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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An entropy-like measure for a pure state? [closed]

It came to my mind that can we describe the flatness of a pure quantum state $\psi$ in terms of an entropy-like measure? Obviously, a Dirac delta -like distribution $|\psi|^2$ has a low "entropy&...
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QM and a variant of Cauchy-Schwartz inequality

Does the following variant of Cauchy-Schwartz inequality have an uncertainty relation like interpretation in quantum mechanics? The result is from Mathematical Inequalities: A Perspective by Cerone &...
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Maximum Entropy Principle with inequality constraints

It is well known that, maximizing the entropy of the joint distribution $P(x_1,...,x_n)$ of a random vector $(X_1,...,X_n)$ subject to equality constraints for the mean vector ($\mu$) and the variance ...
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Intuitively, why does Quantum Mechanics involve a sum over all possibilities?

I understand that one can just mathematically derive the path integral from the Schrodinger equation. I'm looking for an intuitive explanation in contrast with classical mechanics. Consider a ...
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Geometric Brownian Motion versus Ornstein-Uhlenbeck process

The Geometric Brownian Motion model is a continuous-time stochastic process in which a particle move according to a random fluctuations (Wiener process) and a drift term. The corresponding stochastic ...
1 vote
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Changing sign at a Ornstein-Uhlenbeck process: mean, variance and likelihood

I am working with a multivariate Ornstein-Uhlenbeck process and its statistical properties (likelihood, expected values and variance). The Ornstein-Uhlenbeck process can be described as a random walk ...
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Beta-decay of a Tritium [closed]

Calculate the probability of a Tritium beta decay into ground state of a Helium ion with perturbation theory. What should I start with? I lack any ideas.
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Examples when wavefunction is changing but probability density is constant?

What are examples of wavefunction that changes with time but the square of wavefunction is constant?
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Multiplication of probability in quantum mechanics

Consider a ket-space spanned by the eigenkets of an observable $A$ and let $B$ be an additional observable on the same ket-space. We can build a filter that only lets an eigenvalue $a$ of $A$ through ...
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Is there real physical possibility for a "macroscopic" object to undergo quantum tunnelling? [duplicate]

According to quantum mechanics, there is fantastically (astonishingly, astronomically, infinitesimally, ridiculously etc.) small probability for a book on a table to quantum tunnel through the table. ...
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Will the probability for tunnelling go completely to zero? [closed]

According to quantum mechanics, the probability for quantum tunnelling (of an object) never become completely zero, no matter how "big" is the height and the thickness of the barrier. ...
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Are the probability amplitudes in the continuous case, always the coefficients of the eigenstates of the wave-fun. expansion?

Assume that we have a Hamiltonian eigenvalue problem with continuous energy eigenvalues $E$. Griffiths says that the inner product of an eigenstate $ψ$ with the total wavefunction $Ψ$ gives the ...
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Photon absorption in different layers

Let's suppose that space is divided into different regions: for $x < 0$ there is just vacuum space; for $x$ between 0 and $t_1$ space is filled with a material of type 1; for $x$ between $t_1$ ...
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What would have to happen for atomic subshells to have even orbitals instead of odd orbitals?

The orbitals in the subshells, due to ℓ and mℓ, are always odd: 1, 3, 5… • ℓ = 0 equals 1 orbital (s) of mℓ = 0. • ℓ = 1 equals 3 orbitals ($p_z$, $p_x$ and $p_y$) of mℓ ={–1, 0, 1}. • ℓ = 2 equals 5 ...
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Reference for Classical probability theory implemented on General Relativity

I am looking for a reference for something like : a probability density put on the ADM phase space, thereby making the metric probabilistic in the classical probability sense. But not exactly this. I ...
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1 answer
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Higher order (order > 2) derivatives of free energy - higher cumulants in statistical mechanics

The first derivatives of free energies generally give relationships between thermodynamic conjugate pairs, like entropy $S$ & temperature $T$ pressure $P$ & volume $V$ and so on. The second ...
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Magnetic field virtual photons

