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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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Unit of Spectral Emissive Power

I understand Spectral Emissive Power as the total amount of energy carried by photons having the same wavelength (energy), and it has the unit of (W/m2.um). I could not grasp the physical ...
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1answer
49 views

Why quantum map must be hermitian?

Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $ the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$ Why $\mathcal{...
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Number of Isotopes created with decay <-> Chain Yield

The chain yield (or fission yield) states how many isotopes with a certain mass $A$ are created with the decay of $^{235}$U. But how do we know the fractions of specific Isotopes that are created ...
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4answers
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How is quantum mechanics consistent with statistical mechanics?

Let's say we have an harmonic oscillator (at Temperature $T$) in a superposition of state 1 and 2: $$\Psi = \frac{\phi_1+\phi_2}{\sqrt{2}}$$ where each $\phi_i$ has energy $E_i \, .$ The probability ...
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Understanding the radial distribution function [duplicate]

I am confused why the maximum of the radial distribution function for 2p orbital is closer to the nucleus than that for 2s orbital. Doesnt this mean that there is a higher chance of finding 2p orbital ...
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122 views

What's the difference between coin flipping and quantum physics?

Watching this TED video, which explains about Quantum Physics, I'm confused, please let me ask a question. There is one coin whose state is "Head" on the table. First, the computer decides whether ...
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1answer
31 views

Probability of a system in the canonical ensemble

In the canonical ensemble, we have the state of system $x_s$ and the state of the environment $x_e$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states ...
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1answer
211 views

Dirac Delta Function and Position [duplicate]

How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
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1answer
163 views

Interpretation of the Boltzmann factor and partition function

$$p_i = \frac{ \exp\left(-\frac{\epsilon _i}{k_BT} \right)}{Z} $$ $$ Z= \sum_{i} \exp\left(-\frac{\epsilon _i}{k_BT} \right)$$ A) Is $p_i$ the probability of the system having an energy equal to $\...
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1answer
72 views

Does the wavefunction probabilities have to sum to 1? [duplicate]

In quantum mechanics we are often told that $\int |\psi(x,t)|^2 dx^3 =1$. i.e. the probabilities have to sum to 1. And that this implies the time evolution operator is unitary. But can't we define ...
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1answer
242 views

What is an example of a hidden variable model that meets the bound of Bell's inequality?

Following https://en.wikipedia.org/wiki/Bell%27s_theorem: The best possible local realist imitation (red) for the quantum correlation of two spins in the singlet state (blue), insisting on perfect ...
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4answers
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Why does the chain yield in nuclear fission sum up to 200%?

The table of nuclides states the fission yield, or chain yield $Y$, the percentage of decays of $^{235}$U that lead to an isotope with mass number $A$. Why do they sum up to 200%? Wikipedia just ...
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1answer
57 views

Fitting an Ising Model with Probabilities

Question How to use the observations to fit an Ising model? After self-studying for several days, my current guess is: $\theta_{ii} = \log[P(X_{i} = 1)]$ $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$ ...
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3answers
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Interpretation of the magnetic potential ($A$-field) in the quantum mechanical probability of current

The probability of current in quantum mechanics when the is a magnetic potential, A, is defined as: $$\boldsymbol j=\frac{1}{2m}(\psi^*\hat{\boldsymbol p} \psi-\psi\hat{\boldsymbol p}\psi^* -2q{\...
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How could someone calculate the coulomb barrier transparency in 3 dimensions [duplicate]

Transparency is also called probability of transmission, or probability of quantum tunneling.
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2answers
43 views

Definition of continuous distribution (intuitively understanding)

Def: A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve. The probability of observing any ...
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2answers
84 views

Linear combination of 2 spherical harmonic functions

The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
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3answers
1k views

Why can't a particle penetrate an infinite potential barrier?

I am studying basic quantum theory. My question is: Why can't a particle penetrate an infinite potential barrier? The reasoning that I have applied is that particles under consideration have finite ...
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1answer
38 views

Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time?

In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
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1answer
53 views

Finding total flux of probability current through a sphere

For a wavefunction: $$\Psi(\textbf{x}) = e^{ikz} + \dfrac{f(\theta)}{r}e^{ikr}$$ Where $z = r\cos(\theta)$. The probability current $J$ is then given by: $$J(\textbf{x}) = J_1(\textbf{x}) + J_2(\...
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6answers
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What is the probability for an electron of an atom on Earth to lie outside the galaxy?

In this youtube video it is claimed that electrons orbit their atom's nucleus not in well-known fixed orbits, but within "clouds of probability", i.e., spaces around the nucleus where they can lie ...
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2answers
69 views

Understanding entropy, information, and randomness

In a statistical mechanics book, it is stated that "randomness and information are essentially the same thing," which results from the fact that a random process requires high information. More ...
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1answer
65 views

Is a light pulse equivalent to a burst of photons

Reading about photons you get all sorts of weird statements like "time is frozen for a photon", "the photon dies the instant it is born" and "the photon is everywhere and nowhere", etc. Probably these ...
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2answers
63 views

Probability to get an Eigenvalue of Angular Momentum Operator on an Arbitrary Ket

Hello physics SE community, I am currently working on Principles of Quantum Mechanics by Shankar and i get stuck in page 336 (its not even an exercise). It basically said that "we may expand any $\...
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0answers
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Approximate probability of a person's wavefunction collapsing to the moon?

