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 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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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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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
88 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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38 views

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
80 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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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 ...
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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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2answers
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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
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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
26 views

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
321 views

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
39 views

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
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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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3answers
499 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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31 views

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
54 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
132 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
108 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
63 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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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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1answer
35 views

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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33 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
94 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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51 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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1answer
124 views

Why don't electrons “get lost”?

According to quantum mechanics, existence of an electron at a place depends on the wavefunction which in turn gives us the probability of an electron being there. And for a few special places, like ...
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1answer
77 views

How to normalize this wave function? [closed]

My wave function is $$ \Psi = A e^\left({-\frac{\left|x\right|}{2a}- \frac{\left|y\right|}{2b} -\frac{\left|z\right|}{2c}}\right)dx $$ and I need to normalize it. I tried to take an integral of it and ...
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1answer
111 views

Coarse-graining and probabilities in quantum theory

What is the origin of probabilities in quantum theory if it is not postulated? I mean, how can we interpret Born's Rule as a probability and how does it enter in quantum theory? More than this, I want ...
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2answers
57 views

Why is the probability density oscillating in region $x<0$ before the potential barrier?

My understanding was that the standing wave solution is that of a free particle in the region before it enters the classically forbidden region. Does multiplying the wave function by its complex ...
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2answers
76 views

Angular momentum from spin

I have came across the following question: Two particles, each of spin 1, are at rest. It is known that the $z$ component of the spin of each is zero. Show that the probability that the total ...
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1answer
46 views

Simplifying expressions

Is it mathematically valid to simplify the expression $$\left ( \bar \Psi \right)^2 \left(1, \ -1\right)$$ to 1 if $$\Psi = \begin{pmatrix} \cos(x) & 0 \\ 0 & i \sin(x) \end{pmatrix}$$ (...
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1answer
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Basic concept of interaction probability

In William R. Leo's book(Techniques for Nuclear and Particle Physics Experiments) I could not understand the following: $P(x)=$probability of not having interaction after a distance $x$ $w \delta x$=...
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57 views

The justification for the probability of definite energy states in quantum mechanics

In quantum mechanics, if the energy of a system is measured at some $t$ the probability of obtaining the energy eigenvalue $E_i$ is: $$\left| \int_{-\infty} ^{\infty} {\psi_i^* (x)\Psi(x,t)} dx^2 \...
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1answer
32 views

Is there only one state that leads to a collection of probability distributions of observables? [duplicate]

I would like to think of "state of an object" as, classically, the simultaneous knowledge of its position, momentum, rotational angular momentum, orbital angular momentum, kinetic energy, etc. In ...
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1answer
78 views

Cosmology / Interpretation between credibility/confidence_level with bayesian/frequentist approaches

I try to understand the following article : testing general relativity from curvature and energy contents at cosmological scale I don't understand the title of figure 1 : where it is indicated ...
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Probability of emitting a photon proportional to population in final state: problem with normalization?

In section 4-4 the Feynman Lectures reads: The probability that an atom will emit a photon into a particular final state is increased by the factor $(n+1)$ if there are already $n$ photons in that ...

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