Questions tagged [probability]
For questions about probability, probability theory, probability distributions, probability density matrix, expected values and related matters. Purely mathematical questions should be asked on Math.SE.
1,499 questions
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Cross section normalization in $2 \to 2$ scattering
Consider Compton scattering $$p_1 + p_2 \to p_3 + p_4$$ in the laboratory frame. According to Quantum Field Theory and the Standard Model by Schwartz, the relation between the differential cross ...
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Probability of particle settling into potential well
The following question was posed to me by a student I was tutoring.
Consider a one-dimensional potential $V(x)$ with limiting behavior $\lim_{x\to \pm \infty} V(x)=+\infty$ and two "wells" ...
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Is qubit collapse in quantum computing related to decoherence, and how can probabilistic outcomes yield definite answers?
While studying Grover’s algorithm for quantum search, I came across an explanation that a qubit is represented as a state vector in superposition, with amplitudes corresponding to the probabilities of ...
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Observing $L_x$ of a particle which is present in an eigenstate of $L_z,L^2$
I wonder what will be the probability of finding the particle in a particular $L_x,L^2$ state after we know its $L_z,L^2$. Definitely,the $L^2$ value will not change. Here is what I tried:
First of ...
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Compute probability current from stochastic path integral
I would like to compute the probability current associated with a stochastic differential equation, say
$$
\frac{\mathrm{d} X}{\mathrm{d} t}
=
v
+
\sigma \xi(t)
$$
where $v$ is a drift velocity, $\xi$ ...
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How strong is the justification for low probability extreme thermal fluctuations? [closed]
Consider an extreme low-probability event in statistical mechanics, like a marble statue spontaneously moving its hand due to a thermal fluctuation, versus a low-probability sequence of dice rolls, ...
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Energy Conservation for Quantum Systems
This seems like a basic question but I couldn't find the answer when looking online. I'm in an introductory quantum mechanics class and we learned how classical laws can be seen in QM when you look at ...
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Does Heisenberg uncertainty principle and chaos theory create indeterminism? [duplicate]
I’ve been struggling to understand this and hoping for an answer from someone enlightened.
Heisenberg Uncertainty Principle states there is a trade off between how much we can know about position and ...
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Classical Probability in Quantum Mechanics [closed]
I have been reviewing the foundations and formalisms of quantum mechanics in order to better understand some things, and one of my goals is to be able to clearly separate things which are physical (i....
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Is it okay if your signal and noise have the same statistical distribution?
Please bear with me if this seems like a very basic question. Let's say you want to detect a signal by measuring a variable $x\in[0,\infty)$. Let's say you know the variable $x$ follows a probability ...
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Is it possible that the square amplitude law is only approximately correct?
I would like to ask about the law in quantum mechanics whereby the measured probability is the square of the probability amplitude.
Is it possible that the law is only approximately correct, i.e. it's ...
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Why do we use Hilbert Space? [duplicate]
While studying quantum mechanics, I encountered the term Hilbert Space.
As I understand, Hilbert space is an infinite-dimensional complete inner-product vector space.
What physically motivates such a ...
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Physical interpretation of a negative wavefunction?
A wavefunction can be split by separation of variables and solved for the time independent Schrödinger equation:
$$-\frac{(h/2\pi)^2}{2m}\frac{d^2u}{dx^2} + Vu = Eu,$$ from $$\psi(x, t) = u(x)T(t).$$
...
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Evaluating probability using projectors in continuous Hilbert space
Say, for example, there's a system of two particles with spin one-half at state $|\Psi\rangle$. The probability we measure the first particle in a spin up, $|+\rangle$, state in the $z$ direction ...
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Does nature work exclusively on the cause-effect principle?
Does nature work exclusively on the principle of cause-effect or are there situations in which the principle is violated?
Is randomness in probabilistic process truly fundamental or just a reflection ...
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Measuring position on a system of identical particles
From what I understand, for a system of $N$ identical particles in a state $\lvert\psi\rangle$ (symmetric or antisymmetric), the probability density of measuring one particle at position $\vec x_1$, ...
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Likelihood of the data in a Ornstein-Uhlenbeck-like process
I am interested in finding the likelihood for the location of a given number of particles at time = 1 in a process that resemble (or is) a Ornstein-Uhlenbeck (OU) process.
