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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How is the measure for the wavefunction determined in quantum mechanics?

Given some quantum mechanical system described by a lagrangian ${\cal L}=\frac{1}{2}\dot{q}^2-V(q)$, I can imagine solving for the wavefunction $\Psi[q]$ and then using this to compute expectation ...
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Movement of a random walk in the limit (a particle in diffusion)

I asked this question in Math Exchange and MathOverflow and obtained no answer. This question may lack of mathematical rigorous, but I would like to understand why this type of reasoning is sometimes ...
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35 views

Stochastic dynamics of rotation intergral over $d\hat n$?

I am looking into the stochastic dynamics of rotation in which we describe the orientation with a unit vector $\hat n$. If we let $p(\hat n',t)$ denote the probability that $\hat n=\hat n'$ at time $t$...
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Path integral kernel dimensions and normalizing factor

I am currently reading Quantum Mechanics and Path Integrals by Feynman and Hibbs. Working on problem 3.1 made me wonder why the 1D free particle kernel: $$ K_0(b,a) = \sqrt\frac{m}{2\pi i \hbar(t_a - ...
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Probability of photoelectric effect for gamma rays [duplicate]

Why is the probability of the photoelectric effect occurring higher for low energy gamma rays? I'd like a physical answer not just equations.
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Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix? Does a density matrix always have a a basis of eigenvectors?  
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Probability current in free quantum particle wavefunction

Context: In Griffith's book on quantum mechanics, the probability current formula (which indicates the rate of decrease in probability over time, at $x$) is given as: $$ J(x,t):=\frac{i\bar{h}}{2m}\...
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$h^3$ term in probability density function of ideal gas

Other than that we need the $h^3$ to 1) make the units correct and to 2) account for the normalization of the probability distribution, what interpretation can we give to this $h^3$ term in \begin{...
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Probability of finding an energy state of a non-normalisable wave-function

Suppose, say, I have the following wave function It represents the wave function of a free particle. I would want to calculate the probability of finding the particle with energy ħk and energy 2ħk. ...
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Probability measure implies quantum mechanics?

The article "Quantum Logic and Probability Theory," by Wilce, has the following in section 1.4: 1.4 The Reconstruction of QM From the single premise that the “experimental propositions” ...
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Is probability relative?

Suppose you are as massive as earth and you are tumbling down space at a very high speed. You flip a coin. There is a probability function associated with the coin in motion. When it falls back in ...
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Probability of finding a particle in a region in a state given for a wave function plus a constant

This is the problem: A particle is restrained to move in 1D between two rigid walls localized in x=0 and x=a. For t=0, it’s described by: $$\psi(x,0) = \left[\cos^{2}\left(\frac{\pi}{a}x\right)...
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How does finding the $y$-intercept of a graph reduce error compared to finding the average of the data set?

In my particular case I have the equation $$ \frac{1}{d_i} = -\frac{1}{d_o}+\frac{1}{f} $$ and I'm plotting $\frac{1}{d_i}$ against $\frac{1}{d_o}$ to give an intercept of (in theory) -1 and a $y$-...
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Examples of Markov chains [closed]

I will give a talk to undergrad students about Markov chains. I would like to present several concrete real-world examples. However, I am not good with coming up with them. Drunk man taking steps on ...
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1answer
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Meaning of wave function which is in inside of the potential wall? [duplicate]

In particle in square potential barrier problem, we can easily find that some probabilities exist which express how many particles can go beyond of the potential wall. So my question is that, can we ...
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Form of Schrödinger equation for the probability density

Is it possible to formulate the Schrödinger equation (SE) in terms of a differential equation involving only the probability density instead of the wave function? If not, why not? We can take the ...
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Percolation theory: What is the critical amplitude for the “backbone” of a 2-D network?

Disclaimer: I am just learning about percolation theory for the first time, so I am not too familiar with some of the terminology. Suppose you have a 2-D square lattice with bonds connecting sites. ...
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Derive Poisson distribution from probability per time of event

Suppose we have a probability per time $\lambda$ that something (e.g. nuclear decay, random walk takes a step, etc.) happens. It is a known result that the probability that $n$ events happen in a time ...
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Percolation theory: Is there a clear relationship between “probability of connection” and “effective porosity”?

Disclaimer: I am a geophysicist, not a physicist. Sometimes we speak a slightly different language and use different terms. Suppose you have an $N$ x $N$, 2-D rectangular lattice with bonds which ...
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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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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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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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Is there a condition of quantum mechanics that forbids Lorentzian distributions?

Imagine a particular potential that allows a superposition of eigenstates such that in space basis the probability density $|\psi(x)|^2$ is a lorentzian (Cauchy) distribution. The properties of the ...
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Quantum measurement operators defined with Poisson distribution

I have defined a set of quantum measurement operators of the form $$A_C = \sum _M \sqrt{Pr(M|C)} |M \rangle \langle M|$$ where $Pr(M|C)$ is the Poisson distribution $$Pr(M|C) = \frac{e^{-C}C^M}{M!}~~~~...
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How does QFT predict the probability density to find a particle at x?

