# 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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### Computing the probability of a measurement of energy for a free particle [closed]

$\newcommand{\ket}{\left|#1\right>}$ $\newcommand{\bra}{\left<#1\right|}$ $\newcommand{\braket}{\left< #1 | #2 \right>}$ Problem Consider a free particle, with no spin, in one ...
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### In statistical mechanics, why is one "allowed" to treat classical systems probabilistically?

Is the essential argument that these systems are microscopically chaotic enough that we can approximate their evolution as random (vastly simplifying calculations) and still make accurate experimental ...
23 views

### Average probability of sampling all clusters

A system consists of $N$ nodes. The nodes are distributed into $M$ clusters such that each node belongs to a unique cluster. Each cluster $i$ has one unit of weight, $w_i = 1$. Thus the initial weight ...
39 views

### Is there a general construction for three-outcome qutrit POVMs?

For qubits, I can consider the General POVM elements: $M_{\pm} = \frac{1}{2}(I \pm \hat{n}\cdot\overline{\sigma})$ where $\sigma$ is a vector containing the Pauli matrices and $\hat{n}$ a vector with ...
39 views

### Probability density of electrons in double slit experiment [closed]

My professor gave some vague question in the double slit experiment setup which uses electrons instead of light, asked to found out the intensity of electrons on screen as a function of number of ...
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### Why are expectation values of an observable important in QM?

I've been reading that expectation values of an observable is all what we can get and are the key quantities of the theory, but performing the same experiment many times would generate a distribution ...
1 vote
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### Most probable position of finding an electron represented in cartesian and spherical coordinates

Consider the probability of finding an electron within a region: $$P= \iiint |\psi(x,y,z)|^2 dxdydz$$ I would think that The probability of finding an electron at a single point in space would be ...
1 vote
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### Probability of two atoms Bonding [closed]

I am trying to work out an expression for two individual atoms in a vacuum of small volume, $V$, and the probability that they will bond together. I have attempted looking at the quantum approach, in ...
1 vote
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### Is anything truly stochastic? [duplicate]

If everything in the universe happens according to rules, thermodynamic or otherwise, then how would anything (or any choice) ever be stochastic? Multiple choices might be probable, but in any instant ...
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### Boltzmann distribution derivation from Lagrange Multipliers

Context In the derivation of the Boltzmann factor and the canonical partition function based essentially on Lagrange multipliers presented here, the equalities, \begin{align*} p_j &= \frac{1}{Z} e^...
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### Why must the propagator exponent be imaginary?

In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
28 views

### Fokker-Planck: Uncertainty Propagation

I am interested in propagating the probability density function along the sampled trajectories having parametric uncertainty $a$. However I discovered that the fokker-planck equation for that system ...
1 vote
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### Is the Boltzmann distribution 'memoryless'? What is the physical interpretation?

The chance of occupying state with energy $E_i$ is given by the Boltzmann distribution: $$P(E_i)= \frac{1}{Z} \exp \left( \frac{-E_i}{kT} \right)$$ The exponential distribution is a common ...
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### How does indeterminism lead to deterministic laws?

Philosophers and many scientists seem to distinguish between the macro and micro world a lot. Things in the micro world seem to be indeterministic, atleast through the standard interpretation of QM. ...
110 views

### Free particle probability to go from $a$ to $b$ [duplicate]

Feynman and Hibbs write that the probability for a particle to go from $a$ to $b$ is \begin{equation*} P(b,a)=|K(b,a)|^2 \end{equation*} The kernel for a free particle is given as \begin{equation*} K(...
1 vote
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### Is the Stirling approximation fundamental to the derivation of the Boltzmann distribution?

In the derivations of the Boltzmann distribution I've encountered, the Stirling approximation seems to play a pivotal role. From these derivations, I've gleaned that the Stirling formula might be ...
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### Is the path integral emergent?

I have recently read a couple of papers on lattice QCD and found that there is a well-established connection between Boltzmann distribution and the path integral in QFT (disclaimer: I am not a huge ...
30 views

### Gaussian approximation integrated over a unit square, written in terms of error function?

So I am new here, and I apologize for my last question being badly formatted with too much extra information making it just painful to read. I am struggling with getting down a 2D Gaussian estimation ...
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### What is the cross-section impact probability for an extended object moving trough a field of randomly moving particles?

In trying to get an estimate of the probability for an earth orbiting (LEO) satellite to collide with small debris particles. So I need to understand what is the impact probability for an orbiting ...
63 views

### Can Quantum Heisenburg Spin Model be regarded as a random walk?

Can Quantum Heisenburg Spin Model be regarded as a random walk ? We know that the 1D classical Ising Model can be regarded as a 1D random walk. For example, in the simplest case of the 1D Ising model, ...
553 views

### Total cross-section for Bhabha scattering

The Bhabha scattering differential cross section is given by $$\frac{d\sigma}{d\Omega}=\frac{\alpha}{2s}\left(\frac{3+\cos^{2}\theta}{1-\cos\theta}\right)^{2}$$ where $\theta$ denotes the angle of the ...
1 vote
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### What is the interpretation of these hydrogen probability density diagrams?

In the diagram above, what is the interpretation of all of the individual renders? Does the hydrogen atom continuously change between these states? For example, will $(n, l, m_l)$ become $(2, 0, 0)$ ...
955 views

### Is unitary time evolution the same as obeying the Schrödinger equation?

In this question, the answer says that unitary time evolution means that probability is conserved. Is this the same as saying that a system obeys the Schrödinger equation?
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### Distributions "more singular than a Dirac delta" must have negativity

I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
1 vote
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### Expectation Value Definition [closed]

I think I might just not be thinking things completely through here but I've seen the expectation value of a Hermitian operator written in many sources as both \langle A \rangle=\frac{\langle\psi|A| ...
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### Probability distributions of bounded measurement results?

Say that I am measuring a length very inaccurately. For instance, I might measure $1m$ with an uncertainty of $50cm$. If I model the probability distribution as a Gaussian with $\sigma=50cm$, I run ...
1 vote
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### Quantum physics: Probability of first state using dirac notation

I am struggling with this question. Let a two-dimensional Hilbert space H has orthonormal basis vectors |A⟩ and |B⟩. Consider a vector in H ⊗ H: $|v⟩ = α_1 |AA⟩ + α_2 |AB⟩ + α_3 |BA⟩ + α_4 |BB⟩$ . ...
1 vote
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### Probability density function (pdf) of a Wiener process

I am working through a book right now in which there is a short introduction to Brownian motion and Wiener processes. I assume it is not treated nearly as rigorous as in mathematics but still more of ...
1 vote
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### Probability of measuring velocity

In a one dimensional problem in quantum mechanics, does the probability of measuring momentum greater than $p_0$ correspond to the probability of measuring velocity greater than $v_0= \frac{p_0}{m}$?
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Let's say due to a nuclear reaction a radionuclide of half-life $T_{1/2}$ was created. I am trying to find out what will be the probability of that radionuclide undergoing radioactive decay within ...