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Let us first restate the mathematical statement that two operators $\hat A$ and $\hat B$ commute with each other. It means that $$\hat A \hat B - \hat B \hat A = 0,$$ which you can rearrange to $$\hat A \hat B = \hat B \hat A.$$ If you recall that operators act on quantum mechanical states and give you a new state in return, then this means that with $\hat ... 32 Observables do not commute if they can't be simultaneously diagonalized, i.e. if they don't share an eigenvector basis. If you look at this condition the right way, the resulting uncertainty principle becomes very intuitive. As an example, consider the two-dimensional Hilbert space describing the polarization of a photon moving along the$z$axis. Its ... 25 There is a fair amount of background mathematics to this question, so it will be a while before the punch line. In quantum mechanics, we aren't working with numbers to represent the state of a system. Instead we use vectors. For the purpose of a simple introduction, you can think of a vector as a list of several numbers. Therefore, a number itself is a ... 16 UPDATE: the insights and conceptual meanings advocated below in this answer are detailed and technically developed, for example, in these wonderful articles: Fuchs; Peres - Quantum Mechanics Needs No Interpretation, Physics Today (2000), vol. 53, issue 3, p. 70. Englert - On Quantum Theory, Eur. Phys. J. D (2013) 67: 238. Duvenhage - The Nature of ... 15 One problem with the given$3\times 3$matrix example is that the eigenspaces are not orthogonal. Thus it doesn't make sense to say that one has with 100% certainty measured the system to be in some eigenspace but not in the others, because there may be a non-zero overlap to a different eigenspace. One may prove$^{1}$that an operator is Hermitian if and ... 14 Time is not a variable in Quantum Mechanics (QM), it's a parameter — much in the same way as it is in Classical (Newtonian) Mechanics. So, if you have a Hamiltonian, e.g., for the harmonic oscillator, you have$\omega$as a parameter, as well as the masses of the particle(s) involved, say$m$, and you also have time — even though it's not something that ... 13 In non-relativistic quantum mechanics the mass can, in principle, be considered an observable and thus described by a self-adjoint operator. In this sense a quantum physical system may have several different values of the mass and a value is fixed as soon as one performs a measurement of the mass observable, exactly as it happens for the momentum for ... 12 There are observables corresponding to the light going through both slits. You can write down a basis: "it went through slit A + slit B", and "it went through slit A - slit B". Although maybe you can't detect these observables easily with an experiment, they're perfectly good observables, they're orthogonal, and a clever enough experiment should be able to ... 11 A simple example of non-commutativity is rotations in 3D, cf. figure. Physically, the rotations around the$x$- and the$y$-axis are generated by angular momentum operators$\hat{L}_x$and$\hat{L}_y$, respectively, which do not commute. From the mathematical expressions for$\hat{L}_x$and$\hat{L}_y$, you may proceed with the mathematical derivation, ... 11 I don't have a complete answer, but maybe the following is useful for your purposes: Consider the Laplacian$\Delta$on a circular drum of unit radius. As explained on the wikipedia page, the axially symmetric eigenvectors$\Delta u(r) = -\lambda^2 u(r) $are Bessel functions$u(r)=J_0(\lambda r)$. Obviously, the boundary condition requires ... 10 This is a rough version of the summary of the complete answer I posted on math.stackexchange, where more details are discussed in a long digression, in particular mathematical motivations for your points 1., 3. and 4. (Any reader interested in more explanations and a longer updated list of references should check out that other answer in Math.SE). I eagerly ... 10 Mass-squared is a Hermitian linear operator, it's a Casimir operator$\hat{C}_{1}=\hat{P}_{0}\hat{P}_{0}-\hat{P}_{i}\hat{P}_{i}$for the Poincare group. It's Hermitian because the translation generators$\hat{P}_{\mu}$are Hermitian. It commutes with all the generators of the Poincare group and so it's eigenvalues (mass-squared) are constant on each ... 10 Several reasons: Orthogonal functions arise naturally in the study of Sturm-Liouville theory which includes many classical and quantum system mathematical models; More generally, it is the class of normal operators (and an important special case self adjoint operators) which the spectral theorem most readily works and is most complete for. The eigenvectors ... 10 Color charge in the sense of "being blue, red, green" is not a quantum mechanical observable because the$\mathrm{SU}(3)$gauge transformations mix the colors. This means it is meaningless to say "We have a blue particle", because we can perform a gauge transformation and then we "have a red particle". Since physical descriptions related by gauge ... 9 Every observable in the technical or mathematical sense (linear Hermitian operator on the Hilbert space) is, in principle, observable in the physical operational sense, too. That's why it's called this way. Magnetic fields may be measured, for example, by compasses. Analogous methods exist for electric fields, scalar fields, or any other fields. For ... 