Is Solving Schrodinger's Equation the Best Way to Approach Quantum Mechanics?

[houlahound]

Wondering what educators / researchers in the field think is the most important and logical flow of topics.

The way I was instructed was solving Schrödinger's wave equation, SWE, for every possible problem that could be solved with a pen and paper in say 2 or 3 pages max.

The problems were instructive toy problems which I assume are standard.

The next phase was solving similar if not the same problems but slightly less contrived where numerical techniques were used.

That was all great but when I first looked at the literature it did not resemble anything I had ever seen, it was all logic, set theory, abstract algebra...not a differential equation in sight.

The question is what is the progression in terms of topics to go from SWE to abstract algebra and logic/set theory of QM.

Why is SWE the standard (as far as I know) way into QM, can the more abstract approach be done from the start.

 

[bhobba]

The question is what is the progression in terms of topics to go from SWE to abstract algebra and logic/set theory of QM..

That's a tough ask.

Here is the book that does it:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

Be warned - its what mathematicians call non-trivial - meaning it HARD. I have a copy an it stretches my math to the limit.

Personally I would study Ballentine and see if you still want to go down the abstract algebra set theory route. Its our deepest and most penetrating formalism but not really required to do or even understand QM.

Thanks
Bill

 

[houlahound]

Cheers, book sounds intimidating.

I guess I want know is the SWE what pros use to solve real problems. The abstract algebra approach whatever it is seems to be how they talk.

SWE operates in numbers, yes complex problems for sure but really just calculus. Do operations on numbers and get out different numbers.

The advanced work I see there are not even any numbers involved, its hard to see what quantities they are calculating, if the word calculating even applies.

Sorry no well posed question here.

 

[bhobba]

I guess I want know is the SWE what pros use to solve real problems.

Yes.

The set theory logic approach is a foundational approach to penetrate the mathematical essence of QM - not for solving problems.

Thanks
Bill

 

[houlahound]

Thanks, that's good to know and what I was suspecting.

Does the formalism break any new ground for experiment or is it more for foundational proofs like that epic work of Russell proving some basic facts of math that everyone else just takes for granted...referring to Bertrand Russell.

 

[atyy]

I guess I want know is the SWE what pros use to solve real problems. The abstract algebra approach whatever it is seems to be how they talk.

The Schroedinger wave equation you learn first is a specific form of the general Schroedinger equation. The Schroedinger wave euqation is the general Schroedinger equation applied to the specific case of non-relativistic quantum mechanics and when the position basis is used.

The general Schroedinger equation is used throughout quantum theory, even quantum field theory. For example, the general Schroedinger equation is Eq (1.1) in Srednicki's textbook http://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf.

 

[bhobba]

Does the formalism break any new ground for experiment or is it more for foundational proofs like that epic work of Russell proving some basic facts of math that everyone else just takes for granted...referring to Bertrand Russell.

It's a mathematical elegance type thing. When mathematicians get a hold of physical theories it often is unrecognisable to physicists. For example classical physics is put in the language of symplectic geometry
https://www3.nd.edu/~eburkard/Talks/GSS%20Talk%20110413.pdf

QM makes use of that and extends it even further. When put in that language the connection between QM and classical mechanics is very transparent. It's beautiful, elegant and mathematically penetrating - but as far as solving problems goes pretty useless.

Thanks
Bill

 

[houlahound]

So if relativity is not considered would problems in lasers, semiconductors, properties of solids and atomic spectra all be dealt with the SWE, what problems won't it solve?

 

[bhobba]

So if relativity is not considered would problems in lasers, semiconductors, properties of solids and atomic spectra all be dealt with the SWE, what problems won't it solve?

Quantum Field Theory which is also used in condensed matter physics.

Thanks
Bill

 

[houlahound]

Gotcha, last two posts.

Condensed matter is that same as solid state physics??

