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.
1,449 questions
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Is the momentum wave function's square amplitude always time-invariant for a free particle?
I have noticed whenever working with free particles that the square amplitude of the momentum wave function $|\Phi(p)|^2$ ends up being time invariant, so I followed this chain of logic supporting the ...
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How can a quantum particle in an infinite potential well transition between regions separated by nodes? [duplicate]
I don't understand, if the PD describes the probability of finding the particle at a point, how is the probability is non-zero after a node. That means (classically speaking) the particle will have to ...
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Understanding entropy and its connection to probability distributions
Entropy tells us about the "uncertainty" of a probability distribution, i.e. roughly how much information is needed to describe an event that is described by a probability distribution. With ...
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Physical interpretation of the matrix element
In perturbation theory, but also in other scenarios the claim is made that the following expression:
$|\langle f|\hat O||i\rangle|^2$ represents the probability amplitude for a transition of the ...
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How thick would a coin have to be s.t. the probability to land on its "side" is exactly 1/3?
Assume a coin of uniform density with fixed radius $r$ is thrown from a height $x$ with $x>r$, $|\vec\omega|=1$, direction uniformly distributed. Assume the coin "lands" on the side ...
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Conceptually understanding microstates
I had learned about microstates in my thermodynamics class, but I am having trouble understanding the probabilistic property of them. My teacher had told me that every microstate is equally probable, ...
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Difficulty with a scaling argument
I am trying to make sense of an argument in this paper, "Fracture strength: Stress concentration, extreme value statistics and the fate of the Weibull distribution".
The paper deals with how ...
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"Probability of finding a system in a given state" in rigorous quantum mechanics
Feynman and Hibbs pg. 164 includes this paragraph:
The time development of a quantum-mechanical system can be pictured as follows. At an initial time $t_a$ the state is described by the wave function ...
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Raman transition probability for off-resonant light
I have a 3-level system, with the levels being $g_1$, $g_2$ and $e$ and their energies $E_1$, $E_2$, $E_e$ such that $|E_2-E_1|\ll|E_e-E_1|$ ($g_1$ and $g_2$ can even be degenerate, but in general ...
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Dirac delta function as probability density
In Quantum Physics Gasiorowicz states:
Incidentally, had we allowed for discontinuities in $\psi$(x, t) we would have been led to delta functions in the flux, and hence in the probability density, ...
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Could probability amplitude for a path equal a complex number whose length is always 1 and whose angle is the action divided by Planck's constant?
I'm reading "Zee A. - Quantum Field Theory, as Simply as Possible", where near beginning of explanation of QFT he gives what appears to be path integral formulation, he states:
The ...
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Nuclear physics theoretical doubt [closed]
Half life for certain radioactive element is 5 min. Four nuclei of that element are observed at a certain instant of time. After half life i.e. 5 minutes, half of total nuclei will disintegrate. So ...
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Creating a probability version of the time-independent Schroedinger equation
I know, I know ... There is no advantage for creating a probability version of the time-independent Schroedinger equation because 1) it's no longer a linear eigenvalue equation; and 2) it's impossible ...
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Quantum tunneling through step potential: How to calculate transmission coefficient?
In case of a finite potential barrier ($V(x) = V_0$ for $-a<x<a$ and $V(x) = 0$ otherwise). We assume a particle is incident from one side say left and is moving right. In the region $x<-a$; ...
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Proof $| \psi |^2$ is probability density
Let's assume
$$\iiint_{\mathbb{R}^3} | \psi |^2 d^3r=1$$
For some $t=t_0$. Then we want to calculate:
$$\frac{d}{dt}\iiint_{\mathbb{R}^3} | \psi |^2 d^3r=\iiint_{\mathbb{R}^3} \frac{\partial | \psi |^...
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How is Edwin Jaynes' Partition function in "Probability Theory: the Logic of Science" related to partition function in statistical physics?
In Section 9.6 of his book "Probability Theory: the Logic of Science", Edwin Jaynes proposed "partition function" when trying to compute the "multiplicity factors" $M(n,G)...
