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Questions tagged [probability]

For questions about probability, probability theory, probability distributions, expected values and related matters. Purely mathematical questions should be asked on Math.SE.

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Renormalisation and the Fisher-Rao metric

The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be though of as a flux in a space of possible Hamiltonians for a ...
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622 views

How is conditional probability handled in quantum mechanics?

In ordinary probability theory the conditional probability/likelihood is defined in terms of the join and marginal likelihoods. Specifically, if the joint probability of two variables is $\mathcal{L}(...
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100 views

What properties would a physical biased die have?

Imagine a physical biased six-side die. How much and what kind of bias we could possibly introduce by moving it's center of mass? What would be the exact mechanism describing the relation of center of ...
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316 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
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1k views

Probability density of Klein-Gordon equation

This may, perhaps, stir some healthy debate; at least I am having some "fun" thinking about it, hopefully I can solicit some outside views too. It is often regarded that the Klein-Gordon equation ...
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117 views

Help with deriving an asymptotic expression

Note: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a ...
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133 views

Reference for stochastic processes which helps moving from a basic level to a measure theory one

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ...
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307 views

Electron hopping among molecules - Marcus equation

I'm running out of professors to talk to, and I need to clarify a couple of things for the sake of making a realistic model of electron travel through a mesh. This is about calculations of electron ...
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42 views

Is there an open quantum system analogue of the equilibration time bounds for classical ergodic Markov chains?

Background For classical ergodic discrete Markov chains, we can bound the time taken to reach the stationary distribution to the spectral properties of the transition matrix. I will outline this ...
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62 views

A proof for this equivalent version of the Infrared Bound/Gaussian Domination

Consider the Ising Model in the $d$-dimensional discrete torus with side lengh $L$, denoted by $\mathbb{T}_L $, with nearest neighbors interaction (with interaction parameter $J$, no magnetic field, ...
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165 views

What can I expect to see in a oscillator exhibiting bifurcation?

I have a program which aims to simulate a Josephson Bifurcation Amplifier. I am currently trying to obtain a plot of the probability of bifurcation as a function of the ratio between the driving and ...
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110 views

Are frequency and likelihood the same across the multiverse?

My probability text distinguishes between two interpretations of probability values: the frequency of occurrence "as percentage of success in a moderately large number of similar situations" (coin ...
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199 views

How to explain Tsirelson's inequality using extended probabilities?

How to explain Tsirelson's inequality using extended probabilities? Some people have tried explaining the Bell inequalities using extended probabilities. For instance, a pair of entangled photons ...
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49 views

Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...
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96 views

Time and Quantum Mechanics

my question is about the nature of time across classical/macro and quantum scales. I understand that the 2nd law of thermodynamics and entropy increase has a lot to do with our understanding of time ...
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47 views

given an AC field, how many photons are there?

Say you are applying a time varying potential across double quantum dots in the form of $V_{ac}$cos($2\pi f t$). We know that each photon has an energy $E_{photon}= hf$. Is it correct to say that ...
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234 views

Shift a Galton board's distribution by manipulating pegs

My question is related to this one but is simpler and more specific. It's also related to this this question which doesn't answer my question. In a standard Galton board, balls are dropped onto a ...
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63 views

Reconstruct quantum state from probabilities

Given a quantum state, the Born rule lets us compute probabilities. Conversely, given probabilities, can we reconstruct the quantum state? I think the answer is almost trivially positive but how ...
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288 views

Probability density to find the end of a double pendulum

Given infinite time, is there a way to find the probability to find the end of a double pendulum at a given position? I am looking at a system with no damping (constant energy). The bobs are point ...
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214 views

Examples of “Ordered state from disordered state”

A thought experiment - Imagine two containers joined to each other with a stopcock between them. One container initially has only two molecules of a gas and other is empty. Now, if we open the ...
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178 views

Is 'interpretation' in quantum mechanics the same as 'interpretation' in probability?

I heard all 'interpretations' of quantum mechanics give exactly the same answer to every measurement so they are all equally correct. Is that the same 'interpretations' as in probability? Context: ...
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113 views

Interpretation of Characteristic function of probability density function for a classical system

Can the characteristic function ( fourier transform ) of a probability density function in position space for a classical system be related to its momentum space probability density function in any ...
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201 views

Why are isoprobabilty contours circles for uncorrelated functions and ellipse for correlated functions? Kindly explain.

I would like to know how to understand these and to learn how to go from a scatter plot to the isoprobabity contour plots. Thanks is advance!
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90 views

Measure-theoretic maths behind Born's probabilistic interpretation of Schrodinger's equation

I was reading a bit about Quantum Mechanics, Schrodinger's equation and its probabilistic interpretation (found this very insightful intro here https://plus.maths.org/content/schrodinger-1), my ...
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130 views

Applicability of perturbation theory

Consider some system in some initial state $|k^{(0)}\rangle$. The probability that such a state makes a transition to some other state $|m^{(0)}\rangle$ can be computed to various orders in time ...
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274 views

Probability and the propagator

Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to ...
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3k views

Finding the probability that a $1s$ electron in ${}^3$H remains in the $1s$ state after beta decay to ${}^3$He with the sudden approximation

Hydrogen-like wavefunctions have the form: $$R_{10} = \left(\frac{Z}{a_0}\right)^{\frac{3}{2}} 2\space e^{-\frac{Zr}{a_0}}$$ $$Y_{00} = \frac{1}{\sqrt {4\pi}} $$ where $a_0 = \frac{4\pi \epsilon_0 \...
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566 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
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97 views

Statistical Mechanics: Most probable orientation of grain particle in gas chamber?

