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68

The theory of probability used in QM is intrinsically different from the one commonly used for the following reason: The space of events is non-commutative (more properly non-Boolean) and this fact deeply affects the conditional probability theory. The probability that A happens if B happened is computed differently in classical probability theory and in ...


29

The decay phenomenon is a purely quantum mechanical property. This problem is equivalent to a particle in a finite potential well, and a lower potential state that is available outside the well. Classically if the energy of the particle in the well is lower than the potential barrier - it will never get to the lower state. By quantum mechanics, the particle ...


28

Your reasoning demonstrates precisely why formal logic alone is insufficient to study nature. In particular, it lacks the ingredient of inductive inference that is a cornerstone of empirical science. A cosmic ray striking the Earth is not some random act of the gods that can have any imaginable consequence whatsoever. It is a cosmic ray striking the Earth. ...


25

A particle isn't really a point particle; its position is best described by a wavefunction: the probability if finding it in any particular region in space. For annihilation to occur, the wavefunctions of the two particles has to overlap: to the extent that they overlap, there will be a probability that annihilation can occur. The greater the overlap, the ...


21

This question gets to the heart of what makes quantum mechanical amplitudes different from classical probabilities. It is true that if you make measurements in the basis of states $\{\lvert 00 \rangle ,\lvert 11 \rangle\}$ then the two states have the same measurement statistics, and so cannot be distinguished. The interesting thing is that it is possible to ...


16

Rigorously speaking, the probability to find the electron at a distance exactly equal to $r$ from the nucleus is $0$. On the other hand, we can define the probability to find the electron in a volume $d \mathbf{r}$ as $$P(d \mathbf r) =| \Psi(\mathbf r)|^2 d \mathbf r = |\Psi(r,\theta,\phi)|^2 r^2 \sin \theta \ dr \ d\theta \ d \phi$$ where I have ...


15

This question was studied fairly recently by a team at Edinburgh University. Their paper is available here, though I'm not sure if you can get it without having to hand over some cash. The bottom line is that in principle the trajectory of a die can be calculated, but it is a chaotic system and that means tiny inaccuracies in the measured initial conditions ...


15

Speaking loosely, each individual atom has a desire to become stable, but that translates into a probability of decaying. This means, since there are billions and billions of atoms in a macroscopically significat chunk of material, that there are always going to be unlikely holdouts, and these holdouts are responsible for radiation that after the initial ...


14

A wave function is a complex-valued function $f$ defined on ${\mathbb R}^1$ (if your electron is confined to a line) or on ${\mathbb R}^2$ (if your electron is confined to a plane) or ${\mathbb R}^3$ (if your electron ranges over three-space), and satisfying $$\int |f|^2=1$$ (where the integral is defined over the entire line or plane or 3-space). Every ...


13

The total probability of all mutually exclusive alternatives must always be 100%, so it is conserved. The conservation law in the spacetime tend to be "local", so just like for the charge conservation, we may derive the conservation of the probability in Schrödinger's equation from the local continuity equation $$ \frac{\partial \rho}{\partial t} + \mathbf \...


13

If you look closely, you will find that the event "find the particle at $x$" always has zero probability of occuring: Since $\rho(x) = \lvert \psi(x) \rvert ^2$ is a probability density, it must be integrated over some subset of $\mathbb{R}$ to actually gain a probability. The probability to find the particles at one of the positions in a subset $S \subset \...


12

Another use of the SLE approach seems to be (I haven't read the papers below much beyond their abstracts) as a tool to probe for the presence of conformal invariance in various systems, when a direct (numerical or experimental) verification is difficult. In this approach, (i) one extracts suitable non self-crossing paths, (ii) one determines (empirically) ...


12

The theory of quantum mechanics has been developed to explain observations, i.e. measurements. Without observations it is a floating mathematical construct. One of the postulates to connect the mathematics with reality is: To every observable there corresponds an operator which operating on the state function will give an eigenvalue. So the question ...


11

Probabilites are (squares of) probability amplitudes, which can be obtained as inner products of vectors on Hilbert space: $\langle X|Y \rangle$. Under a transformation U, the ket transforms as $$|Y\rangle \rightarrow |Y'\rangle = U |Y\rangle$$ and the bra as $$\langle X| \rightarrow \langle X'| = \langle X| U^{\dagger}$$ So if U is unitary, ...


11

SLEs can be used in order to define a certain kind of QFT. You can check M. Douglas' talk, from page 28 forward: Foundations of Quantum Field Theory (PDF). There's also another great article, Conformal invariance and 2D statistical physics. You may also like SLE and the free field: Partition functions and couplings. Finally, there's an approach to try and ...


11

Having given it some more thought, there is an unambiguous philosophical difference, with practical implications. The two-slit experiment provides a good example of this. In a classical universe, any particular photon that hits the screen either went through slit A or slit B. Even if we didn't bother to measure this, one or the other still happened, and ...


11

In position-space (that is, when your functions are functions of x), the function $\int|\Psi|^2$ gives the probability of finding the particle in a given range. The expectation value of x is where you'd expect to find the particle. It is often essentially the weighted average of all the positions where the probability density, $|\Psi|^2$, is the weighting ...


11

What I don't get is how he ended up with an equation that is a probability distribution. He ended up with Schroedinger's equation, but he did not think it describes probability. He thought more along the lines of de Broglie idea, that the electron is some kind of wave, and $\psi$ - solution to the equation - expresses mathematically shape and other ...


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

With quantum mechanics, you have to ask your questions very, very carefully. Is it possible to have a Sun-sized star in your pocket? It depends on what you mean by "in". Do all of the atoms of the star need to be entirely in your pocket, or is it sufficient that some part of each atom's wave function be inside your pocket? It is possible to have any ...


10

Matty Hoban pointed me to a paper (PDF here) by Itamar Pitowsky from 1991 which looks the geometry of correlation polytopes and their symmetries. I haven't read the paper in full, but glancing through it, on page 400 (page 6 of the actual paper) under the statement of results the author seems to say that the cardinality of the symmetry group is $n! 2^n$ ...


10

This relationship is called the Born rule and it's one of the postulates of quantum mechanics. There have been various attempts to derive it, but none of them having been terribly convincing so at the moment we have to assume it is true. Fortunately experiment supports this assumption. The rule was originally suggested by Max Born (hence the name) in 1926 ...


10

You might like this 110-page paper by me and Alex Arkhipov, which is all about a quantum bosonic analogue of Galton's board (we even use the same graphic you did -- see Section 1.1!). In particular, we gave strong evidence that such a board (with an arbitrary configuration of "pegs," and with multiple entry points for the "balls") is exponentially hard even ...


9

There are two that I know of in the context of state estimation. The first is for estimating the mean of $P$ and is a Metropolis-Hasting MCMC algorithm here: Optimal, reliable estimation of quantum states. The second is also mainly for computing the mean (but can do other functions -- including the characteristic function of the region you are interested ...


9

I think a reasonable first approximation can be made like this: choose an arbitrary orientation for the die, and figure out, if the die were released in that orientation with its lowermost point resting on a surface, which side would it fall on? That can be easily calculated; you just draw a line going straight down from the center of mass, and whichever ...


9

This is something that particle physicists are perfectly well aware of. For any given observed effect, there is always a nonzero probability that the observation will be a false positive that was caused by a random fluctuation. The name of the game is taking enough data that this probability is small enough. In general, the more data you take, the less ...


9

$\mathfrak{R}e$: real part. $A^*$: complex conjugate of probability amplitude $A$.


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 ...



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