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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 $\hat ... 34 Observables don't commute if they can't be simultaneously diagonalized, i.e. if they don't share an eigenvector basis. If you look at this condition the right way, the resulting uncertainty principle becomes very intuitive. As an example, consider the two-dimensional Hilbert space describing the polarization of a photon moving along the$z$axis. Its ... 27 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 ... 18 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 ... 16 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 ... 13 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 ... 12 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 ... 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)$. Obviously, the boundary condition requires$J_0(\lambda)=0$.... 11 This is a rough version of the summary of the complete answer I posted on math.stackexchange, where more details are discussed in a long digression, in particular mathematical motivations for your points 1., 3. and 4. (Any reader interested in more explanations and a longer updated list of references should check out that other answer in Math.SE). I eagerly ... 11 Quantum mechanics is indeed a probability theory, but it is a non-commutative probability theory. So it is not just a matter of having signed/complex measures, but really of having a non-commutative probabilistic framework. Quantum mechanics was developed, historically, before non-commutative probability theories and I think that people in probability ... 10 Mass-squared is a Hermitian linear operator, it's a Casimir operator$\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}$are Hermitian. It commutes with all the generators of the Poincare group and so it's eigenvalues (mass-squared) are constant on each ... 10 Several reasons: Orthogonal functions arise naturally in the study of Sturm-Liouville theory which includes many classical and quantum system mathematical models; More generally, it is the class of normal operators (and an important special case self adjoint operators) which the spectral theorem most readily works and is most complete for. The eigenvectors ... 9 Of course, mass is an observable, although in simple models it is constant. This is already the case classically. One cannot determine the path of as rocket that burns fuel (which forms a large fraction of its mass) without taking into account that the mass is variable. The same holds in quantum mechanics, whenever the mass is not fixed by the modeling ... 9 Every observable in the technical or mathematical sense (linear Hermitian operator on the Hilbert space) is, in principle, observable in the physical operational sense, too. That's why it's called this way. Magnetic fields may be measured, for example, by compasses. Analogous methods exist for electric fields, scalar fields, or any other fields. For example,... 9 OP asks: Is there any physical meaning to this? Yes, the Pauli matrix$\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$can no longer see the sign, so it only has one ... 9 You could certainly model any one quantum observable as a random variable. The problem comes in when you have multiple observables, which you might attempt to model as classical random variables with some joint distribution. From this joint distribution, you can compute various probabilities (like$\textrm{Prob}(Y\neq X)$, for example), according to the ... 8 I'll just supplement Prof. Kalitvianski's answer by adding that the mantra « observables as operators » only applies to the special kind of measurement called a « quantum measurement », which always involve amplification. Other kinds of measurement, such as measuring physical constants, are not modelled by observables either, like the speed of light, the ... 8 As Lubos has mentioned$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\}$is unitless quantity ) In QM observables are required to be hermitian operators (so that they can have real eigenvalues). In ... 8 If I try to measure the field at one point in spacetime, I should get a real value which should be an eigenvalue of the quantum field, right? I guess the eigenvectors of the quantum field also live in Fock space? Yes, that's basically correct. If the value of the field at a point is observable, the eigenvalues of the operator representing it are the ... 8 There is quite a lot of very important information hidden in the term hermitian. For an operator$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$. Moreover, one can show that the eigenvalues corresponding to these eigenvectors ... 8 The reason operators correspond to measured values has to do with what happens when you connect a measurement apparatus to the system under observation. Suppose the Hamiltonian of the system by itself is$H_S$and the Hamiltonian of the measurement apparatus by itself is$H_M$. When$M$is physically connected to$S$, we get a additional "interaction" term$...

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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 ...

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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 ...

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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. ...

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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 ...

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I'm going to try to explain why and how density operators in quantum mechanics correspond to random variables in classical probability theory, something none of the other answers have even tried to do. Let's work in a two-dimensional quantum space. We'll use standard physics bra-ket notation. A quantum state is a column vector in this space, and we'll ...

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There doesn't exist any procedure to uniquely associate a Hermitian operator $L$ to a function of the phase space $f(x,p)$. Quantum mechanics is a theory that exists independently of classical physics. Quantum mechanics is not just a cherry on a classical pie that needs the classical theory to exist at every moment. If we want to define a quantum theory, we ...

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