In quantum mechanics, the probability density of a particle's position is $$\rho(x)=|\langle x|\psi\rangle|^2$$ What is the corresponding expression in QFT to predict this distribution? Since $\rho(x)...
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Where does the factor of half come from, boltzmann distribution for bandgap energy

I have found that it is possible to calculate the conductivity of a semiconductor using the Boltzmann distribution: The source is a slideshow presentation and doesn't list much information. The ...
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157 views

Probability density for velocity in mechanical energy

To be sure my basic physics isn't rusty... Consider a 2D bowled shaped classical potential well within which a classical particle of mass m is rolling. In this system the conservation of energy holds ...
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322 views

Lorentz distribution in emission peaks

Sometimes spectral peaks are fitted using a very pathological function named the Lorentz/Cauchy distribution. The Lorentz distribution has the property of not having a defined mean nor standard ...
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1answer
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Quantum uncertainty affecting classical object? [duplicate]

As far as I know, the probability of a quantum object being in a certain position depends on the wave function value for each position. That raises a question: Is this probability strictly greater ...
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3answers
257 views

Quantum harmonic oscillator in thermodynamics

I'm trying to understand the microcanonical ensemble in thermodynamics using the quantum harmonic oscillator. The Hamiltonian of the whole system is given by $$ H = \hbar\omega\sum\limits_{i=1}^N \...
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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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590 views

Probability vs radial probability density

When we want to talk about the most probable radius to find the electron in $1\text{s}$ orbital, why do we talk about the radial density and not the probability itself? For instance, the probability ...
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Positioning of the root-mean-squared speed, average particle speed and most probable speed on the Maxwell-Boltzman distribution curve

I'm inquiring if there is any intuitive reasoning behind the positioning of $v_{rms}$, $v_{mp}$, and $ v_{avg}$ in their positionings on a Maxwell-Boltzman distribution such as the one below: I was ...
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Why do correlation functions diverge in field theory?

A correlation between two random variables is generally defined as $$\text{corr}(X,Y)=\frac{\text{cov}(X,Y)}{\sigma_X \sigma_Y}$$ In particular this says that $\text{corr}(X,X)=1$. However, in physics ...
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Can observation affect the probability distribution of an event to occur?

It's already well know that in quantum mechanics the act of observation affects the outcome of an experiment making the wave function to collapse.Now in order to clear up my confusion about this topic ...
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228 views

How does one calculate conditional probability using bra-ket notation?

For example, given two spin-1/2 particles in basis $|++\rangle,|+-\rangle,|-+\rangle,|--\rangle$. If the first measurement of spin is carried out and gives $1/2$ for particle one, what is the ...
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Does the first part of the measurement postulate imply the second?

Common formulations of the general measurement postulate of quantum mechanics have two parts, one establishing the probability of the outcome $m$ corresponding to an observable $M_m$ for a system in ...
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What is the Schrodinger equation's dynamics when having classical and statistical mechanics in mind?

Assume I take the following PDE without QM associated with it. $$ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},...
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How many ways to arrange energy packets shown in system A and B?

I tried very hard to solve this. I have attached the solution which I tried. Help me out to find the exact way to solve this
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380 views

Particle in a box in quantum mechanics [duplicate]

In case of the solution of Schrödinger equation in particle in a box we find that the probability of finding the particle at the middle of the box is always zero (for $n$ even) and the probability ...
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A question in Taylor series related to Quantum mechanics

I'm fairly new to QM and was reading about the Taylor series expansion of $V'(x)$ from Chapter 6 (The Classical Limit) of Shankar's Principles of Quantum Mechanics, pages 182-183. The terms in the ...
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Symmetry transformations on a quantum system; Definitions

We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a ...
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How do we choose the standard probability current?

In quantum mechanics, the probability current is defined as $$\mathbf{J} \propto \text{Im}(\psi^* \nabla \psi)$$ and satisfies the continuity equation $$\nabla \cdot \mathbf{J} = - \frac{\partial \...
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How to calculate the probability of Hamiltonian Operators being in a certain state?

So, I'm reading through my Quantum Mechanics textbook and I stumbled upon a bit of maths that I'm not entire sure how they got: \begin{align*} \mathcal{P}_{a_1} &= \left| e^{-iE_1t/\hbar} \...
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1answer
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Classical random walk equation - confusion

I have always thought that a (classical) on a straight line is governed by the binomial distribution: $$ {{n}\choose{k}} p^{k}(1-p)^{n-k},$$ $n$ being the number of tries and $k$ the number of ...
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What is the deal with the Schrodinger's cat? Why is it considered a paradox?

What is the deal with the Schrodinger's cat? Why is it considered a paradox? Cat is a macro object. He can be only in 2 states - he's either dead or alive, the fact that you don't have the ...
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What is the proper way to model diffusion in inhomogeneous media (Fokker-Planck or Fick's law) and why?

I'm quite confused with the following problem. Normally a one-dimensional Fokker-Planck equation is written in the following form: $$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\...
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Can the complex square of the wave function be interpreted as shape of the particle? [closed]

We know that for wave function of a photon or an electron, the complex square of the wave function is understood as the probability density of finding point like particle in the location. Can anyone ...
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Validity of Born Rule [closed]

The Born rule has been very successful in quantum mechanics. However, the interesting fact about this rule is that it only allows pairwise interference. In other words, there are no interference terms ...
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Why set the probability density for finding a falling stone as proportional to the time interval it spends at a given location? [closed]

Today a friend of mine, who has started out with Griffiths, asked me a question about one of the examples (1.1) in Griffiths' QM book. The example basically says that a rock is dropped from a cliff, ...