# The original postulates of quantum mechanics

## Main Question or Discussion Point

I am trying to find the basic list of postulates that lay the foundation for QM, but i see a different list of postulates in different textbooks and different places.
In MIT lectures, Prof Allan Adams gives 3 basic postulates: 1. State of a system given by Wave function. 2. Mod squared psi gives probability density. 3. superposition principle.
In quantum mechanics by Shankar, i see 4 postulates: 1. Wave function and its meaning. 2. Definition of position and momentum operators and other operators. 3. Measurement of observable gives one of the eigenvalues of its corresponding operator. 4. Wave function satisfies Schrodinger equation.
Online I see in may websites a list of 6 postulates which include statements on the orthogonality of eigenstates and method of finding expectation values, for instance
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html
This is confusing as opposed to other subjects,say, special theory of relativity which always starts with 2 postulates from which everything else is built up. What is the most basic list of postulates in QM from which everything else will follow?

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atyy
Different people give different postulates. Nielsen and Chuang's text on quantum computation https://www.amazon.com/dp/0521635039/?tag=pfamazon01-20 (section 2.2) contains a list, as does Matteo Paris's The modern tools of quantum mechanics http://arxiv.org/abs/1110.6815 (Postulates 1-3 on p3, with further refinements of Postulate 2 as Postulates II.1 - II.5 on p9).

More traditional versions of the postulates can be found in the textbook of Cohen-Tannoudji, Diu and Laloe, in John Preskill's notes http://www.theory.caltech.edu/~preskill/ph219/chap2_13.pdf (section 2.1) and in Bram Gaasbeek's An Introductory Course on Quantum Mechanics http://arxiv.org/abs/1007.4184 (section 4.3.2). The traditional versions differ from the ones given by Nielsen and Chuang and Paris mainly in how rule for state reduction or wave function collapse is stated. The traditional version of the state reduction postulate cannot be easily generalized to continuous variables, whereas Nielsen and Chuang and Paris's version of state reduction can be generalized to continuous variables.

The above postulates are the framework of quantum mechanics. For any particular system, additional postulates are needed to specify the Hilbert space, observables and the Hamiltonian. The analogy is that in classical mechanics, Newton's 3 laws are a framework, but additional postulates are needed to specify the form of the forces, eg. the law of universal gravitation.

It is also important to understand how the postulates are applied to physical situations. For that I recommend Landau and Lifshitz and Weinberg's text on quantum mechanics.

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bhobba
Mentor
Nugatory
Mentor
I am trying to find the basic list of postulates that lay the foundation for QM, but i see a different list of postulates in different textbooks and different places.
....
This is confusing as opposed to other subjects,say, special theory of relativity which always starts with 2 postulates from which everything else is built up. What is the most basic list of postulates in QM from which everything else will follow?
You will have noticed that this thread is continuing the trend :)

There's some history here. Special relativity can be traced back to a single paper (Einstein, 1905, "On the electrodynamics of moving bodies") in which Einstein stated two postulates and then derived the theory from them. Thus, it was clear from the very beginning that all of special relativity was based on those two postulates.

Quantum mechanics, on the other hand, emerged from the efforts of many people approaching the problem from different directions with different starting assumptions. The theory had emerged before anyone attempted to axiomatize it, and as a result there is no single historically preferred axiomatization.

Quantum mechanics, on the other hand, emerged from the efforts of many people approaching the problem from different directions with different starting assumptions. The theory had emerged before anyone attempted to axiomatize it, and as a result there is no single historically preferred axiomatization.
Ok. I get how the same theory can start from different assumptions, like in classical mechanics, where you may either start with Newton's 2nd law for Newtonian mechanics, or you can start from the principle of least action for the Lagrangian formalism. But these 2 statements are equivalent. So even if there are different starting assumptions, shouldn't they be equivalent to one another. I would have guessed that at least the number of axioms would be the same, which is not the case here. Shouldn't there be, at least in theory, a basic list of axioms which form the philosophy behind the whole theory? Different people may state them differently, but shouldn't the content at least be the same?
For example the orthogonality of eigenstates given as a postulate in some books is not a basic postulate (correct me if i am wrong). It is a result of the postulate that the wave function satisfies Schrodinger equation. Can't we exclude them from the list?
Of all the lists I have seen, I felt that the one given by Shankar (mentioned in starting post) to be the most basic one, but it still feels to me like we are laying down the mathematics and not the theory. I don't claim to be an expert, I am just asking.

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Get a copy of Ballentine and read the the first three chapters
I have to get that book. Have heard many things about it (good and bad).
Its really from just two axioms - Schrodinger's equation etc comes from symmetry - which is itself profound.
Interesting. Can you tell me what the 2 axioms are?

bhobba
Mentor
Interesting. Can you tell me what the 2 axioms are?
Axiom 1
Associated with an observation we can find a Hermitian operator O, called the observations observable, such that the possible outcomes of the observation are its eigenvalues.

