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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Quantum Mechanics: How to compute how fast must a function go to zero at infinity? [closed]

We say that the wave function must go to zero at infinity faster than $1/x^{0.5}$ in order for it to be normalizable. What about other quantities like the probability current? What is the general rule ...
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1answer
69 views

Is the probability current an observable?

Is the probability current in Quantum Mechanics an observable? If so, how can it me measured (directly or indirectly)?
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106 views

The Theoretical Minimum: Probability/Spin Question

Background In The Theoretical Minimum, it states that any spin state can be represented by a linear combo of the basis vectors $|u\rangle$ and $|d\rangle$ It then goes on to show how this is done ...
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59 views

Why is the reflection coefficient in quantum mechanical scattering defined this way?

In Griffiths' "Introduction to Quantum Mechanics, second edition" section 2.5.2, p. 73, he states: For the delta-function potential, when considering the scattered states (with $E > 0$), we have ...
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1answer
102 views

Fermi's golden rule and infinite probablity?

I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: ...
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5answers
223 views

Quantum Theory just lazy physics?

Is it true that physicists decided that because they couldn't predict with certainty the location of an electron at any given time, that they just created equations using probability instead, still ...
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1answer
92 views

What is the probability of two bullets to get clashed? [closed]

I was surfing on Instagram, and I found this amazing proto whose description is "the probability of such an event to happen is incredibly small, so this is a really curious finding". Well.. I'm ...
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1answer
52 views

Bell's Theorem - Why does $\lambda$ have a probability?

I'm reviewing Bell's theorem. In his proof by contradiction, he assumes the world is deterministic and defines a vector $\lambda$ as the set of all hidden variables which play a role in determining ...
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1answer
50 views

Probability of finding electron in a spherical shell

In the book Arthur Beiser - Concepts of modern physics, probability of finding an electron in Hydrogen atom in the spherical shell between $r$ and $r+dr$ is given as \begin{equation} P(r)dr = r^2|R(r)...
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1answer
86 views

Layman Question on Probability Amplitudes and Probabilities

Important Note: For "layman" read "next to zero understanding of QM mathematics" I am reading Quantum Mechanics: The Theoretical Minimum. In Chapter 2 on Quantum States, the following is presented: $...
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164 views

What does the continuity equation for probability in quantum mechanics mean?

In quantum mechanics, the continuity equation $-{d\rho}/{dt}=\nabla\cdot{J}$ holds for a probability density $\rho$ and probability current $J$. But what does it mean, from a physical point of view? ...
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Why are isoprobabilty contours circles for uncorrelated functions and ellipse for correlated functions? Kindly explain.

I would like to know how to understand these and to learn how to go from a scatter plot to the isoprobabity contour plots. Thanks is advance!
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125 views

Has Jaynes' argument for quantum mechanics as a possible theory of inference been debunked?

To my understanding, there is currently no scientific consensus on which interpretation of quantum physics is the correct one, if any. The most famous one, perhaps for historical reasons, is the ...
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42 views

Proving Probability Current and Momentum relationship

I am trying to show that $\mathbf j = \frac{\hbar}{2mi}\left(\Psi^* \mathbf \nabla \Psi - \Psi \mathbf \nabla \Psi^{*} \right) \,, $ simplifies to $\mathbf j = \frac{\mathbf p}{2m} (|\Psi|^2) $. I ...
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1answer
86 views

Expectation Value of Unitary Time Evolution Operator in Quantum Mechanics

Does the expression $\langle \Psi_i|U(t)|\Psi_i\rangle$ have a specific meaning, where $U(T)$ is the unitary time evolution operator of $\Psi$, and $\Psi_i$ is the initial state of $\Psi$? If so, ...
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2answers
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Microstates, Distribution of Particles, and the Probability of an Empty Compartment

If I have a closed system composed of $N$ particles and $p$ compartments, the total number of microstates available to that system is $$ p^N $$ Now say I want to find the probability that any one of ...
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42 views

Is the conservation of probability in the Schroedinger's equation unique?

