Why quantum mechanics?

Question

Imagine you're teaching a first course on quantum mechanics in which your students are well-versed in classical mechanics, but have never seen any quantum before. How would you motivate the subject and convince your students that in fact classical mechanics cannot explain the real world and that quantum mechanics, given your knowledge of classical mechanics, is the most obvious alternative to try?

If you sit down and think about it, the idea that the state of a system, instead of being specified by the finitely many particles' position and momentum, is now described by an element of some abstract (rigged) Hilbert space and that the observables correspond to self-adjoint operators on the space of states is not at all obvious. Why should this be the case, or at least, why might we expect this to be the case?

Then there is the issue of measurement which is even more difficult to motivate. In the usual formulation of quantum mechanics, we assume that, given a state $|\psi \rangle$ and an observable $A$, the probability of measuring a value between $a$ and $a+da$ is given by $|\langle a|\psi \rangle |^2da$ (and furthermore, if $a$ is not an eigenvalue of $A$, then the probability of measuring a value in this interval is $0$). How would you convince your students that this had to be the case?

I have thought about this question of motivation for a couple of years now, and so far, the only answers I've come up with are incomplete, not entirely satisfactory, and seem to be much more non-trivial than I feel they should be. So, what do you guys think? Can you motivate the usual formulation of quantum mechanics using only classical mechanics and minimal appeal to experimental results?

Note that, at some point, you will have to make reference to experiment. After all, this is the reason why we needed to develop quantum mechanics. In principle, we could just say "The Born Rule is true because its experimentally verified.", but I find this particularly unsatisfying. I think we can do better. Thus, I would ask that when you do invoke the results of an experiment, you do so to only justify fundamental truths, by which I mean something that can not itself just be explained in terms of more theory. You might say that my conjecture is that the Born Rule is not a fundamental truth in this sense, but can instead be explained by more fundamental theory, which itself is justified via experiment.

Edit: To clarify, I will try to make use of a much simpler example. In an ideal gas, if you fix the volume, then the temperature is proportional to pressure. So we may ask "Why?". You could say "Well, because experiment.", or alternatively you could say "It is a trivial corollary of the ideal gas law.". If you choose the latter, you can then ask why that is true. Once again, you can just say "Because experiment." or you could try to prove it using more fundamental physical truths (using the kinetic theory of gases, for example). The objective, then, is to come up with the most fundamental physical truths, prove everything else we know in terms of those, and then verify the fundamental physical truths via experiment. And in this particular case, the objective is to do this with quantum mechanics.

Answer 1

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 minimal physical input, then how about this: take the uncertainty principle as a postulate. Then you know that the effect on a system of doing measurement $A$ first, then measurement $B$, is different from doing $B$ first then $A$. That can be written down symbolically as $AB \neq BA$ or even $[A,B] \neq 0$. What kind of objects don't obey commutative multiplication? Linear operators acting on vectors! It follows that observables are operators and "systems" are somehow vectors. The notion of "state" is a bit more sophisticated and doesn't really follow without reference to measurement outcomes (which ultimately needs the Born rule). You could also argue that this effect must vanish in the classical limit, so then you must have $[A,B] \sim \hbar $, where $\hbar$ is some as-yet (and never-to-be, if you refuse to do experiments) undetermined number that must be small compared to everyday units. I believe this is similar to the original reasoning behind Heisenberg's matrix formulation of QM.

The problem is that this isn't physics, you don't know how to predict anything without the Born rule. And as far as I know there is no theoretical derivation of the Born rule, it is justified experimentally!

If you want a foundations viewpoint on why QM rather than something else, try looking into generalised probabilistic theories, e.g. this paper. But I warn you, these provide neither a complete, simple nor trivial justification for the QM postulates.

Answer 2

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 on Nucleus which would cease the existence of atom within a fraction of second (you can tell them to calculate). But, as we know, atoms have survived billions of years. How? Where's the catch?

Answer 3

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 reference to experimental facts, albeit more of a gedanken nature. As others have commented, some link to experiments is, and should be, unavoidable.)

Answer 4

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 before any experiments: he tried to make physics as geometrical as possible.

3) From Lie algebras people knew that the translation operator has something to do with the derivative. From momentum conservation people knew that translations have something to do with momentum. It was not that strange to associate momentum with the derivative.

Now you should just mix everything: merge statistical mechanics with the Hamiltonian formalism and add the key ingredient which was obvious to radio-physicists: that you can not have a short (i.e, localized) signal with a narrow spectrum.

Voila, you have quantum mechanics.

In principle, for your purposes, Feynman's approach to quantum mechanics may be more "clear". It was found long after the other two approaches, and is much less productive for the simple problems people usually consider while studying. That's why it is not that popular for starters. However, it might be simpler from the philosophical point of view. And we all know that it is equivalent to the other approaches.