Axiom 2 - called the Born Rule
Associated with any system is a positive operator of unit trace, P, called the state of the system, such that expected value of of the outcomes of the observation is Trace (PO).

And the link I gave basically derives axiom 2 from axiom 1.

Thanks
Bill

cgk
I, too, find the axioms very confusing, because of this:
[...]The above postulates are the framework of quantum mechanics. For any particular system, additional postulates are needed to specify the Hilbert space, observables and the Hamiltonian. [...]
It seems to me like all the postulates are entirely pointless, because they cannot be used to describe anything without the additional input given here. bhobba's two axioms are a good example for this: So we have axiomatized that observables are represented by hermitian operators, and that the physical state can also be, to some degree, specified by operators. How does this define any physics? Given an atom with charge Z and two electrons, what would these two axioms tell you about how to come up with its energy spectrum? It also sounds like one could make up countless dynamical and non-dynamical systems which observe these two axioms and which are not the quantum mechanics we know.

So if situation-specific additional input is required in any case (and this input is highly non-trivial! Not just because of the Hamiltonian/forces, also because of the state space), why even bother with axioms at all? Is this really better than just giving the time-dependent Schrödinger equation, the Born rule, and some heuristic rules on how to make its Hamiltonian and state space, and saying "here, this is what we experimentally found to work"?

 (and let's not even get into the fundamental mathematical issues. Like "eigenvalues" of non-compact Hermitian operators...)

• TrickyDicky
atyy
It seems to me like all the postulates are entirely pointless, because they cannot be used to describe anything without the additional input given here.
How about the analogy I gave, that it's like Newton's 3 laws? Newton's 3 laws also cannot describe anything without additional input. In particular, one has to specify the form of F, eg. $F= GMm/r^{2}$ for gravity, or $F=\mu N$ for static friction. In the same way, the Hilbert space, observables and Hamiltonian have to be specified by additional postulates for each particular application of the quantum mechanical framework.

An alternative is to just add the Lagrangian of the standard model of particle physics :D (just kidding, physicists and experiments believe in it, but there are some mathematical problems with axiomatizing the standard model, an issue best discussed elsewhere ...)

OK, but why axioms at all? Are the axioms just a type of language? Yes, they are. They are a type of probabilistic language that is self-consistent. The axioms of classical probability are Kolmogorov's axioms. Quantum mechanics does not obey the Kolmogorov axioms, but interfaces consistently with them. There are ways to make at least some forms of quantum mechanics conform completely to Kolmogorov's axioms, eg. Bohmian Mechanics can be (notionally) taken as a more fundamental theory from which non-relativistic quantum mechanics can be derived as an effective theory. However, in most cases, many more degrees of freedom have to be introduced, and the degrees of freedom and their dynamics are not unique. Also, it is not known whether we can "Kolmogorize" or "classicalize" all forms of quantum mechanics. So at the moment, we stick to quantum mechanics without additional degrees of freedom.

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bhobba
Mentor
It seems to me like all the postulates are entirely pointless, because they cannot be used to describe anything without the additional input given here. bhobba's two axioms are a good example for this: So we have axiomatized that observables are represented by hermitian operators, and that the physical state can also be, to some degree, specified by operators. How does this define any physics?
The dynamics follows from symmetry considerations - specifically those implied by Galilean relativity - see chapter 3 - Ballentine.

That symmetry underlies the dynamics is one of the most beautiful and profound revelations of modern physics. Wigner got a Nobel prize for it and its spirit now permeates all of physics:
http://www.pnas.org/content/93/25/14256.full

To really appreciate it you need to see a treatment of classical mechanics based on symmetry:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20
'If physicists could weep, they would weep over this book. The book is devastingly brief whilst deriving, in its few pages, all the great results of classical mechanics. Results that in other books take take up many more pages.'

Thanks
Bill

bhobba
Mentor
Newton's 3 laws also cannot describe anything without additional input.
Yes.

Newtons laws, just like the dynamics in QM, also follows from symmetry. They put constraints on that additional input.

In QM the symmetries are applied to the axioms I gave. In classical mechanics its applied to the principle of least action (PLA) which follows from the two axioms via the sum over histories approach.

When I first came across this stuff it had a deep effect on me, as I think most people exposed to it feel.

Thanks
Bill

cgk
How about the analogy I gave, that it's like Newton's 3 laws? Newton's 3 laws also cannot describe anything without additional input. In particular, one has to specify the form of F, eg. $F= GMm/r^{2}$ for gravity, or $F=\mu N$ for static friction. In the same way, the Hilbert space, observables and Hamiltonian have to be specified by additional postulates for each particular application of the quantum mechanical framework.
That is not quite the same thing, because there are really only two kinds of fundamental forces which occur in classical physics: Electromagnetic ones and gravitation. Everything else can, technically, be derived from those and the Newton axioms using microscopic physics (and possibly some additional axioms regarding statistical physics). That is: In some sense, the Newton axioms give a complete account of classical mechanics physics at the microscale, while the QM postulates do not.

