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Which quantum mechanics formulation is popular: Schrödinger's wave mechanics or Heisenberg's matrix mechanics? I find this extremely confusing: Some post-quantum mechanics textbooks seem to prefer wave mechanics version, while quantum mechanics textbooks themselves seem to prefer matrix mechanics more (as most formulations are given in matrix mechanics formulation.)

So, which one is more preferred?

Add: also, how is generalized matrix mechanics different from matrix mechanics?

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5 Answers

It is partly a specialty thing. Non-relativistic QM mixes the two approaches quite often (as do relativistic QM), but quantum field theory leans very heavily towards Heisenberg, as well as the path integral formalism.

There are more approaches, but they tend to be scarcely used. The algebraic approach is getting some use, especially in QFT on curved spacetimes.

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Current standard textbooks teach a mixture of both wave mechanics and matrix mechanics although the emphasis is put more in the wave formulation because this is much more easy for most quantum problems. Matrix mechanics is simpler when dealing with the harmonic oscillator.

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Ever since Dirac came up with transformation mechanics (kets and bras), and showed both matrix and wave mechanics are special cases of it, physicists have worked with kets and bras. With von Neumann later, this was reinterpreted as Hilbert spaces.

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This is not an answer to the question. One always uses the Hilbert space formalism, independently of whether you are working in the wave or matrix mechanics framework. –  Benjamin Hodgson Oct 12 '12 at 17:16
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If you do a "physics for non-physicists" course (in my case as part of a Physical Chemistry degree) you'll almost certainly be taught the Shrodinger equation because it requires less mathematical sophistication to use. For most chemical applications it's also a lot easier to use.

I'm not sure if anyone uses matrix mechanics these days. I get the impression it's regarded as a worthy but slightly clumsy step along the way to quantum field theory, but that may simply mean I mix with the wrong type of scientist :-)

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I surely use matrix mechanics. Many others do. See e.g. this solution to the hydrogen atom motls.blogspot.cz/2011/11/… which doesn't mention wave functions at all. They're not really observables. Physics is about observables and in QM, they're operators – which means that in any particular basis, they are matrices. At the same time, the equivalence of the pictures is so manifest that it doesn't make much sense to say that we're only using matrix mechanics or only wave mechanics. –  Luboš Motl Sep 23 '12 at 9:06
    
Speaking as someone for whom less mathematical sophistication is most welcome, solving the Schrodinger equation for the hydrogen seems much easier, and I get a wavefunction that can easily be used to extract observables. Plus when I move on to multi-electron atoms, or indeed molecules, I can use the hydrogen wavefunctions as a basis for a variational calculation. I doubt you would find many theoretical chemists who would use matrix mechanics in anger. –  John Rennie Sep 23 '12 at 9:22
    
matrix mechanics = QM in the Heisenberg picture. Lots of stuff (in particular almost all QFT) belongs to this category. Only the old terminology is no longer used. –  Arnold Neumaier Sep 23 '12 at 20:01
    
In quantum chemistry, the Heisenberg picture is of little use as most of quantum chemistry is about the ground state. But in spectroscopy, you need the spectrum of H, which (in each basis) is a matrix. Thus once you need a large part of the spectrum, you are doing matrix mechnaics, as the Schroedinger equation is then of little help. –  Arnold Neumaier Sep 23 '12 at 20:04
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The true fathers of quantum mechanics – Heisenberg, Born, Jordan, and later Bohr – started with matrix mechanics; it was the picture in which the classical equations of motion were easier to be understood as a limit of the new theory. That was in 1925. That was how quantum mechanics was born for the first time.

Within a year, wave mechanics was born and shown to be physically equivalent to matrix mechanics, mostly by Dirac and partly by Schrödinger. Wave mechanics instantly became popular, perhaps more popular than matrix mechanics, due to its mathematical similarity to classical field theory which is why it looked and still looks simpler to many. However, this mathematical similarity doesn't mean that the physical interpretation is the same. That's why we may blame many widespread misconceptions about quantum mechanics on the popularity of the wave mechanics.

Today, we usually use the terms "Schrödinger picture" and "Heisenberg picture" of quantum mechanics for the historical concepts wave mechanics and matrix mechanics, respectively.

It's not clear what you mean by "generalized matrix mechanics". It may mean many things, e.g. some obscure and totally inequivalent theory

http://arxiv.org/abs/hep-th/0203007

with more than two indices per "operator".

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