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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26 views

Is it possible to have a negative density matrix, and how would you calculate entropy?

I am trying to calculate the Von Neumann Entropy of a quantum state. Given a state $ | \psi \rangle$, I am calculating the Von Neumann entropy by doing the following: $$ S = -\mathrm{tr}(\rho \ln(\rho)...
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Hyperbolic probability interference : $P=|\psi_1|^2+|\psi_2|^2+2|\psi_1||\psi_2|\cosh \theta $

In quantum mechanics, the complex amplitude is modulus squared to produce a probability via the Born rule. $$ P=|\psi_1|^2+|\psi_2|^2+2|\psi_1||\psi_2|\cos \theta \tag{1} $$ In this paper https://cds....
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Feynman's random walk (6.3)

In chapter 6 of volume 1 of the Feynman lectures on physics, Feynman's elaborates on the random walk. https://www.feynmanlectures.caltech.edu/I_06.html Amidst his lecture, he says this: The expected ...
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How to calculate the probability of an enclosed gas migrating to one side of an enclosed box

Non-physicist asking. I have a hollow enclosed 1cm-sided cube. It is filled with N mols of an ideal gas. (1) What is the probability that for one infinitesimal moment, all the gas will occupy half of ...
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What conclusions can we draw from Thomas Breuer self-reference theorem?

This question is based on the another question of mine. Can we conclude one of the following statements from the results of Breuer? (1) The universe constantly receives information (influence) from ...
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What is the probability that two chemical species of binding energy $E$ will be bound?

This may seem like a very simple question, but I've been agonising over it for days. What is the probability, $p$, that two chemical species with binding energy $E$, will be bound. My first instinct ...
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55 views

Feynman's random walk (6-3)

How does Feynman get to $D^2_{N-1}$? The expected value of $D^2_N$ for $N>1$ can be obtained from $D_{N−1}$. If, after $N−1$ steps, we have $D_{N−1}$, then after $N$ steps we have $D_N=D_{N−1}+1$ ...
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Boltzmann distribution for non-energy models

My question is rather general and it regards the possibility to associate the Boltzmann probability distribution to some energy model. What are the general assumptions that an Energy-based model needs ...
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Probabilistic determination of syzygy

Is there a framework for describing the probabilistic confidence of an $N$-body syzygy? Forgive me if I'm misusing terminology: I'd like to know how physicists/astronmers model the confidence of "...
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Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
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Probability for each momentum in one-loop diagram equal?

Why is the probability for each momentum in a loop (e.g. vacuum polarization) equal? Why has a infinite momentum the same probability to occur than a virtual particle with low moment. I know - these ...
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Doubt regarding Schwinger pair production derivation done in Schwartz (Errata checked)

In chapter $33$ Schwartz discusses Schwinger pair production using Euler-Heisenberg Lagrangian. My doubt starts in the paragraph above $(33.92)$. The calculation starts with Schwartz using optical ...
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Are classical probability a special case of quantum probabilities? How to show it?

First, let's say I have a classical system involving throwing a fair coin. There are two possible events $\{\text{head},\text{tails}\}$. Their respective probabilities are: $$ P(\text{head})=\frac{1}{...
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Position and momentum of an electron in a hydrogen atom [closed]

What can be expected as results for a set of position and momentum measurements of an electron in a hydrogen atom? I believe that for the position, we can expect something depending on the radial ...
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Trouble understanding the double slit probability measure

The probability of a particle going into either of two slits is given, classically, by $$ P=P(\text{slit}_1)+P(\text{slit}_2)=|\psi_1|^2+|\psi_2|^2=1 \tag{1} $$ This probability must be equal to $1$ ...
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What are the benefits and reasons behind considering a probability density distribution for electrons as opposed to a mere density distribution?

Before now, I had never questioned this matter. Why must we assume a probability density distribution (pdd) as opposed to a general density distribution (gdd)? Perhaps I have a misunderstanding. My ...
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Probability current density confusion

As we all know, the probability current density in quantum mechanics is defined as: $$\textbf{J}=\dfrac{\hbar}{2mi}(\Psi^* \nabla \Psi-\Psi \nabla \Psi^*)$$ For simplicity let us work in one dimension ...
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Why is $\Delta x$ or $\Delta p$ constant for a particular $\psi_n$?