The dynamics follows from symmetry considerations - specifically those implied by Galilean relativity - see chapter 3 - Ballentine.
Your two axioms contained no mention of symmetry, or of how to apply it to derive dynamical equations. I am of course aware of how symmetry can be used to motivate (not derive!) various equations. But my point is: If you want to use it in a consistent manner in an axiomatic system, you better make axioms for it! In an axiomatic system you cannot just summon additional information from elsewhere when needed. The complete rules of the game must be given!

 Or the state space and its restrictions. I'd be most interested in how to derive the spin-statistics theorem from Gallilean(!) symmetry. Especially since it is absolutely essential in describing all real-world matter. Of course, one could also just postulate the anti-symmetry of the Fermionic wave function... (and the form of how it is coupled to other degrees of freedom...)

bhobba
Mentor
Your two axioms contained no mention of symmetry, or of how to apply it to derive dynamical equations. I am of course aware of how symmetry can be used to motivate (not derive!) various equations.
It is derived - not motivated eg Schroedinger's equation follows. However there are deep issues of how you apply it due to the operator ordering issue. Even the deepest formalism of geometrical QM doesn't resolve that.

By definition classical mechanics is when Galilean relativity applies and you end up with standard QM, Schroedinger's equation etc. By definition relativistic mechanics is when the Lorentz transforms apply and you get QFT. In either case the two axioms apply.

Thanks
Bill

bhobba
Mentor
That is not quite the same thing, because there are really only two kinds of fundamental forces which occur in classical physics: Electromagnetic ones and gravitation
Hmmm, frictional forces, Van Der Wall forces come to mind. Then we have the forces of solidity itself that results from the Pauli exclusion principle.

Why, in classical mechanics, are fundamental forces conservative? It is the classical limit of Schrödinger's equation which is constrained by symmetry to be of that form.

Thanks
Bill

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bhobba
Mentor
Or the state space and its restrictions. I'd be most interested in how to derive the spin-statistics theorem from Gallilean(!) symmetry. Especially since it is absolutely essential in describing all real-world matter. Of course, one could also just postulate the anti-symmetry of the Fermionic wave function... (and the form of how it is coupled to other degrees of freedom...)
You cant - that requires QFT which is based on Einsteinian relativity. But the two axioms still apply.

Thanks
Bill

Nugatory
Mentor
Hmmm, frictional forces, Van Der Wall forces come to mind. Then we have the forces of solidity itself that results from the Pauli exclusion principle.
I expect that most people most of the time would consider all of these to be manifestations of the electromagnetic force. The exclusion principle does control the configuration of electrons around the nucleus, but (until you're in the non-classical realm of degenerate matter) solidity still comes from the electromagnetic interactions between these electrons.

bhobba
Mentor
The exclusion principle does control the configuration of electrons around the nucleus, but (until you're in the non-classical realm of degenerate matter) solidity still comes from the electromagnetic interactions between these electrons.
From dim memories of chemistry classes my recollection is Van Der wall forces are at least partly caused by quantum effects of electron sharing. Again I dimly remember friction is partly caused by adhesion which also depends on things like electron sharing. But I would really like to hear the views of those more conversent in such things.

I am a bit more familiar with the solidity issue. My understanding is Dyson showed solidity is due to quantum degeneracy pressure rather than electrostatic repulsion.

Thanks
Bill

Nugatory
Mentor
From dim memories of chemistry classes my recollection is Van Der wall forces are at least partly caused by quantum effects of electron sharing. Again I dimly remember friction is partly caused by adhesion which also depends on things like electron sharing. But I would really like to hear the views of those more conversent in such things.

I am a bit more familiar with the solidity issue. My understanding is Dyson showed solidity is due to quantum degeneracy pressure rather than electrostatic repulsion.
You may be right... I'm relying on my own dim memories as well. I'll shut up now and wait for someone who knows more to speak up.

bhobba
Mentor
You may be right... I'm relying on my own dim memories as well. I'll shut up now and wait for someone who knows more to speak up.
I think its wise for me to do the same - its really getting into stuff way off the beat and track to my usual interests.

A point needs to be made as well, and I should have said it right from the outset - saying that QM follows from two axioms and you will find the detail in Ballentine - its the usual physicists conception of such things which is a lot different to a mathematicians view. Such a treatment can be found in say - Varadarjan - Geometry Of Quantum Theory. Studying that text though leads to a whole heap of hurt - its as the euphemism says - mathematically non trivial. I know - I tried reading it one time (even bought my own copy) - groan.