As a sort of introductory concept to quantum mechanics, I've heard many explain that there's a small but nonzero probability of unlikely events happening: your hand quantum tunnels through the desk, ...
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3answers
112 views

Observer in Many Worlds Interpretation [duplicate]

Something has been bothering me about the Many Worlds Interpretation. Proponents of it (e.g. Sean Carroll) often claim that it does away with the observer, or at least the paradox-inducing status the ...
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1answer
64 views

Statistics of 1D discrete random walks

I have already asked this question in Math.SE. Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a ...
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1answer
46 views

Finding the uncertainty from a probability distribution?

If you have two properties, $A$ and $B$, that do not commute, and thus have a commutator $C$, and the uncertainties $\Delta A$ and $\Delta B$ obey the relation $$(\Delta A)(\Delta B)\geq \frac{1}{2}|C|...
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49 views

Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...
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2answers
254 views

Can the entropy of mixing be negative? [closed]

There is a general notion that the entropy of mixing should always be positive (or zero if we are mixing exactly the same stuff). However, I have a seeming counterexample at hand. Consider a box ...
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3answers
302 views

Probability of finding a particle in 1D box from classical view

The probability of finding a particle in a 1D-Box (from the classical view, the particle behaves as a particle but not a wave) is the same everywhere in the box. This seems to me quite ...
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2answers
119 views

Heisenberg uncertainty principle

Is it true that an electron is fundamentally probabilistic in nature? That the Heisenberg uncertainty principle is not describing our limited ability to measure the particle's position and momentum ...
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1answer
178 views

Write the dimension of 1D wave function? [closed]

I want to know how to find the dimension or unit of one-dimensional wave function
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208 views

What are the units of probability density?

Units of probability density? If bound electron is thought of as a cloud of charge, and it's charge density is proportional to the probability density. Then coulombs /m3 proportional to?
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3answers
163 views

Normalisation of quantum states: why?

We all learn that quantum states need to be normalized, as they are associated to probabilities which needs to sum up to one. However, I would like to know whether you have other valid reasons to ...
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32 views

Why the distribution of Fluctuationg force in brownian motion has gaussian distribution?

I am reading the Zwanzig's book and I have a confusion about the average of the fluctuating force and its distribution. As it says $F(t)$ is a random variable that means it has a probability ...
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1answer
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Where is the difference to a quantum mechanic system with collapsing wavefunction in this experimental setup?

Let’s use a disc throwning system, where two porcelain plates are thrown at once and in different directions, one with its axis horizontal and the other vertical. To catch the plates we use a detector,...
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1answer
62 views

Difference between Wavefunction collapse and throw of dice

It might be a really stupid question and I think I am trivializing it. I am not able the understand the big issue with collapse of wave function. So we have set of probabilities and when you measure ...
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1answer
80 views

Intuitive understanding of a wave function

Looks like wave function is an abstract mathematical object. I was trying to see if there is a simple way to visualize this. Can someone please help with that? I was thinking may be we can think that ...
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1answer
77 views

Implementing a Monte Carlo Simulation for the Gaussian Model

I want to implement a Monte Carlo simulation of the 1D Gaussian Model (the continuous generalisation of the Ising Model). That is the statistical mechanical model with the following Hamiltonian: $$ H ...
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3answers
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Is there a traditionally accepted threshold probability at which highly unlikely becomes impossible? [closed]

Many events which by any practical definition are impossible have extremely low but nonzero probability of occurrence. For instance, the positions of oxygen molecules in a room are basically random ...
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1answer
44 views

Orbitals, shapes and wave function relation

Are the shapes of orbitals like $s$ which is sphere due to wave function or due to square of the wave function?
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24 views

Correlation function and power spectrum of discrete time Gaussian noise summed with a time delayed version of itself

Suppose we have a process $\zeta(n) = \xi(n) + \xi(n + 1)$ Where $\xi(n)$ is discrete time white noise process, where the values taken at different times are from identically distributed Gaussian ...
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1answer
77 views

Small psi in the time-independent Schrödinger equation

I'm a total beginner in quantum mechanics, and I am learning about time-independent Schrödinger equation. we separate the wave function into two functions $$\Psi(x, t) = \psi(x)\phi(t).$$ Does the ...
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1answer
133 views

Why can we predict any experimental outcome, given a probability density over quantum states?

There is a very interesting answer given by Peter Shor in this website here. However, I admit I don't fully understand it. In particular, I don't understand: If we have a probability density μv on ...
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1answer
45 views

Probability of $\frac{-1}{\sqrt2} S_x + S_z $

I have a State $\left|\Psi\right>=\frac{\left|1\right>+\left|0\right>}{\sqrt{2}},$ in the $z$-Spin basis and want to calculate the probability of this state for the eigenvectors of the ...
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1answer
102 views

Angle of particles hitting a wall - Kinetic theory of gases

In Blundel's "Concepts of thermal physics" it says that the number of particles hitting a wall with speeds between $v$ and $v+dv$ and angle between $\theta$ and $\theta +d\theta$ to the normal of the ...
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2answers
225 views

Energy of room. Ideal gas law

I have been following Blundel's "Concepts of thermal Physics" and I got to the derivation of the ideal gas law. And it all made sense, we made a couple of assumptions and approximations, but then I ...
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4answers
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Are probabilities of a measurement in quantum mechanics absolute?

Consider an experiment in which an electromagnetic wave whose polarization is along an angle $a$ with the $x$ axis is sent through a polarizer whose polarizing direction is along $y$ axis. The ...
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0answers
60 views

Why does $\langle x' | x \rangle$ give the Dirac delta distribution? [duplicate]

I'm having difficulty understanding why the following is true: $$ \int_\mathbb{R} \langle x' | x \rangle dx = \int_ \mathbb{R} \delta(x-x')dx$$ where $\delta(x)$ is the delta distribution. Are we ...