I'm in an introductory statistical mechanics course, and we've been posed the following situation: Long-shaped dust particle (so imagine something like a grain of rice) is placed in a gas chamber (so ...
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29 views

List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
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36 views

Converting a discrete statistical energy distribution to a continuous version

The probability of finding a particle at a particular energy level when energy is considered discrete is according to Boltzmann: $$P(E_j) = \frac{g_j\cdot e^{-\beta E_j}}{\sum_{j=1}^\infty g_j \cdot ...
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35 views

Calculating the probability of one particle being in a certain state in two-particle system

Let's say I have the two-particle state $$|\psi\rangle=\frac{|H\rangle_a|H\rangle_b+|V\rangle_a|V\rangle_b}{\sqrt{2}}$$ where $H$ is horizontally polarized and $V$ is vertically polarized. And I ...
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49 views

Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?

Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
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28 views

Notation in a question on probabilities and particle counting

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home) A particle counter records counts with an ...
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25 views

Discrepancy regarding Husimi Probability distribution calculation

I am trying to simulate a system of j qubits and for visualization of the dynamics considering the Husimi distribution of the state. To carry out the projection onto coherent states I have proceeded ...
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66 views

A hydrodynamic theory for systems with rich microscopic detail?

I've been looking at various models of stochastically interacting particle systems. Let's take for example the totally asymmetric simple exclusion process on a 1D lattice with some initial conditions. ...
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62 views

If a wave function represents knowledge, what does a density matrix mean, then?

I'm curious about this. I've heard of interpretations of quantum theory in which the wave function $\psi$ is taken as representing knowledge, or information (e.g. "Quantum Bayesianism"), about the ...
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49 views

Can local hidden variables models be ruled out without invoking expectation values?

Bell's inequalities, in their standard form, are a statement about the limitations faced by a probability distribution that can be written as $$p(a,b|x,y)=\sum_\lambda p(\lambda) p(a,b|x,y,\lambda)=\...
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110 views

Expression of stress-energy tensor with density function

I try to get the following expression defining the stress-energy tensor in General Relativity ($\big< \big>$ means average) : $$T^{\mu\nu} = \int \dfrac{\text{d}^3p}{E} F(\vec{x},\vec{p})p^{\mu}...
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79 views

Can the density operator be used to describe a continuous probability distribution, analogously to a classical probability distribution function?

I've been reading about Liouville's equation and am now trying to understand Von Neumann's equation, which seems to be more or less the quantum mechanical version. Both cases involve an ensemble of ...
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56 views

Are Ross-Littlewood-sequences or negative probabilities possible in physics?

There are claims that the Ross-Littlewood paradox could be simulated in physics. See https://stats.stackexchange.com/q/315502/ and in particular Paul's answer there. Also a solution of the Einstein-...
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40 views

Proving that Shannon entropy is maximal for the uniform distribution using convexity

I need to show that $-\sum_i{p_i \log{p_i}}$ is maximal iff $p_i=p_j$ for all $i\neq j$ using the convexity inequality: $\phi (\frac{\sum{a_i}}{N})\leq \frac{\sum{\phi (a_i)}}{N}$ I tried expanding ...
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59 views

Measurement and probability

In the $\left\{\left|j_1j_2;jm\right\rangle=\left|ls;jm\right\rangle\right\}$ basis, I've got the state ket $$\left|\alpha\right\rangle=\frac{1}{2\sqrt{2}}\left(\sqrt{\frac{2}{3}}\left|11;20\right\...
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84 views

Does Bell's theorem have anything to say about the locality or realism of Quantum Mechanics?

In the original paper written by Bell, it's clear to me that what he's really trying to answer is what class of Classical Theories (that obey the laws of classical probability theory) can replicate ...
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188 views

Existence of joint probability distribution (hidden variables) for partially compatible measurements

Informal summary: If we have $n$ measurements out of which we apply at most $q$, and we know that up to $q$ measurements commute on a given starting state, is there a hidden variable model that ...
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322 views

How does one generate a random $N \times N$ density matrix with quaternionic entries — with respect to Hilbert-Schmidt measure?

In the article https://arxiv.org/abs/0909.5094 the formula (eq. (1)) \begin{equation} \rho_{HS} =\frac{A A^{\dagger}}{\mbox{Tr} A A^{\dagger}} \end{equation} is given, for generating a random $N \...
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317 views

Distribution of Photon Inter-arrival Times

In case of a coherent light source (eg. laser beam) the number of photons counted in a given time interval can be modeled as a Poisson distribution. From the theory of Poisson processes, it is well ...
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144 views

Counting experiment and standard error

I have a counting experiment: Let's say I have N identical bees. I take one of them, expose it to $\gamma$-radiation and look if it has died or survives. If it survives, I count it as 1 count. If it ...
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776 views

Physical interpretation of a complex potential for a particle in quantum mechanics

In Griffiths' Quantum Mechanics, it is mentioned in a problem that For an unstable particle that spontaneously disintegrates with a lifetime $\tau$, the total probability of finding the particle ...
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208 views

Probability conservation versus Liouville's theorem

This question arises to my mind while studying notes on Kinetic theory written by Prof. David Tong. There he derived the Liouville’s equation. The outline of his derivation goes as follows: Our ...