# Tag Info

52

The ergodic hypothesis is not part of the foundations of statistical mechanics. In fact, it only becomes relevant when you want to use statistical mechanics to make statements about time averages. Without the ergodic hypothesis statistical mechanics makes statements about ensembles, not about one particular system. To understand this answer you have to ...

51

The theory of probability used in QM is intrinsically different from the one commonly used for the following reason: The space of events is non-commutative (more properly non-Boolean) and this fact deeply affects the conditional probability theory. The probability that A happens if B happened is computed differently in classical probability theory and in ...

41

I am late to this party here, but I can maybe advertize something pretty close to a derivation of quantum mechanics from pairing classical mechanics with its natural mathematical context, namely with Lie theory. I haven't had a chance yet to try the following on first-year students, but I am pretty confident that with just a tad more pedagogical guidance ...

34

Be aware that a "mathematical derivation" of a physical principle is, in general, not possible. Mathematics does not concern the real world, we always need empirical input to decide which mathematical frameworks correspond to the real world. However, the Schrödinger equation can be seen arising naturally from classical mechanics through the process of ...

31

As for references to other approaches to the foundations of Statistical Physics, you can have a look at the classical paper by Jaynes; see also, e.g., this paper (in particular section 2.3) where he discusses the irrelevance of ergodic-type hypotheses as a foundation of equilibrium statistical mechanics. Of course, Jaynes' approach also suffers from a number ...

29

Why would you ever try to motivate a physical theory without appealing to experimental results??? The motivation of quantum mechanics is that it explains experimental results. It is obvious that you would choose a simpler, more intuitive picture than quantum mechanics if you weren't interested in predicting anything. If you are willing to permit some ...

21

Answering this requires a bit of a preamble, so bear with me ... Any observer can construct a coordinate system to locate points in spacetime. By construct a coordinate system I mean they choose a ruler for measuring distance and a clock for measuring time, then they choose the directions of their three spatial dimensions. Even before Einstein, coordinate ...

19

Small addition to ACuriousMind's great answer, in reply to some of the comments asking for a derivation of Schrödinger wave equation, using the results of Feynman's path integral formalism: (Note: not all steps can be included here, it would be too long to remain in the context of a forum-discussion-answer.) In the path integral formalism, each path is ...

14

There's two (ultimately related) answers. For the first answer, just forget about $\hbar$ (but say $c=1$), we are doing a classical relativistic field theory. The first is that you can consider the field profile around a static, spherically symmetric source of mass $M$ (you need to add a coupling to the action of the form $g \phi J$, where $J$ is an ...

13

No macroscopic quantum system is described by a pure state. For example, notions like temperature or pressure, which apply to macroscopic systems do not even exist for systems described by a pure state. The description of macroscopic objects (discussed in statistical mechanics) is always in terms of a density matrix (or the essentially equivalent notion of a ...

11

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples: The finite number of states in a compact region of phase space. See e.g. this Phys.SE post. The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular ...

10

You are right, it is wrong to think that in gauge theory "gauge transformations are just a redundancy". This becomes true only if one abandons locality, ignores all boundary effects, all instanton effects, hence most of what is interesting about gauge theory. Of course forming gauge equivalence classes (say of observables) is something one wants to do every ...

9

Well, the problem is still open. Although maybe the axioms were taken as self-evident for mathematics, Hilbert did not really want mathematically self-evident axioms to be the basics for physical axioms. Since Gauß and the hyperbolic space, it is well known that you can get equally valid models from different assumptions that could all be seen as ...

8

Let me try to convince you that the density operator is a mathematical convenience and not a fundamental aspect of quantum mechanics by describing a very general setup for states and observables in both classical and quantum mechanics. This may not directly answer your question, but hopefully it will settle whatever motivated this question. Briefly, ...

8

An excellent book which does more or less what you ask for is Asher Peres' "Quantum theory:concepts and methods". It starts from the Stern-Gerlach experiments and logical reasoning to develop the basic principles of quantum mechanics. From there, it develops the necessary algebra. Another interesting book for an approach of the conceptual side of quantum ...