In particular, I am ...
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In classical statistical mechanics, is the number of microstates countable or uncountable?
I initially thought that the number of microstates of a system is uncountably infinite, because the momentum and position of a particle can take any value in a continuous range of values in classical ...
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What proportions are necessary for a rectangular dice to roll a square side up 20% of the time?
I heard that an ancient version of the game Mia might have been played with cut knucklebones as dice, and be a little bit rectangular.
Is there a way to model dice face proportions with a rectangular ...
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What does unitarity violation in QFT translates to? [duplicate]
In QM the probability is violated when:
$\int \rho(\vec r,t)\vec dr\neq 1$ for a quantum mechanical system in an arbitrary state $\psi(\vec r,t)$.
In this case we know that $\psi(\vec r,t)$ is the ...
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Probability density matrix in $4D$ CFT in curved spacetime
In this essay named "Shape Dependence of Holographic Renyi Entropy in Conformal Field Theories", the author claim that the Renyi entropy in $4D$ CFT in curved spacetime can be calculated ...
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Can the partition functions of two identical distributions be different?
To my understanding, we can recover the canonical distribution from the grand canonical distribution, if we set all the microstates of the grand canonical distribution to have the same number of ...
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Statistical mechanics and the distribution of phases of oscillation
Imagine an elliptical region of phase space $\{(q,p)\in \mathbb{R}^2:p^2/2m + kq^2/2=E\}$ of constant energy $E$ for a 1D harmonic oscillator. Since all points on the ellipse have the same energy, ...
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How the Kramers escape rate relates to the diffusion?
Consider a Brownian motion in a one-dimensional potential $U(x)=U_0\cos\left(\frac{2\pi}{a}x\right)$.
What is the diffusion constant of this process?
My first thought is to find the Kramers escape ...
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What is the probability that the Solar system encounters a black hole?
What is the probability that the Solar system encounters a black hole?
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Relationship between momentum and probability current in Dirac equation
I would expect that for a given eigenstate with energy $E$, the density of momentum $\mathbf{p}$ and the probability current $\mathbf{j}$ at any point would be related as:
$\mathbf{p}=\frac{E}{c^2}\...
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What's the probability distribution of when the ball will fall from Norton's dome?
Suppose our world is completely described by Newtonian mechanics. All the materials are rigid and can be cut infinitesimally.
There exist scenarios where the future is intrinsically probabilistic. ...
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Are fields in particle physics some wave functions, that is amplitude of probability?
In particle physics, there are fields for fermions, typically labelled $\psi(x)$, and fields for scalar, typically labelled $\phi(x)$, which are functions of the space-time coordinate $x=(t, x, y, z)$....
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What does it mean by pdf ( probability density function) integrated minimum and maximum value of a progress varaible?
I am reading a paper on combustion and I come across this staement which I don't understand how they arrived it ?
A progress variable is generally defined as a linear combination of species mass ...
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Probability a system at temperature $T$ has more than a certain energy
In the Feynman Lectures I, Ch. 42, it is stated that the probability of a system at temperature $T$ having energy $W$ above the average energy is $e^{-W/k_BT}$. This is used to derive the rate of ...
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What causes amplitudes to interfere?
I have started working through Quantum Mechanics section of the Exercises for the Feynman Lectures on Physics, but I got stuck on one of the questions. The question is about an idealized version of an ...
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Characterization of Markovianity, Gaussianity, and color for noise processes
Consider a noise process $\xi(t)$ that has some statistics in time. There are various ways to characterize such a process, 3 being Markovianity (independence from history), Gaussianity (Gaussian ...
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Clarification on equation 5 in Feynman's 1948 path integral paper
In the 1948 paper "Space-Time Approach to Non-Relativistic Quantum Mechanics", Feynman presents his formulation of quantum mechanics in terms of path integrals. I've been reading through it ...
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Why randomness taking a specific outcome at that time? [duplicate]
Im a 16 year old and this question stuck in my mind. I cant find apt answer.
Imagine we tossed a coin, we have 50-50% chance to get heads or tails. We say its random but truly we can predict the ...
