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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What is the physical significance of the value of wave amplitude being $1$?

In Feynman Lectures Vol.1, it is written that: First of all, we know that the new way of representing the world in quantum mechanics - the new framework - is to give an amplitude for every event ...
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If we have $kx-\omega t$ equal to a specific value then can the probability of finding a particle be one? [closed]

If we have kx-wt=-63.434948823, in the equation, ψ =Cos(kx-wt)+iSin(kx-wt), then we get the following value, 0.44721359549874+i(-0.894427191000525). When we continue to find the probability by ...
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Quantum Bayesianism and predictions

My question is about the fact that in QBism every physicist has different subjective prior probabilities but I am not sure if the reach the same predictions at the end as occurs for standard ...
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Probability of finding single particle state in QM and QFT

I was reading "Quantum Optics" by Walls and Milburn, and in chapter 10.1 describing about atom-radiation interaction, it says normal QM equation for single electron in an atom (here $\psi_j(x)$ is an ...
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1answer
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Calculating One-Dimensional Particle Separation Probabilty Density

Question Today I am inquiring how one would calculate the particle separation probability density for 2 particles in a square well, for 2 distinguishable particles. We are given both particles wave ...
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Confirm process to find quantum probability [closed]

I am trying to find a probability distribution for a state: $$\psi = e^{a + ib} $$ The probability would be $$| \psi |^2$$ Would the solution be: A) $[ e^{a + ib} ]^2$ B) $[ e^{a - ib} * e^{a +...
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Indefinite integral of a density function

Suppose that $\rho(x)=\frac{dm}{dx}$ is the linear density of a rod. Can we find the mass at each point of the rod by integrating $\rho(x)$, so that:$$m(x)=\int\rho(x)dx.$$ Can we do the same with ...
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General question about Galton Board

I’m new here, and most notably NOT a Scientist, but I do have a question about the Galton Board (which I happily stumbled upon today, thanks to YouTube and ‘the quarantine’) So, as I understand it, ...
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Interaction of neutron with polonium [closed]

Let put some quantities of $Po^{209}$ and throw it by neutron My question what is the formula that we should use it to compute the probability of interacting the neutron with nuclei ? I find this ...
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1answer
43 views

How is the Maxwell-Boltzmann Distribution a Chi-square Distribution?

This Wikipedia states that the MB Distribution in terms of energy is a Chi-square Distribution with 3 degrees of freedom. I know that the probability density formula of a Chi-square Distribution with ...
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What is the physical meaning of the probability current density in Quantum Mechanics?

How can we physically interpret the probability current density? Also, what is the physical sense that it's value being zero. I know that this is zero for the real wavefunction but also, the real ...
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1answer
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Determinisitic system with a probablisitic initial condition [closed]

Consider a deterministic system like a spring mass damper. Lets say we do not know the exact initial condition but we are given a probability distribution function (PDF), $p(x,v,t = 0)$ of the mass's ...
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Quantum mechanics: Probability current density in terms of velocity vs. in terms of continuity equation

For simplicity, consider a one-electron system. Some sources tell that the probability current density can be written in terms of the velocity operator $\mathbf{v} = -i[\mathbf{r}, H]$ as $$ \mathbf{j}...
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How to calculate the statistical probability/weight of a crystal with n atoms being defects? [closed]

We have a crystal where all the lattice sites as well as the interstitial sites create a cubic lattice. The crystal is comprised of N atoms which are only of one kind (monoatomic) and n atoms being ...
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On the adiabatic theorem

In the Adiabatic theorem explanation on Wikipedia it says: Diabatic process: Rapidly changing conditions prevent the system from adapting its configuration during the process, hence the spatial ...
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1answer
85 views

How does one write conditional expectation in bra-ket notation?

In physics, the expectation of a random variable under the bra-ket notation is $\langle u \rangle$, how do I write the conditional expectation of $u$ on another variable $v$? $\langle u | v\rangle$ ...
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Is the probability of an electron being somewhere zero?

So recently I've been reading "How to teach Quantum Mechanics to your Dog" by Chad Orzel. In chapter 3, he says, if I understood this right, that electrons can only exist in specific quanta - that is ...
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Can we have a Non-Gaussian Likelihood and which are the conditions or examples?