Answer 5

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 radiation with infinite power. This is a fundamental theoretical problem, and there is no need to appeal to any experimental results to understand it, a theory which predicts infinite emitted power is wrong.

The quantization of light solves the problem, and historically this played a role in the development of quantum mechanics.

Of course this doesn't point to any of the modern postulates of quantum mechanics you're looking to justify, but I think it's still good to use the UV catastrophe as one of the motivations to look for a theory beyond classical physics in the first place, especially if you want to appeal as little as necessary to experimental results.

Answer 6

Classical mechanics is not final theory from the one side and is not further decomposable from the other. So you can't improve it, it is given as is.

For example, you can't explain why if moving body is disappearing from previous point of it's trajectory it should reappear at infinitesimal close point but can't appear a meter ahead (teleporting). What does constraining trajectory points into continuous line? No answer. This is axiom. You can't build a MECHANISM for constraining.

Another example: you can't stop decomposing bodies into parts. You can't reach final elements (particles) and if you do, then you can't explain why these particles are indivisible anymore. The matter should be continuous in classics while you can't imagine how material points exist.

Also, you can't explain, how the entire infinite universe can exist simultaneously in it's whole information. What is happening in absolutely closed box or what is happening in absolute unreachable regions of spacetime? Classics tends us think that reality is real there too. But how it can be if it is absolutely undetectable? Scientific approach says that only what is measurable does exist. So how can it be reality in absolutely closed box (with cat in it)?

In classic mechanic you can't reach absolute identity of building blocks. For example, if all atoms are build of protons, neutrons and electrons, these particles are similar, but not the same. Two electrons in two different atoms are not the same in classic, they are two copies of one prototype, but not the prototype itself. So, you can't define really basic building blocks of reality in classics.

You can't define indeterminism in classics. You can't define unrealized possibilities in classic and can't say what have happened with possibility which was possible but not realized.

You can't define nonlocality in classics. There are only two possibilities in classics: one event affects another (a cause and effect) and two events are independent. You can't imagine two events correlate but don't affect each other! This is possible but unimaginable in classics!

Answer 7

So far as I understand, you are asking for a minimalist approach to quantum mechanics which would motivate its study with little reference to experiments.

The bad. So far as I know, there is not a single experiment or theoretical concept that can motivate your students about the need to introduce Dirac kets $|\Psi\rangle$, operators, Hilbert spaces, the Schrödinger equation... all at once. There are two reasons for this and both are related. First, the ordinary wavefunction or Dirac formulation of quantum mechanics is too different from classical mechanics. Second, the ordinary formulation was developed in pieces by many different authors who tried to explain the results of different experiments --many authors won a Nobel prize for the development of quantum mechanics--. This explains why "for a couple of years now", the only answers you have come up with are "incomplete, not entirely satisfactory".

The good. I believe that one can mostly satisfy your requirements by using the modern Wigner & Moyal formulation of quantum mechanics, because this formulation avoids kets, operators, Hilbert spaces, the Schrödinger equation... In this modern formulation, the relation between the classical (left) and the quantum (right) mechanics axioms are

$$A(p,x) \rho(p,x) = A \rho(p,x) ~~\Longleftrightarrow~~ A(p,x) \star \rho^\mathrm{W}(p,x) = A \rho^\mathrm{W}(p,x)$$

$$\frac{\partial \rho}{\partial t} = \{H, \rho\} ~~\Longleftrightarrow~~ \frac{\partial \rho^\mathrm{W}}{\partial t} = \{H, \rho^\mathrm{W}\}_\mathrm{MB}$$

$$\langle A \rangle = \int \mathrm{d}p \mathrm{d}x A(p,x) \rho(p,x) ~~\Longleftrightarrow~~ \langle A \rangle = \int \mathrm{d}p \mathrm{d}x A(p,x) \rho^\mathrm{W}(p,x)$$

where $\star$ is the Moyal star product, $\rho^\mathrm{W}$ the Wigner distribution and $\{ , \}_\mathrm{MB}$ the Moyal bracket. The functions $A(p,x)$ are the same than in classical mechanics. An example of the first quantum equation is $H \star \rho_E^\mathrm{W} = E \rho_E^\mathrm{W}$ which gives the energy eigenvalues.

Now the second part of your question. What is the minimal motivation for the introduction of the quantum expressions at the right? I think that it could be as follows. There are a number of experiments that suggest a dispersion relation $\Delta p \Delta x \geq \hbar/2$, which cannot be explained by classical mechanics. This experimental fact can be used as motivation for the substitution of the commutative phase space of classical mechanics by a non-commutative phase space. Mathematical analysis of the non-commutative geometry reveals that ordinary products in phase space have to be substituted by start products, the classical phase space state has to be substituted by one, $\rho^\mathrm{W}$, which is bounded to phase space regions larger than Planck length--, and Poisson brackets have to be substituted by Moyal brackets.