Thanks
Bill

cgk
On Pauli repulsion: Pauli repulsion/the "exchange interaction" is indeed what causes the solidity of matter---but it is not real. The exchange interaction is a particular aspect the Coulomb interaction for Fermionic systems. This is why I insisted that the anti-symmetry of the wave function is essential for all real-life matter, even in the strongly non-relativistic case. Without it you can't even describe a single Helium atom in empty space. And this is one of the (many) things which simply cannot be derived from the two axioms given before.

On van der Waals forces: Some are electromagnetic (static multipole interactions), some are covalent (H-bonds), some are, basically, classically electrodynamic with a quantum touch (London dispersion forces/Casimir forces).

Maybe I should not have said that essentially all of classical mechanics can be derived from gravitational and electromagnetic forces. One can also make up additional forces (e.g., harmonic spring forces or Lennard-Jones forces), which are then considered as empirical "effective" forces (resulting forces from some lower-level microscopic interactions, possibly to some perturbative order). But, in any case, you *only* need to specify the force law (either microscopic or effective) in order to make sense of Newton's axioms and derive lots of physics from it, without necessarily invoking additional axioms.

And this is precisely what the two quantum mechanical "axioms" do not allow. They do not provide any information on how to describe any kind of real physical system using them. Not even the free point particle in empty space. bhobba said that QM can be derived from these axioms alone, and later specified that symmetry laws need to be invoked additionally. My point was that symmetry laws cannot be invoked unless they are axiomized, too---if other information can be invoked at will, there is no point in using axioms to begin with. But even with the additional symmetry laws, I do not see how the axioms could possibly be nearly enough to fully specify quantum mechanics to any useful level. That there must be some ambiguity should follow even from the fact alone that classical physics, free-particle quantum mechanics, interacting-particle quantum mechanics, and various forms of both interacting and non-interacting quantum field theories can all be formulated in a way which is compatible with those axioms, and all of them imply different behavior. And these are only the theories physicists actually consider to make sense in some way---there are many QFT Lagrangians compatible with the symmetries one could possibly write down (including many which do not correspond to the dynamics of any real quantum mechanical system). How do you know which one is right?

 Assuming them is anything but trivial in my opinion. Why axiomatize a bunch of complicated symmetry laws and ways of how to invoke them (which, with many additional assumptions allow one to propose forms of dynamic laws which are compatible with them), if one could just as well axiomatize the dynamic law itself (the Schrödinger equation), from which the symmetry laws can then be derived without any complications? We do the same thing in classical mechanics, too, and it seems much more straight-forward to me, personally.

• bhobba and ShayanJ
atyy
And these are only the theories physicists actually consider to make sense in some way---there are many QFT Lagrangians compatible with the symmetries one could possibly write down (including many which do not correspond to the dynamics of any real quantum mechanical system). How do you know which one is right?
I don't think we actually disagree on anything, since it was my own post which first stated in this thread that the axioms given must be filled in with specifics about the Hilbert space, observables and Hamiltonian. Also, certainly in classical physics, it is the equation of motion that is the real axiom because of its uniqueness, while the Lagrangian corresponding to a given equation of motion is not necessarily unique - classical general relativity furnishes several examples, eg. the Hilbert, Palatini and Holst actions all give rise to the same field equations.

However, I think it is important that in principle we do believe that almost all observations (except massive neutrinos and dark matter) are captured by the standard model Lagrangian which is specified by symmetry, and that within the framework of Wilson's renormalization group, one writes down every term consistent with the given symmetry.

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dextercioby
Homework Helper
There's a list of axioms that can be given, so that a connection between functional analysis and physics is done easily. It's precisely known as the "Dirac quantization" or "canonical quantization" and tells people that one needs to have a classical Hamiltonian description of a system, then "quantize" it using a certain prescription:

$$[,]_{PB} \mapsto \frac{1}{i\hbar} [,]$$

where on the left hand side one has the Poisson bracket of any Hamiltonian observables, while on the right one has the Hilbert space commutator of the operators which describe the observables in the formalism of QM.

But this "canonical quantization" has its limitations, as shown by Groenewold and van Hove at the end of the 1940s, http://arxiv.org/abs/math-ph/9809015v2

bhobba
Mentor
My point was that symmetry laws cannot be invoked unless they are axiomized
OK - it looks like we are going to have to disagree on this.

My point is by definition classical mechanics is defined by the symmetries of the Galelian transformation ie time is an absolute ie non-locality is assumed from the start. That is what leads to Schroedinger's equation etc. The reason, for me, it's not an axiom is its the definition of what classical dynamics is. You can assume the Lorentz transformations and get QFT. The two axioms are the same regardless.

Thanks
Bill