I am working on Fisher formalism and MCMC method. It seems that Fisher formalisme assumes that posterior is always Gaussian. So if I find with MCMC a gaussian posterior, I validate the results of ...
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Is it possible to observe a quantum probability distribution?

Is it possible to observe the probability distribution of a quantum particle in real time? So not to observe A state, which would collapse the wavefunction, but observe the whole wave and its ...
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1answer
64 views

Fokker-Planck linear potential

I am struggling with finding a solution to the following Fokker Planck equation with linear potential: $$\partial_{t}P(x,t)=k\partial_{x}P(x,t)+D\partial_{x}^{2}P(x,t)$$ Can anyone help me please? ...
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Expressing the CHSH inequality in terms of probability

Let's say we have a distribution which gives the probabilities $\mathbb{P}(a,b|x,y)$, where $a,b$ are the outputs of a quantum channel for Alice and Bob respectively, and $x,y$ are the inputs (also ...
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1answer
27 views

Basis for F. Mandl's interpretation of the amplitude of a plane wave

I'm going through Mandl's Quantum Mechanics and I'm having trouble understanding some of the moves he makes when discussing the finite potential barrier. He begins by interpreting the plane wave $Ae^{...
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54 views

Quick way of showing visibility of HOM interference for weak coherent states?

It's shown in this paper how weak coherent states, can at-most have a visibility (using the "special visibility" of HOM interference) of 1/2 (as compared to single photons which have a visibility of 1)...
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2answers
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Confusion about the interpretation of the wave function

I have just been introduced to the wave function in my lectures. The way my book and lecturer motivate the wave function is by analogy with the light double slit experiment. The first thing that is ...
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How does entropy relate to energy, conceptually?

Conceptually, I've always understood entropy to be a statistical idea. For example, if you have a vacuum inside of a box and you place a handful of gas atoms on one side, the molecules have a higher ...
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Most Probably Path Length - Statistical Mechanics & Thermodynamics

I’m currently attempting to solve a question set by my lecturer to find the most probable path length of molecules of a substance. The question does not give any values, but I’m assuming it is asking ...
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What does it mean to integrate with respect to matrices?

In Random matrix theory, the following definition of a partition function for an ensemble is common. $$Z=\int dM \exp [-N Tr(M^2)]$$ where $M$ is a Random matrix of dimension $N \times N$. In general,...
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Simulating the spatial distribution of water droplets from a dripping tap

I saw this pattern under a leaky tap. (Recreated images) The pattern was interesting because it looked like a probability distribution. Bigger droplets lie in the centre, and smaller ones scattered ...
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3answers
147 views

Quantum Probability, what makes quantum characteristic functions quantum?

I'm trying to understand how $[Q,P] \neq 0$ leads to the conclusion that no probability distribution can be established for $A$ and $B$. Classically if we had two random variables $Q$ and $P$ we ...
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Statistical mechanics, partition function, and probability

I would like to confirm if the distribution $\rho$ of natural parameter $\varphi$ written as $\rho$ =$\frac{1}{Z}$$H(\beta)$$e^{(\varphi T(\beta))}$, where $Z=\int H(\beta)$$e^{(\varphi T(\beta))}d\...
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1answer
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Are the measurement outcomes of an observable gaussian distributed?

Suppose in an experiment we perform $n$ independent measurements to find the true value of an observable $X$. Let the outcomes of $n$ measurement are denoted by $x_1,x_2,...x_n$. If $n$ is ...
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1answer
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The Physics behind “The Wall” Game show ball drop

In "The Wall" game show, the slots and also the diverters are designed symmetrically and also identically. So when a ball is dropped from a particular slot number it should end up in a particular ...
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1answer
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Generating random change in photon frequency $\Delta \nu$ from angle-averaged isotropic redistribution function

I am implementing the Monte Carlo Simulation Code for the case of photons being scattered by electrons. The paper I'm referring (Hillier 1991, Appendix B) suggests that after each scattering, the ...
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Fokker-Planck equation for 2D SDE

