$\newcommand{\braket}[2]{\left<#1|#2\right>}% \newcommand{\bra}[1]{\left<#1\right|}% \newcommand{\ket}[1]{\left|#1\right>}% $As an undergraduate who's trying to learn quantum mechanics it's frustrating to not be able to distinguish between what is a postulate and what is additional information. Griffiths's Introduction to Quantum Mechanics, for example, does a poor job of making a distinction. I'm reading Quantum Mechanics by McIntyre and he does a good job of explicitly outlining postulates but they seem to lack information. For instance, I'm quite sure that operators representing observables ought to be linear.

I think it would be constructive to have on this page a complete set of quantum mechanics postulates that lack no information and have no additional information so please contribute to this if you're knowledgeable on the subject. Let's build off of McIntyre's postulates:

  1. The state of a quantum mechanical system, including all the information you can know about it, is represented mathematically by a ket $\ket{\psi}$.
  2. A physical observable is represented mathematically by an operator $A$ that acts on kets.
  3. The only possible result of a measurement of an observable is one of the eigenvalues $a_n$ of the corresponding operator $A$.
  4. The probability of obtaining the eigenvalue $a_n$ in a measurement of the observable $A$ on the system in the state $\ket{\psi}$ is \begin{equation} P_{a_n} = |\braket{a_n}{\psi}|^2, \end{equation} where $\ket{a_n}$ is the normalized eigenvector of $A$ corresponding to the eigenvalue $a_n$.
  5. After a measurement of $A$ that yields the result $a_n$, the quantum system is in a new state that is the normalized projection of the original system ket onto the ket (or kets) corresponding to the result of the measurement: \begin{equation} \ket{\psi^\prime} = \frac{P_n\ket{\psi}}{\bra{\psi}P_n\ket{\psi}}. \end{equation} The time evolution of a quantum system is determined by the Hamiltonian or total energy operator $H(t)$ through the Schrödinger equation \begin{equation} i\hbar\frac{d}{dt}\ket{\psi(t)} = H(t)\ket{\psi(t)}. \end{equation}
  • $\begingroup$ Correction to 4: that's the probability of being in that eigenstate, which may be less than the probability of the eigenvalue (it may be "degenerate", i.e. a repeated eigenvalue). See e.g. en.wikipedia.org/wiki/Degenerate_energy_levels $\endgroup$ – J.G. Apr 12 '18 at 10:54
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    $\begingroup$ I imagine you would want to add the tensor product postulate. Besides that it looks fine to me! $\endgroup$ – knzhou Apr 12 '18 at 11:01
  • $\begingroup$ have a look here hyperphysics.phy-astr.gsu.edu/hbase/quantum/qm.html , second page. I think the language is geared towards measurments rather than abstract functions only. $\endgroup$ – anna v Apr 12 '18 at 11:05
  • $\begingroup$ See Wikipedia. $\endgroup$ – Qmechanic Apr 12 '18 at 12:54
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    $\begingroup$ I disagree, it's simply not explicit. $\endgroup$ – user113773 Apr 17 '18 at 2:24

Here's a list of axioms for QM :

  1. The theory is described by a separable complex Hilbert space $\mathcal H$.
  2. Physical states are represented by rays in that Hilbert space (an equivalence class of vectors of finite norm related by a phase)
  3. Physical observables are represented by self-adjoint linear operators
  4. Given a set of commuting observables $\{A_i\}$, they define a Kolmogorov probability theory $(\Omega, \Sigma, P)$ where $\Omega$ is the sample space (which is the projective Hilbert space), $\Sigma$ the $\sigma$-algebra and $P$ the probability, such that the observables $A_i$ define random variables $X_i$ with probabilities of measuring the value in $E \in \sigma(A_i)$, the spectrum of the observable, defined by the projection-valued measure $$P(X_i \in E) = \mu^{A_i}_\psi(E) = \langle \psi, \mu^{A_i}(E) \psi \rangle$$The details on how to define $\mu^A$ are part of spectral theory.
  5. The old chestnut about time evolution where $$i\hbar \partial_t \psi = \hat H(t) \psi$$for some self-adjoint operator $\hat H$.

I think that should be enough to have the whole theory defined properly, although there are subtleties to consider (the collapse of the wavefunction corresponds to the fact that if the observables do not commute we have to use a different probability theory and apply it to the state as we measured it using conditional probabilities). In particular :

  • Observables cannot be any operators, since they need to have a real spectrum.
  • Observables do not necessarily have eigenvalues.

This does not include everything related to symmetries, the spacetime underlying the theory, commutation relations, superselection rules and so forth, as those come closer to being more about specific quantum theories, but I can include them if necessary.

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