The Schroedinger's equation can be viewed as a diffusion equation with imaginary constants $a$ and $b$ satisfying, $$(1) \quad \Psi_t=a \cdot \Delta \Psi-b \cdot V(x,t) \cdot \Psi$$ However if $a$ ...
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1answer
39 views

Probability of nuclear decay of small staring number of atoms

I came across a rather dubious question that a teacher had put in a power point. It said something like,"Given a sample of 100 atoms of isotope x, after one half life of the said isotope, how many ...
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1answer
44 views

Why wavefunction is sometimes multiplied by the radius to get probability density?

When solving 1d particle in a box, the probability density is said to be proportional to $|\psi|$, but when solving 3d orbitals, the probability density is said to be proportional to $|\psi|^2 r^2$. ...
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3answers
363 views

Does quantum mechanics break causality? [duplicate]

If quantum mechanics is probabilistic, there is no reason for a particle to be in one place and not the other, but particles do make up their minds... but how?
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25 views

how to use generating function to solve coupled linear master equations?

I am trying to solve a two dimensional continuous time and discrete state master equation. The master equation is linear and looks as follows, $\frac{\partial P_A(x,y,t)}{\partial t} = k_{11} P_A(x-1,...
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22 views

Eigenvalues for correlation matrix which have the form of an harmonic function

I am trying to understand the written in the picture below. I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ace^{-\...
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0answers
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Why does the preservation of transition probabilities imply the preservation of all quantum probabilities?

I have a question about symmetries in quantum mechanics. Let $H$ be a Hilbert space, and $\mathbb{P}H$ the corresponding projective Hilbert (ray) space. In quantum mechanics, a symmetry is usually ...
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64 views

How does one intrepret probabilites in the many-worlds interpretation?

Let's say I flip a coin, and don't look. From the copenhagen interpretation, the state of the coin is: $\frac1{\sqrt2}(i|\text{heads} \rangle - |\text{tails}\rangle)$ If I observe the coin, there is ...
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2answers
59 views

How to interpret irreversibility in time?

I'll quote Feynman's Lectures, chapter 52 (Symmetry in Physical Laws) of volume 1: [...] If we see the egg splattering on the sidewalk and the shell cracking open, and so on, then we will surely ...
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1answer
48 views

What does it mean that the Rutherford's cross section is infinite?

I'm studying elastic scattering and I read that the Rutherford's differential cross section is defined as: $$\left( \frac{d \sigma}{d \Omega} \right)_R = \frac{Z^2}{4} r_o^2 z^2 \frac{(m_ec / \beta p)...
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42 views

What is the probabililty that a fair coin lands on its side?

This is a popular gag in movies, but I wonder how likely it really is. What is the probability that a uniform cylindrical coin (with radius $1$ and height $h$) lands on its side? If the ground were ...
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1answer
126 views

What is a quasi-probability distribution?

I have some questions about the quasi-probability distribution. What is it? And why it is important in quantum mechanics? And what does "quasi" mean?
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28 views

Entropic forces in Brownian motion

Reading Entropic forces in Brownian motion I'm having trouble to understand how the author makes a computation. He needs to calculate the number of ways a particle that is released from the origin can ...
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30 views

Boltzman equation for a collisionless medium?

In the derivation of the Boltzmann equation (link to Wikipedia) for a collisionless gas it is assumed that: $$f(\vec r+\frac{\vec p}{m} \Delta t, \vec P + \vec F \Delta t, t + \delta t)=f(\vec r, \...
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1answer
123 views

In the algebraic formulation of Quantum Mechanics, how do probability amplitudes naturally arise?

In the algebraic formulation of quantum mechanics, consider $\mathcal{B}(\mathcal{H})$ as the set of all bounded operators on $\mathcal{H}$ (with involution, norm, etc.), which form a C*-algebra $C$. ...
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Confusion about radioactivity

The following question is from General Problems on Physics by I.E Irodov 6.220. Find the decay constant and the mean lifetime of $^{55}\operatorname{Co}$ radionuclide if its activity is known to ...
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Probability current: what's a good definition?

