# 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.

745 questions
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### I've read that $\langle a | b\rangle$ is a probability amplitude but $\langle a | a\rangle$ is a probability. Why the inconsistency?

I'm studying elementary quantum mechanics, and I've read that $\langle a \vert b \rangle$ is the probability amplitude of a transition from state $a$ to state $b$. Thus, $|\langle a | b \rangle|^2$ ...
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### Can someone explain the concept of 'Negative Probabilities' in an intuitive manner? [duplicate]

Can someone explain the concept of Negative Probabilities in an intuitive manner? I can't seem to understand this concept. I hope someone can explain this concept in an intuitive manner.
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### How to find this probability with the microcanonical ensemble?

When dealing with isolated systems we are dealing with the microcanonical ensemble. In that case, we suppose that each individual microstate has the same probability. So if $\Omega(E)$ is the number ...
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### Are Ross-Littlewood-sequences or negative probabilities possible in physics?

There are claims that the Ross-Littlewood paradox could be simulated in physics. See https://stats.stackexchange.com/q/315502/ and in particular Paul's answer there. Also a solution of the Einstein-...
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### Why are wave packets constructed so that the maximum probability occurs at K0 (the average wave number)?

The definition of a wave packet I have been given is that it is "a superposition of many plane waves, with wave numbers grouped around an average value $k_0$". I was told that, for a particle we want,...
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### Topological entropy in Markov chains

Given a finite Markov chain, how do I find the topological entropy $h_T$? Furthermore, I should compare it with the Shannon entropy $h_S$ and show that $h_T\leq h_S$. Is this a general fact? This ...
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### How deterministic nature of our world emerges?

Quantum mechanics shows that nature is non-deterministic. But the world we see around us seems deterministic. Take for an example: harmonic oscillator when $n$ becomes very large the probability ...
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### Probabilities with density matrices

Given two mixed states $\rho$ and a $\sigma$, does it make sense to say that the probability of $\rho$ being in the state $\sigma$ is given by $Tr(\rho \sigma)$? It seems to me that the answer must ...
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### Postulates of inner product

In quantum mechanics, two fundamental properties of inner products (J.J Sakurai) Chapter 1.2, are: $\langle \alpha|\beta\rangle = \langle \beta|\alpha\rangle^*$ $\langle \alpha|\alpha\rangle \ge 0$ ...
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### What's the distribution of scattering angles for hard spheres with random impact parameter?

I am modelling the scattering of hydrogen atoms against each other. In this model, one hard sphere scatters elastically off another hard sphere, they are identical with radius $r$. They meet with ...
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In one of my homework in Quantum mechanics, I was asked to find $|Ψ(x,t)|^2$, where \begin{align} Ψ(x,t)&=1/\sqrt{10}[3ψ_1(x)e^{-iE_1t/ħ}-ψ_3(x)e^{-iE_3t/ħ}]\, ,\\ &=1/\sqrt{10}[3\sqrt{2/a} \...
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### Normal Distribution vs Poisson Distribution

I am not sure that I understand when to use Normal Distribution and when to use Poisson distribution! For example, in RF communication the channel noise is mainly modeled as Normal Gaussian ...
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### Probability of a specific energy state

We consider the normalized wave function: $$\psi(x,t) = \sqrt{\frac{2}{3}}\psi_0(x)\exp\left(\frac{-iE_0t}{\hbar}\right) + \sqrt{\frac{1}{3}}\psi_1(x)\exp\left(\frac{-iE_1t}{\hbar}\right)$$ To ...
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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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### 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 $t$...
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 - ...