolation" obtains very different meaning in above cases. In the conceptual case, it is the continuous theory which approximates the fundamental lattice theory. The problem is to derive this approximation correctly. In the case of number crunching, the number of nodes we can use in the computer is far far less than the number of ether atoms used in any fundamental lattice theory. So, it is the lattice theory which approximates the fundamental theory, and it does not matter what one thinks about the fundamental theory. Anyway, even if one thinks that the fundamental theory is a lattice theory, the lattice for the numerical computation would have to approximate the large distance limit of the fundamental lattice theory, which is the continuous theory.

So, in one case, the continuous theory approximates the lattice, in the other one, the numerical, the lattice theory approximates the continuum.

 

[vanhees71]

Renormalized perturbative QFT is also a well-defined theory, but with limited applicability to low-energy QCD, where the coupling is large. Fortunately there's lQCD with also limited applicability but with complementary calculational possibilities, where it is applicable (e.g., for the hadron spectrum and thermal-equilbrium quantities like the equation of state of strongly interacting matter).

For me the aim of theoretical physics is to "get the numbers out" which lead to predictions that can be compared to observations in nature to test the theory. Of course, the theory should also be as well defined as possible in a mathematical sense, but I'm a physicist not a mathematician ;-)).