I am doing scattering problems with a really weird hamiltonian (e.g., only first-order in derivatives). I don't know how to define a probability current: I've looked at answers on this site, and they ...
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2answers
126 views

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom?

How do we decide whether an electron orbital has a non-zero or zero probability of lying inside the nucleus of an hydrogen atom? It is mostly from the radial function, as to what I think but how ...
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3answers
242 views

Can a photon be absorbed by a proton?

When incident light passes through a hydrogen gas, for example, does it have 50% chance (since it's a 1:1 ratio of protons to electrons) of getting absorbed by the proton? Any chance at all? If no, ...
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47 views

Probability that a measurement will be in some set

Let $\mathcal{H}$ be the Hilbert space of a quantum system and $A$ one observable in $\mathcal{H}$. If $A$ has discrete spectrum $\{a_n : n \in \mathbb{N}\}$ for simplicity, then by the postulates of ...
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1answer
92 views

Many worlds probability of getting cancer [closed]

My understanding of probabilities in many worlds is following: If I would decide to start smoking and we know that 10% of smokers get cancer that means that in 10% of all worlds during my lifetime I ...
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0answers
56 views

Measure-theoretic maths behind Born's probabilistic interpretation of Schrodinger's equation

I was reading a bit about Quantum Mechanics, Schrodinger's equation and its probabilistic interpretation (found this very insightful intro here https://plus.maths.org/content/schrodinger-1), my ...
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1answer
72 views

Demonstration that the $\langle f(x)\rangle$ of an odd function $f(x)=-f(-x)$ of position $x$ in a symmetric potential well $V(x)=V(-x)$ is null

Consider a potential infinite well, which borders are $x=-a$ and $x=a$. I pretend to demonstrate that the expected value of a odd function $f(x)$, i.e., $\langle f(x)\rangle$, is null. We have the ...
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1answer
162 views

Is there a mathematical basis for Born rule?

Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the ...
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39 views

Pair annihilation - can annihilation be moderated?

I recently asked this question: How close does a particle-antiparticle pair need to be for annihilation to happen? And that received a good answer. But there was a second part to my question that ...
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3answers
299 views

Can we predict throwing a dice?

What happens if we throw a dice from same position, with same force, by creating a vacuum environment on earth? Will it be predictable now i.e. will the dice have same results all the time? If answer ...
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1answer
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How close does a particle-antiparticle pair need to be for annihilation to happen?

I've most often seen the statement that the annihilation of a particle and its antiparticle occurs when they 'collide' with one another. So in other words when they get very close to one another right?...
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1answer
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Spin sums in cross sections. Summing amplitudes or probabilities?

The context: I'm calculating the cross section for a scalar particle to decay into a fermion-antifermion pair in Yukawa theory, at tree level. In doing this, when calculating the amplitude from ...
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2answers
54 views

In what cases and with what method does one find a time dependent probability density for a quantum system in an infinite square well?

How can one find the time dependent probability density function of a quantum system given $\Psi(x,t=0)$? Say, $\psi(x) \sim x^4$ for $0 < x < L$. How can one find the time dependant probability ...
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Problem in understanding Feynman's explanation of the Dirac-Delta function [duplicate]

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
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4answers
633 views

Is quantum physics truly random or does it just appear that way because of Heisenberg uncertainty principle

The behavior of an electron (and other tiny things) is said to be probabilistic because we can't say where an election will be when we measure it, but only where it will probably be. As I understand ...
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2answers
352 views

Probability to find a particle in a particlar state $\psi_{n}$ [closed]

I have a problem to understand the probabilities in QM. In particular, if I have a particle in state $\psi_{n}$, then we change the system and we ask for the probability to find the particle in a ...
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2answers
137 views

Do electrons always have a probability of being somewhere?

In the same way as when they surround a nucleus? How about when electrons go through wires or are ejected as beta particles? Do they still only have probabilities of being somewhere, or...?
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How is the probability density function in equilibrium state made equal to a dirac delta function?

While studying statistical mechanics, I stumbled upon the introduction of the dirac delta function in defining the probability density of the microstates and hence the conclusion of equal priori ...