Consider the following two-dimensional SDE \begin{align*} \mathrm{d}\mathbf{X}(t) &= {\mathbf{f}(\mathbf{X}(t))}\mathrm{d}t+\mathrm{d}\mathbf{W}(t)\\ \end{align*} where $\mathbf{X}(t)=\begin{...
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Order parameter fluctuations in the mean field model for ferromagnetism (mathematical approach)

I'm a math student taking first steps into statistical mechanics and... I need help! Consider the Curie-Weiss model (i.e. the classical mean field model for ferromagnetism). If $N$ is the number of ...
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1answer
32 views

Simulating photons on detector statistically

I am simulating a simple detector, imaging a very faint astronomical source (so that some dozens of (uncorrelated) photons are detected in every frame) Currently I am first sampling from Poisson-...
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487 views

Can we determine the probability of finding an electron without a potential well?

Can we determine the probability of finding electron in a region of space if there is no boundary condition? For example, if the wall of a one-dimensional potential well is infinitely far away then ...
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Can the Dirac equation for a free particle accommodate the physical scenario of one single particle in space?

The plane wave solution for the Dirac equation for a free particle is of the form $ \psi = U e^ {i (xP_x + yP_y + zP_z -Et)/\hbar}$, where $\psi$ is the Dirac spinor with four components and $U$ is a ...
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2answers
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Reference request for a mathematical motivation for the Born rule

I was reading the popular science book The Hidden Reality by Brian Greene. My question is about a part in the notes at the end of the book. It is chapter 8, note 9. Brian Greene describes a ...
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1answer
52 views

Lifetime definition

In Walter Thirring book Quantum mechanics of Atoms and Molecules he says that the probability that a initial state $\Psi$ be again measured at later time is $|\langle \Psi|\exp(-iHt)\Psi\rangle|^2$ ...
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2answers
126 views

Isn't causality a disproof for the Copenhagen interpretation [duplicate]

In QM, each measurement result cannot be predicted, in other words, that is random. This being random has a great implication: if there is no rule to specify the results, so there is cause. This lack ...
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1answer
105 views

Probabilistic stacking of blocks [closed]

This is a variation on the stacking problem. A block is a 1D object of length L and uniformly distributed mass. (with some negligible thickness). A stack of size n is a series of n blocks placed flat ...
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Substitution problem in boltzmann factor

The question is about Boltzmann factor. Under continuous energy, the following equality holds. $$\int_{E=0}^{\infty} \; \frac{1}{kT} \; e^{-E/kT} \; dE \; = \;1 \tag 1$$ $$ < E > \; = \; \...
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1answer
59 views

Brownian Motion

I’m currently interested in learning some topics about the Brownian motion and the random walk (in general, from a pure statistical and probabilistic way). For that, I would like to ask you if you ...
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0answers
32 views

Probability for a group of atoms to have a certain magnetization

The paramagnetism of materials can be explained by the behavior of the alignment or not of the magnetic dipole moments of their atoms. Simply put, in a convention where the Cartesian axis $z$ points ...
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About the width of the distribution of the kinetic energy in a gas

In this lecture about statistical mechanics, page $10$, the author said that the kinetic energy $E$ of a gas can be viewed as a random variable (because it is a sum of squared velocities, which ...
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1answer
71 views

According to quantum uncertainty, can an object transform into another object even with extremely low probability?

First of all I would like to point out that when it comes to quantum physics, I have very poor knowledge so please excuse me if I misuse some words to describe what I mean. My question is based on ...
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30 views

Electron tunneling: transmission>1? [duplicate]

Assume a simple electron tunneling scheme from medium 1 into medium 2. By applying the boundary conditions (continuity of the wave function and its derivative at the interface), it is straightforward ...
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4answers
83 views

In QM, what causes a particle to have more probability to be somewhere else when it's found in a less probable position?

For the position of a particle, there's a probability which, somewhere, there's highest probability. And as you move away from it, the probability reduces. But the particle can be anywhere. It's often ...
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48 views

Moving Balls Thought experiment - Are all microstates equally likely? - Boltzmann distribution

I have got a question relating to an exercise from the book "Concepts in Thermal Physics" by Stephen J. Blundell and Katherine M. Blundell. The exercise is about the following thought / computer ...

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