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Chances of X-rays or Gamma rays to cause bit flips
When an X-ray or a gamma ray strikes a memory cell like in DRAM and causes electron-hole pairs to be created due to effects like ionization of electrons or transferring energy into the electrons, is ...
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Is there a connection between quantum state probability and Bayes' theorem? [closed]
In statistical physics, the probability that an electron is in the i-th quantum state is given by the following equation (eq. 1):
$$
p_i = \frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}
$$
This formula ...
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Clarification on when to apply the Born Rule in quantum mechanical measurement problems
One of the postulates of quantum mechanics says that if $A$ is an operator corresponding to some observable, with eigenbasis $\{ |a_i\rangle \}_i$, and we measure some state $|\psi\rangle$ to be in ...
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Radial distribution of ideal gas in a cylinder
today I have a computational doubt and a theoric one.
Starting with the theoric question, suppose I have a generic ideal gas in a cylinder and each particle is subject to a potential dependent on the ...
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Uncertainty principle probability [closed]
How are positional probabilities and momentum probabilities determined in the principle of uncertainty?
For example, if the position probability is 30, does it work like the momentum probability is 70....
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Laplace's demon and quantum mechanics paradox of not influencing our world? [duplicate]
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all ...
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Does the gradient of the probability density tell me in which direction the system is most likely moving?
Suppose that a system is described by a probability density $\rho$ on the state space $S$. Of course we need to assume some additional structure on $S$ to define the gradient $\nabla\rho$. I am ...
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Number of particles in a given energy interval
While reading statistical mechanics from beiser,I encountered statements like *If $n(E)$ is the number of particles with energy $E$,then the number of particles with energy between $E$ and $E+ dE$ is $...
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Time dependency of Wave function and its probability density function (PDF) [closed]
When we study the Schrodinger wave equation, we have a time dependent wave function $\Psi(x,t)$, and when we deduce its Probability Density function we come to know there is no time dependence in the ...
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Drake Equation with Random Walks
Goal
I'd like to use the Drake Equation with random walk theory to estimate the probability of aliens reaching Earth.
Drake Equation
The Drake Equation estimates the number of advanced civilizations ...
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Quantum Mechanical Current Normalisation
Consider an electron leaving a metal. The quantum-mechanical current operator, is given (Landau and Lifshitz, 1974) by
$$
j_x\left[\psi_{\mathrm{f}}\right]=\frac{\hbar i}{2 m}\left(\psi_{\mathrm{f}} \...
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What is the physical meaning of the normalization of the propagator in quantum mechanics?
Suppose we have a quantum field theory (QFT) for a scalar field $\phi$ with vacuum state $|\Omega\rangle$. Then, in units where $\hbar = 1$, we postulate that the vacuum expectation value (VEV) of any ...
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What Does Feynman Mean When He Says Amplitude and Probabilities?
In Feynman lectures on gravitation section 1.4, he tries to debate over whether one should quantize the gravitation or not.
He provides a two-slit diffraction experiment with a gravity detector, which ...
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Quantum: Which improbable macroscopic events are possible?
Basically, the title. Web search had not found pages in top results with similar QA.
E.g. I understand nuclear blast can just end at any time because random chain-reaction has probability of not ...
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Probabilistic behavior of quantum mechanics [closed]
In a hypothetical scenario, if I were to measure the quantum spin of an electron and it showed "up," and then I traveled back in time without changing the initial conditions, would measuring ...
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Is the zero vector necessary to do quantum mechanics?
Textbook quantum mechanics describes systems as Hilbert spaces $\mathcal{H}$, states as unit vectors $\psi \in \mathcal{H}$, and observables as operators $O: \mathcal{H} \to \mathcal{H}$. Ultimately, ...
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Physical meaning of each term of the square modulus of a wave function
The expression below is the square modulus of the wave function of a harmonic potential ($V=\frac{1}{2}m\omega^2 x^2$) in which it's stated that the probability of finding the particle in the $\psi_0$ ...
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Factorization on increments in Markov chain
I'm trying to show the following property for a Markov chain:
$$\left<[x(t+\tau)-x(t)][x(t'+\tau)-x(t')]\right> =\left<x(t+\tau)-x(t)\right>\left<x(t'+\tau)-x(t')\right> $$
Where $t\...