8

If I would be designing an introduction to quantum physics course for physics undergrads, I would seriously consider starting from the observed Bell-GHZ violations. Something along the lines of David Mermin's approach. If there is one thing that makes clear that no form of classical physics can provide the deepest law of nature, this is it. (This does make ...

7

You should use history of physics to ask them questions where classical physics fail. For example, you can tell them result of Rutherford's experiment and ask: If an electron is orbiting around nucleus, it means a charge is in acceleration. So, electrons should release electromagnetic energy. If that's the case, electrons would loose its energy to collapse ...

7

I searched for "mixing" and didn't find it in other answers. But this is the key. Ergodicity is largely irrelevant, but mixing is the property that makes equilibrium statistical physics tick for many-particle systems. See, e.g., Sklar's Physics and Chance or Jaynes' papers on statistical physics. The chaotic hypothesis of Gallavotti and Cohen basically ...

7

I suppose there are two scientific reasons to look into the foundations of QM: As part of checking in finer and finer detail that indeed the world is governed by standard quantum physics. The towering example here is Bell's theorem. From inspection of the foundations this makes some prediction which can be and has been checked by experiment. As part of the ...

7

Having given it some more thought, there is an unambiguous philosophical difference, with practical implications. The two-slit experiment provides a good example of this. In a classical universe, any particular photon that hits the screen either went through slit A or slit B. Even if we didn't bother to measure this, one or the other still happened, and ...

7

I think there are two answers to this, one emprical and one theoretical. First, the theoretical one: What you describe is essentially induction, the belief that we can generalize from a subset of a class events/situations to the whole class of events/situations. This belief is, by necessity, unprovable, only falsifiable, since proving it would require ...

7

When a gamma ray hits a direct conversion detector, I get a current. I could convert the current into a sound (frequency) and listen. At what point did "position" come into it - other than "the particle was close enough to my sensor to interact"? At no point did a position determination tell me about the energy - yet I have information about the energy. Or ...

7

A fundamental postulate of QFT establishes that the theory admits a strongly continuous representation of (orthochronous proper) Poincaré group $\cal P$. A certain one-parameter subgroup of $\cal P$ describes time evolution (with respect to an inertial reference frame) which, as a consequence, turns out to be unitary since it is part of a larger unitary ...

7

According to Richard Feynman in his lectures on Physics, volume 3, and paraphrased "The Schrodinger Equation Cannot be Derived". According to Feynman it was imagined by Schrodinger, and it just happens to provide the predictions of quantum behavior.

6

Mathematical proof is to physics roughly what syllogism (or some other fundamental inference rule) is to logic. Namely, it begins from assumptions modelling our conception of some physical reality and shows what must be so if the assumptions hold, but it cannot say anything about the underlying assumptions themselves. A simple example was given by dmckee in ...

6

There are several forms of discreteness in quantum theory. The simplest one is the discreteness of eigenvalues and the associated countable eigenstates. Those arise similarly to the discrete standing waves on a guitar string. The boundary conditions only allow certain standing waves that nicely fit into the enforced region in space. Even though the string is ...

6

Though there are many good answers here, I believe I can still contribute something which answers a small part of your question. There is one reason to look for a theory beyond classical physics which is purely theoretical and this is the UV catastrophe. According to the classical theory of light, an ideal black body at thermal equilibrium will emit ...

6

All the key parts of quantum mechanics may be found in classical physics. 1) In statistical mechanics the system is also described by a distribution function. No definite coordinates, no definite momenta. 2) Hamilton made his formalism for classical mechanics. His ideas were pretty much in line with ideas which were put into modern quantum mechanics long ...

5

My impression from the literature is that physicists are still divided on this question. The Quantum Information Theory camp says the latter, but the Quantum Optics people say the former. A related, but distinct, issue is whether one regards the concept of «open system» as a mere mathematical convenience, or as a fundamental concept. This issue has ...

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