We were asked to calculate $\Delta x \Delta p$ for the $\psi_0,\psi_1$ of the harmonic oscillator.And so we calculated the answers and verified that $$\langle T \rangle +\langle V\rangle = (n+1/2)\...
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Why does the correlation depend linearly on the angle, in the classical case of Bell experiments?

I am struggling to understand why we would predict the decay in correlation between the two measurements in a Bell Inequality experiment to be linear (see red line in this image from Wikipedia). I ...
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Examples of macroscopic systems with exponentially distributed lifetimes?

I was considering the statistical lifetimes of various light bulbs at first. However, upon further reading it seems that they tend to be approximately Weibull distributed with a shape parameter $k \...
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How does quantum theory allow a rock to turn suddenly into a duck? [closed]

Quantum theory does not allow a rock to turn suddenly into a duck. It does not allow any other bizarre transformation to happen either. This idea is a myth perpetuated by people who misunderstand ...
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What are we measuring in a quantum field when we square the wavefunction?

Suppose we are doing a measurement in a particular quantum field, i.e electron field. Are we looking for the probability of the electron to show up at that spot we are measuring or are we measuring ...
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263 views

Is the Uncertainty Principle a mathematical consequence or a physical consequence or both? [duplicate]

I am currently exploring the mathematical structure of Quantum Mechanics on an introductory level. A couple of books and online sources (including this website) stated how the Uncertainty Principle is ...
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What's the relationship between the definition of the uncertainty principle using standard deviations vs using $\Delta x$ and $\Delta p$?

So I've heard two different explanations of the uncertainty principle, both of which make sense on their own, but I'm having a hard time figuring out how they're connected. The first is that the ...
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Is it possible to create a consistent quantum theory such that the probability is the complex norm, instead of the square complex norm?

In quantum physics, the relation $$ \int_{-\infty}^{\infty} (\psi[x,t]^*)(\psi[x,t]) dx=1 \tag{1} $$ is paramount. What would the consequence be of defining the normalization condition as $$ \int_{-\...
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Why don't the gas molecules accumulate at one corner of an isolated box?

If for an isolated system in thermodynamic equilibrium all the accessible microstates are equally probable, why do the gas molecules in an isolated container, never accumulate at one corner of the box ...
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Probability of a quantum particle [duplicate]

Now recently I have started quantum mechanics and I understood the wavefunction but I don't understand why $|Ψ|^2$ gives the probability density of a quantum particle. Is there a reason or perhaps ...
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If superposition is just uncertainty, then how come quantum computers work? [closed]

If superposition is just uncertainty due to a particle changing on observation and not literally 2 things at once, how come quantum computers work while having qubits that are literally 1 and 0 at the ...
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Is the probabilistic nature of QM simply just randomness (does it exclude causality)? [closed]

I have read this question: What is the reason that Quantum Mechanics is random? where Puk says in a comment: I see. I would call both "random", with the degree of "randomness" ...
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How do we expand Bayes theorem to account for probability amplitudes?

The Bayes theorem simply states: $$ P(B | A) P(A) = P(A | B) P(B) $$ I wonder if there is something that can be meaningfully said as generalization of this relationship when the probabilities in ...
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Understanding superposition principle

Does the superposition principle actually tell us about our inability to predict what happens during the course of the experiment? Does it tell that, since an experiment has multiple outcomes ( i.e , ...
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556 views

Cannot understand the Probability current of quantum mechanics

In Quantum mechanics we have a concept of probability current. But I can't understand what it means that 'probability flows'. All I can know is that at a fixed point the probability of finding the ...
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Average distance of a freefalling body in random intervals

This is taken from the first example in Griffith's intro to QM: Suppose I drop a rock off a cliff of height h. As it falls, I snap a million photographs, at random intervals. On each picture I ...
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I'm confused about two aspects of quantum mechanics: 1.probabilistic nature of quantum 2.time evolution operator [duplicate]

first >> we know that quantum mechanics works with a probabilistic nature so that we can't say " what will happen? " but " what might happen? " second >> we can ask how ...
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Continuity equation in QM

I found this question in a quantum mechanics exam: What is the physical interpretation of the continuity equation $\frac{\partial\rho}{\partial t}+\frac{\partial j}{\partial x}=0$? Here $\rho(x,t)$ is ...
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What is a probabilistic physical theory?

What do we mean when we say that we have a probabilistic theory of some phenomenon? Of course, we know from experience that probabilistic theories "work", in the sense that they can (somehow)...
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Why do we describe probability amplitude rather than probability itself in quantum mechanics?