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Why is the PDF in Maxwell Distribution of Speeds simply NOT the derivative of the Boltzmann Distribution?
It is written in Atkin's Physical Chemistry
It implies that the fraction of
molecules with velocity components $v_x$, $v_y$, $v_z$ is proportional to an exponential
function of their kinetic energy, $...
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Can quantum effects that result in an object taking infinitely many possible paths apply to macroscopic objects?
If according to quantum physics, a particle moving from point A to point B takes every possible path. Is it possible to apply this to macroscopic objects?
For instance if I throw a ball up in the air, ...
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Is it true that quantum mechanics technically there is a 50% chance of any event happening? [closed]
This is most definitely a very stupid question. According a friend of mine, there is a 50% chance of any event happening according to quantum superposition and quantum probabilities.
My counter ...
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Correlation matrix of spin glass model
Consider a mean-field spin glass model (e.g., the Sherrington-Kirkpatrick model or the spherical spin glass):
$$ H_J(\sigma)= -\frac{\beta}{\sqrt N} \sum_{i,j=1}^N J_{ij}\,\sigma_i\,\sigma_j - h \sum_{...
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Fermi's golden rule for total transition rate [duplicate]
Im following Mark Thomson's Modern Particle Physics section 2.3.6 but I have question in the proceedure followed from equation 2.46 to 2.48
Starting from equation 2.46:
$$d \Gamma_{fi} = \frac{1}{T} ...
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Probability of Finding an Electron at Distant Position
The electron can be found everywhere except the Nodal Plane.
The probability of finding an electron is maximum in the orbitals and decreases as farther as we move, but it is Never Zero.
So what about ...
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Sampling from a Boltzmann distribution
Suppose I have a potential energy function $V(x)$ and I'd like to draw random samples of the position of the particle, $x$, from the Boltzmann distribution
$$P(x) = \frac 1 Z e^{-\beta V(x)}$$
at ...
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Zero curl of probability current of arbitrary dimensions particle according to Schrödinger equation
$$\vec{v}=\frac{\vec{J}}{\rho}=-\frac{i\hbar}{ 2m}\left(\frac{\nabla \psi}{\psi}-\frac{\nabla\psi^{*}}{\psi^{*}}\right)=-\frac{i\hbar}{2m}\nabla\left(\ln\frac{\psi}{\psi^{*}}\right)$$
then curl equals ...
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Continuous Sample Paths for distributions determined by Schrodinger equation?
Let us first recall how Brownian motion is described in pure mathematics. For all $t\in [0,\infty)$, we have a probability distribution $\mathscr D_t:= \text{Normal}(0,t)$, and random variables $B_t$ ...
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Does the density inside the $2\ \rm s$ node correspond to $2\ \rm s$ electron or $1\ \rm s$ electron?
If we see orbital density of $2\ \rm s$ orbital we see that its divided by a node and the part of probability density inside the node is at where $1\ \rm s$ is supposed to be does it means that the ...
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Is there a particularly useful analogy that helps to understand what a wave function is? [duplicate]
Is there a particularly useful analogy that helps a lay person to understand what a wave function is?
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Dimensional analysis and integration
In dimensional analysis, the argument of an analytic function cannot have dimensions, because if you were to expand the function in a Taylor series, you'd end up adding terms e.g. $1+x+\frac{x^2}{2}+\...
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Is an exponentially large state space unique to quantum physics?
The exponential growth in the dimension of a many-body Hilbert space with increasing particle number is often presented as something that makes quantum physics very different from classical physics. ...
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If QFT is relativistic approach of QM, how does the concept of probability distribution of location of a particle evolve along the way?
When I first found out that QFT says if the value of the field associated with some particle is high at some point in space it means a particle will be localized there, more specifically, will be ...
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Derivation of the Fokker-Planck equation
A nonlinear dynamical system is considered
$$ dx/dt = f(x) + z(x)p(t), $$
where $p(t)$ Gaussian noise with zero mean and exponential correlation function.
How I can derivation of the Fokker-Plank ...
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Do relativistic propagators give probability amplitudes?
I find myself confused as to whether relativistic propagators can be understood as probability amplitudes or not.
This Wikipedia article for example, under 'Relativistic propagators', states that:
In ...
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