When an electron travels through a magnetic field, it alters course and by doing so it emits synchrotron radiation. What we call magnetic field, as I understand it is a mathematical "...
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Amplitude of $e^-+\gamma\to e^-+\pi^0$ scattering in QFT [closed]

The exercise The goal is to compute the transition probability of the process $$e^-+\gamma\to e^-+\pi^0 $$ according to the interaction Lagrangian $\mathcal L=g\varepsilon^{\mu\nu\rho\sigma}\partial_\...
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Probability of finding a number of system particles with a given energy

Consider a system composite of $n$ independent particles. Let the number function $N: [0, +\infty) \rightarrow \{0, 1,..., n\}$. Given an energy $E \in [0, \infty)$, the number $N(E)$ is the number of ...
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Probability of branching times under a Ornstein–Uhlenbeck-Yule process

According to Edwards, 1970 the probability density of the branching times in a Brownian-Yule branching process can be expressed as: \begin{equation} P(\mathbf{u'},n|\lambda,n_0,T)=\lambda^{n-n_0}\frac{...
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Corollary of Wiener process and the the appearance of $\sqrt{t}$

One of the properties of a Wiener process is given by (taken from https://en.wikipedia.org/wiki/Wiener_process), A corollary useful for simulation is that we can write, for $t_1 < t_2$: $$W_{t_2} =...
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Why it is a longstanding challenge to reproduce Born rule in Everettian QM?

I'm reading this Sebens and Carroll's paper on Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics. Where they presented their derivations of the Born rule and ...
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Questions on the double slit experiment

I understand how in the original double slit experiment light goes through two slits and you get an interference pattern because of constructive and destructive interference. But when you put an ...
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On the statistical meaning of density of states (DOS)

According to the so-called law of the unconscious statistician: The expected value $\langle \cdot \rangle$ of a measurable function of ${\displaystyle X}$, ${\displaystyle g(X)}$, given that $X$ has ...
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Accidental coincidence formula

Suppose you have a setup with $n$ scintillators coupled with $n$ PMTs. These signals are passed to a discriminator. I am observing some signal, let's say this signal is coming from cosmic rays and I ...
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Generally, Probability of given level. Particularly, Particle in a box probability of a given level 0?

I had the horrifying realization that I don't fully understand how to find probability of a state with a given energy. Referring to this post (Calculating the probability of a given energy) and ...
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Two subsystems in thermal contact in the microcanonical ensemble, confusion in the terminology of the microcanonical and canonical ensemble

I am confused about the terminology used in an example illustrating the microcanonical ensemble. The example is found in chapter 4 (p.176) of Gould and Tobochnik's manual Statistical and Thermal ...
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Does the 'Equal a priori probability' statement apply to every physical system?

I'm currently studying Statistical Mechanics, but I'm having trouble with grasping the concept of 'equal a priori probability' and especially the results that stem from it. So, the concept of equal a ...
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Does the difference between the definition of probability in physics and probability in mathematics really matter?

The realistic definition of transition probability in physics is well defined and constrains the probability to rational numbers. The abstract definition of probability in mathematics is also well ...
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Is there any continuous distribution (except Gaussian) that can be characterized by their first and second moments only? [closed]

I am trying to figure out why Gaussian distribution is uniquely significant in random physical processes. I think the answer lies in its characterization by only two moments. In a way it is the ...
1 vote
2 answers
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Probability density of fermions in QFT

I was following the book 'Quantum Field theory and the Standard model' written by Mattehw D. Schwartz and I found something unconvincing in p.174 of the book. In that page, it argues that the zeroth ...
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What is the meaning of the wavefunction?

I'm taking an introductory quantum mechanics course (at the undergraduate level), for which we're following the Griffiths text. I'm doing well in the course, and can perform calculations fairly ...
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Given resulting isotope number, probability of a certain atom

A single atom of francium-211 decays with a resulting isotope with an atomic mass number of 207. The probability of how the atom will decay is in the table attached. I want to ask if question a) is ...
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2 votes
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About the nature of particle in a box

Suppose we have a particle in a box, which indicates an infinitely deep potential well and we have our particle in an unequal superposition of the first two energy states, which we can write as $$a\...
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Trouble with Einstein coefficients - what is the meaning of the transition probability?