9 Of course, mass is an observable, although in simple models it is constant. This is already the case classically. One cannot determine the path of as rocket that burns fuel (which forms a large fraction of its mass) without taking into account that the mass is variable. The same holds in quantum mechanics, whenever the mass is not fixed by the modeling ... 9 OP asks: Is there any physical meaning to this? Yes, the Pauli matrix$\sigma_j$represents (up to a proportionality factor) the spin in the$j$th direction of a spin$\frac{1}{2}$system. Such system has only two spin states:$\uparrow$and$\downarrow$, with opposite eigenvalues. The square$\sigma_j^2$can no longer see the sign, so it only has one ... 8 There is quite a lot of very important information hidden in the term hermitian. For an operator$A$on a finite-dimensional Hilbert space$\mathcal H$, one can show that there exists an orthonormal basis for the Hilbert space consisting of eigenvectors of the operator$A$. Moreover, one can show that the eigenvalues corresponding to these eigenvectors ... 8 As Lubos has mentioned$QP-PQ=i\hbar$is one of the basic requirements of quantum mechanics. Classically observables are functions of variables$q$, and$p$and Poisson bracket relation read$\{q,p\}=1$(note that$\{q,p\}$is unitless quantity ) In QM observables are required to be hermitian operators (so that they can have real eigenvalues). In ... 8 I'll just supplement Prof. Kalitvianski's answer by adding that the mantra « observables as operators » only applies to the special kind of measurement called a « quantum measurement », which always involve amplification. Other kinds of measurement, such as measuring physical constants, are not modelled by observables either, like the speed of light, the ... 8 The reason operators correspond to measured values has to do with what happens when you connect a measurement apparatus to the system under observation. Suppose the Hamiltonian of the system by itself is$H_S$and the Hamiltonian of the measurement apparatus by itself is$H_M$. When$M$is physically connected to$S$, we get a additional "interaction" term ... 8 Not equal, but equivalent, in the sense that they have the same effect on the wavefunction in question. More precisely, the book is using a slight abuse of terminology. Taking momentum as an example, it's not really the case that the dynamical quantity of momentum is equivalent to the operator$\frac{\hbar}{i}\frac{\partial}{\partial x}$, because a number ... 8 Non commuting observables means that a so called measurement is capable of changing the state of the system. For instance, when there are two observables A and B that fail to commute, there is an eigenvector of A that it is not an eigenvector of B. When you interact with A then A then B the two results of the A interaction always agree with each other. ... 7 This is a supplement to freude's correct answer: Hamiltonian is the infinitesimal generator of time translation defined as $$\mathrm{\hat{U}}(\mathrm dt)= 1- \frac{i}{\hbar} \mathrm{\hat{H}}(t)\mathrm dt\;.$$ Time-Evolution Operator: Let the system be at$|\phi\rangle\;.$Now, let's wait for some time..... What is the probability amplitude of finding ... 7 In QM, a real valued observable$A$is mathematically represented by a projector valued measure over$\mathbb{R}$,$P^{(A)}$, i.e., if$E$is a Borel subset of the real line, then$P^{(A)}(E)$is a projector representing the proposition "the outcome of measuring$A$falls in$E$". In principle, that's all you need for representing, mathematically, ... 7 The eigenvectors of a quantum field are states with a definite value of the field: $$\hat{\phi}(x) \left| \Phi \right> = \phi(x) \left| \Phi \right>$$ These are not states which have a definite number of particles, i.e., they are superpositions of Fock space states with different numbers of particles. The easiest way to see this is to write the ... 7 If I try to measure the field at one point in spacetime, I should get a real value which should be an eigenvalue of the quantum field, right? I guess the eigenvectors of the quantum field also live in Fock space? Yes, that's basically correct. If the value of the field at a point is observable, the eigenvalues of the operator representing it are the ... 7 It's a very good question but the answer is No, there is nothing such as "it went through both slits" observable (i.e. no linear operator that would correspond to this Yes/No question). The reason is that such "information" cannot be observed, not even in principle and not even statistically. Much more generally, there don't exist any observables that would ... 7 The search for a quantum mechanical theory could be done in a mathematical systematic fashion, and it starts from observables. So the answer to the OP last question is that the process is inverted: the relevant observables are given first (and as we will discuss below, they are justified by observations); then you find the space of states where these ... 6 There is no time operator in quantum mechanics. At least, there's no nontrivial time operator. You could have an operator whose action is just to multiply a function by$t\$, but time is a parameter in QM, so the operator will never do anything more complicated than that. Its eigenfunctions wouldn't be terribly useful either because they would just be delta ...