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Boltzmann distributions on atomic orbitals: infinite degeneracy?
The (unnormalized) Boltzmann probability distribution of states as a function of energy and temperature is given by
$$P(\epsilon_i) \propto g_i\exp\left(\frac{-\epsilon_i}{k_BT}\right)$$
with $P(\...
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What is the connection between moments in probability theory and the moment of inertia?
My question arises as the moment of inertia (MOI) has been described as a second moment.
In my understanding if the MOI is indeed a second moment of a distribution of mass, this suggests the MOI could ...
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Fermi's Golden Rule: Interpreting the Dirac Delta in Transition Probabilities [duplicate]
I am trying to understand an aspect of Fermi's golden rule in the case of a constant perturbation, $V$. The formula for the transition probability from an initial state $i$ to a final state $f$ is ...
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Is there a name for the number of '9's in numbers such as 0.999 (where it would be 3)?
I am doing an optics simulation involving transmission and reflection coefficients very close to 1, such as 0.999. While I was an undergraduate student, a professor mentioned that, in certain fields, ...
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How to deal with the divergence in tree-level diagrams? where the propagator momentum is on-shell
Only consider the interaction term between electron Higgs and $Z$ boson
$$
\mathcal{L}_{h ff}=-\frac{Y_f v}{\sqrt{2}} \bar{\psi} \psi\left(1+\frac{h}{v}\right)
=-m_f \bar{\psi} \psi\left(1+\frac{h}{...
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How are quantum systems different from dice?
I've had this question for a while:
Is a state space $\mathcal{H}$ for a quantum system just a sample space in a probability space?
The question arises because i can't really tell a difference between ...
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Equation for probability of quantum tunneling
I am looking at fusion reactions in stars and came across how particles will bypass the Coulomb barrier through quantum tunneling. I was wondering if there is an equation for the probability of a ...
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Questioning the Probability Expression for Neutrino Oscillation in Griffiths' "Introduction to Elementary Particles"
In Griffiths' book, Introduction to Elementary Particles (Griffiths, D. (2020). John Wiley & Sons, p. 390), the author defines the pure electron and muon neutrino states as:
$$|ν_{e}\rangle=-\sinθ|...
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How are uncertainties of the quantum state/wavefunctions themselves modeled?
This question might be confusing so let me try to clarify this carefully.
The wavefunction is a tool that allows us to calculate probability distributions that model uncertainties. Thus makes sense.
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Why a probability distribution in RHS in deriving Bell's Inequality?
Why is there typically an integral over a probability distribution in the RHS of a derivation of Bell's inequality
$$|P(\boldsymbol{a}, \boldsymbol{b}) - P(\boldsymbol{a}, \boldsymbol{c})| \leq \int{p(...
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Why the transition probability in the master equation approach just the rate$*dt$ for a simple birth process?
I am modeling a process of an exponentially growing population of cells as $\frac{dn}{dt}=\lambda n$. To account for the intrinsic noise in the birth process of these cells, I write down the ...
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The role of probability in the many-worlds interpretation [duplicate]
A quantum system can transition to one of two states, with probabilities 30% and 70%. The many worlds interpretation says that the universe splits into two, one for each state. If so, what do the 30% ...
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How is the inner product of two quantum states related to their associated Bloch vectors?
I have a doubt about how two equivalent ways of calculating the inner product between two states seem to not be actually equivalent, as they should. In particular, I'm interested in the case where the ...
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Fermi-Dirac Distribution for Multiple Species
If I have a system containing two types of fermions, what is the probability of a state of energy $E$ being occupied? Is it just the sum of two standard Fermi probabilities for each type of fermion?
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A question about time evolution of position distributions
If I have two probability distributions $P$ at $t$ and $P’$ at $t’$ separated by some time interval. Then, can I describe the transform between the two distributions as $$P’(x) = \int P(a) D(a, x-a, t’...
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How to go from probability distribution to transitions probability distribution?
For the past few days I have been studying Advanced statistical mechanics. I am studying a Wiener process in general. Such a process is a non-stationaty time-independent Gaussian process. The ...
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