 

[atyy]

To add to bhobba's point - the non-relativistic quantum mechanics of many identical particles is given by the SWE - this has an exact reformulation as a non-relativistic quantum field theory - they are two ways of saying exactly the same thing.

 

[Demystifier]

Condensed matter is that same as solid state physics??

The latter is a branch of the former. Condensed matter can be either solid or liquid.

 

[kith]

So if relativity is not considered would problems in lasers, semiconductors, properties of solids and atomic spectra all be dealt with the SWE, what problems won't it solve?

For the Stern-Gerlach experiment, you need to replace the Schrödinger wave equation by the Pauli equation (where the wavefunction is replaced by a so-called spinor wavefunction). This is an example where the abstract treatment is simpler because for most quantities of interest, you can disregard the wavefunction and only consider the spin degrees of freedom.

If you are familiar with wavefunction but not so much with the abstract formalism, I recommend Sakurai's book "Modern Quantum Mechanics". He directly starts with a clear, short and simple introduction of the abstract formalism. I also like how he connects it with the wavefunction approach later on.

 

[A. Neumaier]

That was all great but when I first looked at the literature it did not resemble anything I had ever seen, it was all logic, set theory, abstract algebra...not a differential equation in sight.

The question is what is the progression in terms of topics to go from SWE to abstract algebra and logic/set theory of QM.

Why is SWE the standard (as far as I know) way into QM, can the more abstract approach be done from the start.

There are two basic ways to do quantum mechanics: Either as matrix mechanics (differential equations for operators), which is done in the Heisenberg picture, or as wave mechanics (differential equations for the state vector), which is done in the Schroedinger picture. For many purposes (namely except for the study of time correlations) both are equivalent, but field theory is typically done in the Heisenberg picture, while few-particle problems are typically done in the Schroedinger picture.

Simple problems are easiest solved in terms of the Schroedinger equation. But once one goes beyond the introductory part one needs much more machinery, and abstract algebra helps a lot to get the computational complexity down. It is also essential in quantum information theory, which is essentially multilinear algebra applied to quantum problems.

 

[vanhees71]

No! Quantum mechanics is independent on the choice of the picture or time evolution. Whether you have wave mechanics or matrix mechanics depends only on whether you choose a complete orthonormal eigenbasis of a complete set of compatible operators with an entirely continuous or and entirely discrete set of (generalized) eigenvalues. The usual case are mixed representations. The choice of a basis, however is independent of the choice of the picture of time evolution (as long as you don't use explicitly time-dependent observables to define your basis, which is fortunately not necessary for the usual applications).

You claim the treatment of "time correlations" (whatever you mean by this) leads to different predictions for whether you use the Heisenberg or Schrödinger picture. Can you elaborate on this, because it cannot be true due to the basic mathematical structure of quantum theory, which you can formulate entirely independent of the choice of the picture of time evolution in manifest covariant (meaning here invariant under arbitrary time-dependent unitary transformations, which is just another way to formulate picture-independence) form, and any physically observable fact is thus picture-independent?

 

[A. Neumaier]

You claim the treatment of "time correlations" (whatever you mean by this) leads to different predictions for whether you use the Heisenberg or Schrödinger picture. Can you elaborate on this

I am not claiming that they lead to different predictions! Instead I claim that time correlations are meaningless without the Heisenberg picture. To define a time correlation $\langle A(s)A(t)\rangle$ one needs a family of operators $A(s)$ that depend on time, hence the Heisenberg picture. One can convert the expression into one in the Schroedinger picture, but the resulting expression has no meaning without its interpretation in the Heisenberg picture!

 

[vanhees71]

Admittedly in such cases the Heisenberg picture is more convenient than the Schrödinger picture.

 

[A. Neumaier]

Admittedly in such cases the Heisenberg picture is more convenient than the Schrödinger picture.

... and the only one where the definition of the time correlation makes sense! This is why I consider the Heisenberg picture fundamental, and the Schroedinger picture just a very useful special case when only a single time is involved.