Perhaps it seems to be a not very intelligent question, but I am unfortunately not able to understand what the probability per second that a molecule will absorb a photon is, as part of the theory of ...
1 vote
1 answer
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What is the probability distribution the angular component of the electron in the hydrogen atom?

The angular component of the electron in a hydrogen atom is the family of spherical harmonic functions, Y(θ,Φ). I have seen the angular function probability distribution graphically represented as the ...
5 votes
1 answer
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In Fermi's Golden Rule, does the transition probability increase linearly with time or quadratically with time?

When deriving Fermi's Golden rule, we get that the probability of a quantum system transitioning from an initial state $|i\rangle$ to a final state $|f\rangle$ is $$P_{i\rightarrow f}(t)=\frac{|V_{fi}|...
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Combining multiple measurements: how to calculate the uncertainty in this case? [closed]

Let's say I have an experiment where I measure some variable $D$ (in the experiment I am grading it is the diffusion constant but I want to keep the discussion general). There are two runs of ...
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Moments of coordinates of uniform distribution on unit sphere

Suppose $(\alpha_1, \ldots, \alpha_n)\in\mathbb{C}^n$ are drawn uniformly at random from the unit hypersphere. Since $|\alpha_1|^2 + \cdots + |\alpha_n|^2 = 1$, by symmetry it is clear $\mathbb{E}[|\...
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DART crash on Dimorphos: computation of orbital period change

What is the distribution of expected changes in the period of Dimorphos' orbit around Didymos when the spacevehicle DART crashes against it?
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How can you modify the odds of winning at a wheel spin?

So lets say you have a wheel divided in x sector You want to rig the wheel to decrease the chances of someone landing on the green circle. Now lets say you do it by putting magnets in the red object ...
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How can I show that the negative region of Wigner quasi-probability distribution is small enough than $h$?

Wigner quasi-probability distribution is the main tools used in the formulation of Quantum mechanics on phase space which is equivalent to usual formulations. This distribution can have negative ...
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How to efficiently get the largest probabilities / amplitudes of a quantum state stored as an MPS?

Let's say, that we have the following pure, superposition state $$ |\psi \rangle = \frac{1}{\sqrt{2}}|000001 \rangle + \frac{1}{2}|101101 \rangle + \frac{1}{2}|100100 \rangle $$ stored in the MPS form....
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How can I prove this identity? [closed]

I'm considering the stochastic tree. When root node is activated, the $n$ child nodes get the signal. But the probability of activation is $p$. $d$ is index of layer and starts from 1 that is ...
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Question about Shannon information entropy

Where can I find a graphical representation or a detailed explanation illustrating the relationship between Shannon information entropy, probability and available information?
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How can we generalize the Poincaré recurrence time to other sets of events?

I know that the Poincaré recurrence tells us that every given physical arrangement in a finite physical space will eventually recur given enough time. However, as I was thinking about it, even if my ...
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Other than preserving probabilities, what further conditions are needed for a ray transformation to be a symmetry?

In Weinberg's QFT volume 1, section 2.2, he talks about "further conditions" for a ray transformation to be a symmetry. After going through Chapter 3, I still don't see what he means by this....
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Evolution of a position state in an infinite well potential

Let the potential be $$V = \infty \hspace{3cm}(0>x, x>L)$$ $$V = 0 \hspace{3.7cm}(L>x>0).$$ Now, we measure the position of a particle and discover it is located at $L/4$. What is the ...
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Transition from discrete case to a continuous case with regards to the Born's Rule

I learned that given that the eigenvalue equation is $$ \widehat{A}\left|u_{n}^{i}\right\rangle=\lambda_{n}\left|u_{n}^{i}\right\rangle $$ where $ i \in\{1,2, \ldots, g_n\} $, and that the state $ |\...
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Probability of a state "being found in another" [closed]

I'm sorry if something gets lost in translation, as my professor wrote all questions in portuguese, but what does it mean to ask the probability of finding "state $|\alpha \rangle$ in state $|\...

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