In the quantum mechanics, the dynamics of quantum system are described in terms of probability amplitude. However, we want to calculate the probability in the end which can be measured. Why don't we ...
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Angular instead of radial probability density in spherical coordinates

By the Born rule in Quantum Mechanics, a state's complex wave function $\Psi(x,t) \in L^2$ gives probabilities when we take its complex norm $\overline\Psi(x,t)\cdot\Psi(x,t) = |\Psi(x,t)|^2$. In this ...
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Small time solution to Fokker-Planck equation

In reference to this note, a specific Focker-Planck equation with initial condition $W(\rho, t=0)=\delta(\rho-1)$ have the solution $$W\left(\rho,t\right)=\dfrac{e^{-\frac{t}{4}}}{\sqrt{\pi}t^{\frac{3}...
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Perturbing PDF with spatial dependent perturbation

Let's consider a PDF $\rho(x)$, with normalization 1. Let's perturb it in the following way: $$ \rho(x+\varepsilon F(x) ), $$ with $\varepsilon$ small. I impose that the perturbed PDF is again a PDF ...
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Is the collapse of a classical probability density a puzzle analogous to the wave function collapse puzzle in quantum mechanics?

Famously, the collapse of the wave function is considered one of the biggest puzzles of quantum mechanics and motivates people to take ideas like the many-worlds interpretation seriously. Something ...
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Intuitive derivation of the Husimi Q function from the Wigner function

The Wigner function is given by $$W(\alpha)=\frac{1}{\pi^2}\int \text{e}^{\alpha \beta^*-\alpha^*\beta}\text{Tr}\left(\hat \rho \hat D(\beta) \right) \text{d}^2\beta,$$ where $\hat D(\beta)=\text e^...
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Interpretation of probability current density of hydrogen atom wave function

I'm trying to understand the physical interpretation of the value of the probability current density of an electron in a hydrogen atom: $$j_r=0$$$$j_{\theta}=0$$ $$j_{\phi}=\frac{h}{rsin\theta}|\psi_{...
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Effect of commuting observables on the probability of measuring a certain value [closed]

Say you can measure $3$ observables $(A, B, C)$ and you do the measurements in two different ways. $\newcommand{\ket}[1]{|#1\rangle} \newcommand{\bra}[1]{\langle#1|} \newcommand{\braket}[2]{\langle#1|#...
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What is the probability of QED interactions?

In Feynman diagrams of QED vertices of interaction are often labelled by the amplitude sqrt(alpha) where alpha is the e.m. fine structure constant. When higher order diagrams are constructed, each ...
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Verifying the Gaussian Transformation of $exp\left\{\frac{1}{2}\sum_{i,j} S_i J_{ij} S_j\right\}$ from “Advanced Mean Field Methods”

The book Advanced Mean Field Methods mentions the following equation as a result of a "simple gaussian transformation". $$ exp\left\{\frac{1}{2}\cdot\textbf{s}^T \cdot \textbf{J} \cdot\textbf{s}\...
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What is the probability of finding a specific value of energy? [closed]

knowing that energy is given by $E_{n}=\frac{n^{2}\pi^{2}\hbar^{2}}{2ma^{2}}$ and that $$|\psi(t=0)\rangle=\frac{1}{\sqrt{6}}|\phi_{1}\rangle+\frac{1+i}{\sqrt{12}}|\phi_{2}\rangle+\frac{1-i}{\sqrt{4}}...
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“Probabilities are the ghosts of quantum mechanical amplitudes”

I came across this quote today; [Quantum computers] process information using quantum mechanical amplitudes. And probabilities are sort of the ghosts of amplitudes after they have been degraded to ...
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Why are probabilities for each micro-state equal within a micro-canonical-ensemble?

This question is about statistical mechanics: Why does it make sense to postulate, that in thermal equilibrium all micro-states with fixed U within a micro-canonical ensemble are equally probable? ...
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How do you randomly draw samples from the probability density function of the quantum harmonic oscillator in MATLAB?

The Quantum Harmonic Oscillator in the ground state is specified by the following Gaussian PDF in two dimensions: $$p(x,y)= \frac{M \omega_x}{\pi h}\sqrt{ \frac{\omega_y}{\omega_x}} e^{-\frac{M}{h}(\...

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