Although this minimalist approach cannot be obtained by using the ordinary wavefunction or Dirac formalism, there are three disadvantages with the Wigner & Moyal approach however. (i) The mathematical analysis is very far from trivial. The first quantum equation of above is easily derived by substituting the ordinary product by a start product and $\rho \rightarrow \rho^\mathrm{W}$ in the classical expression. The third quantum equation can be also obtained in this way, because it can be shown that

$$ \int \mathrm{d}p \mathrm{d}x A(p,x) \star \rho^\mathrm{W}(p,x) = \int \mathrm{d}p \mathrm{d}x A(p,x) \rho^\mathrm{W}(p,x)$$

A priori one could believe that the second quantum equation is obtained in the same way. This does not work and gives an incorrect equation. The correct quantum equation of motion requires the substitution of the whole Poisson bracket by a Moyal bracket. Of course, the Moyal bracket accounts for the non-commutativity of the phase space, but there is not justification for its presence in the equation of motion from non-commutativity alone. In fact, this quantum equation of motion was originally obtained from the Liouville Von Neuman equation via the formal correspondence between the phase space and the Hilbert space, and any modern presentation of the Wigner & Moyal formulation that I know justifies the form of the quantum equation of motion via this formal correspondence. (ii) The theory is backward incompatible with classical mechanics, because the commutative geometry is entirely replaced by a non-commutative one. As a consequence, no $\rho^\mathrm{W}$ can represent a pure classical state --a point in phase space--. Notice that this incompatibility is also present in the ordinary formulations of quantum mechanics --for instance no wavefunction can describe a pure classical state completely--. (iii) The introduction of spin in the Wigner & Moyal formalism is somewhat artificial and still under active development.

The best? The above three disadvantages can be eliminated in a new phase space formalism which provides a 'minimalistic' approach to quantum mechanics by an improvement over geometrical quantisation. This is my own work and details and links will be disclosed in the comments or in a separated answer only if they are required by the community.

Answer 8

I always like to read "BERTLMANN'S SOCKS AND THE NATURE OF REALITY" * by J. Bell to remind myself when and why a classical description must fail.

He basically refers to the EPR-correlations. You could motivate his reasoning by comparing common set theory (e.g. try three different sets: A,B,C and try to merge them somehow) with the same concept of "sets" in Hilbert spaces and you will see that they are not equal (Bell's theorem).

• http://hal.archives-ouvertes.fr/jpa-00220688/en/

Answer 9

It seems to me your question is essentially asking for a Platonic mathematical model of physics, underlying principles from which the quantum formalism could be justified and in effect derived. If so, that puts you in the minority (but growing) realist physicist camp as opposed to the vast majority of traditional instrumentalists.

The snag is the best if not only chance of developing a model like that requires either God-like knowledge or at least, with almost superhuman intuition, a correct guess at the underlying phenomena, and obviously nobody has yet achieved either sufficient to unify all of physics under a single rubrik along those lines.

In other words, ironically, to get at the most abstract explanation requires the most practical approach, rather as seeing at the smallest scales needs the largest microscope, such as the LHC, or Sherlock Holmes can arrive at the most unexpected conclusion only with sufficient data (Facts, Watson, I need more facts!)

So, despite being a fellow realist, I do see that instrumentalism (being content to model effects without seeking root causes, what might be compared with "black box testing") has been and remains indispensable.

Answer 10

I highly recommend this introductory lecture on Quantum Mechanics: http://www.youtube.com/watch?v=TcmGYe39XG0

Answer 11

Thomas's Calculus has an instructive Newtonian Mechanics exercise which everyone ought to ponder: the gravitational field strength inside the Earth is proportional to the distance from the centre, and so is zero at the centre. And, of course, there is the rigorous proof that if the matter is uniformly distributed in a sphere, then outside the sphere it exerts a gravitational force identical to what would have been exerted if all the mass had been concentrated at the centre.

Now if one ponders this from a physical point of view, «what is matter», one ends up with logical and physical difficulties that were only answered by de Broglie and Schroedinger's theory of matter waves.

This also grows out of pondering Dirac's wise remark: if «big» and «small» are mereley relative terms, there is no use in explaining the big in terms of the small...there must be an absolute meaning to size.

Is matter a powder or fluid that is evenly and continuously distributed and can take on any density (short of infinity)? Then that sphere of uniformly distributed matter must shrink to a point of infinite density in a finite amount of time.... Why should matter be rigid and incompressible? Really, this is inexplicable without the wave theory of matter. Schroedinger's equation shows that if, for some reason, a matter wave starts to compress, then it experiences a restoring force to oppose the compression, so that it can not proceed past a certain point (without pouring more energy into it).
See the related http://physics.stackexchange.com/a/18421/6432 . Only this can explain why the concept of «particle» can have some validity and not need something smaller still to explain it.