For questions about probability, probability theory, probability distributions, expected values and related matters. Purely mathematical questions should be asked on Math.SE.

learn more… | top users | synonyms

1
vote
0answers
34 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 ...
1
vote
1answer
59 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, ...
2
votes
2answers
61 views

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 ...
3
votes
0answers
34 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 $a$ ...
3
votes
1answer
37 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 ...
1
vote
1answer
41 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 $|\psi|^2 r^2$. ...
1
vote
3answers
194 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?
0
votes
0answers
20 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} ...
0
votes
0answers
17 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}}\\ ...
1
vote
0answers
42 views

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 ...
0
votes
0answers
59 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 ...
0
votes
2answers
50 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
35 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 ...
2
votes
1answer
72 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?
0
votes
0answers
21 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 ...
1
vote
0answers
29 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, ...
2
votes
1answer
113 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 $C$. ...
1
vote
0answers
27 views

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 ...
1
vote
0answers
71 views

Particle in a box problem [closed]

What is the fraction, as a percentage, of the total probability of finding, between points at 0 pm and 30 pm, an electron in the level of a one-dimensional box 150 pm long? Here I found the ...
1
vote
0answers
27 views

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 ...
2
votes
2answers
81 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 ...
2
votes
3answers
140 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, ...
0
votes
2answers
45 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 ...
1
vote
1answer
82 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 ...
2
votes
0answers
45 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 ...
1
vote
1answer
68 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 ...
7
votes
1answer
136 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 ...
0
votes
2answers
33 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 ...
1
vote
3answers
174 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 ...
29
votes
1answer
1k views

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 ...
2
votes
1answer
55 views

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 ...
0
votes
2answers
53 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 ...
0
votes
0answers
89 views

Problem in understanding Feynman's explanation of the Dirac-Delta function

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 ...
1
vote
4answers
234 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 ...
0
votes
2answers
145 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 ...
1
vote
2answers
109 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...?
0
votes
0answers
39 views

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 ...
1
vote
1answer
109 views

Time evolution operator acting on a non-eigenket

I'm taking a course in QM at my university, and I'm trying to work out an assignment given to the class by our professor. The setup is as follows: The problem is about a simplified description of ...
0
votes
2answers
179 views

Why is the probability of finding a particle in a quantum well greatest at its center?

Imagine that in a classical sense, a particle has some total energy $E$ and that its potential energy $U(x)$ varies with $x$ in the shape of a well (see the top-left image). Of all the possible ...
4
votes
1answer
46 views

Probability in Measuring Noncommuting Observables

If I have a particle in a state $\Psi(x) = e^{-x^2}$ could I calculate probability of simultaneously measuring, say, $x > 0, p_x < 0$? I understand that $p_x$ and $x$ don't commute and ...
2
votes
2answers
107 views

How can I make this toy quantum random walk model unitary?

Take a toy $(1+1)$-dimensional lattice model of the universe. A particle begins at $x=0$ at $t=0$. It has an amplitude ${1}/{\sqrt{2}}$ to move one step to the left and amplitude ${1}/{\sqrt{2}}$ to ...
2
votes
1answer
52 views

Derive probability current density - factors of 2 discrepancy [closed]

To derive the probability current density for a particle in an electromagnetic field, we calculate $\dfrac{\partial \rho}{\partial t} = \dfrac{\partial}{\partial t} (\Psi^* \Psi) = \dfrac{\partial ...
1
vote
1answer
135 views

Wavefunction interpretations in QM

From two-slit electron-interference experiment we can infer that there is a wave $\psi(x,t)$ that can be associated with electron. The amplitude at some point is the sum of amplitudes reaching that ...
1
vote
1answer
35 views

Can a photon that matches the energy gap pass through a molecule?

I've heard that atoms and molecules are made up of mostly empty space. I've heard that electrons exist as a probability cloud around and/or between atoms and molecules. My question is, is there a ...
4
votes
0answers
95 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
1
vote
1answer
36 views

From exponential distribution to Poisson distribution

We know that the exponential distribution characterises the probability distribution for the waiting time between two consecutive Poisson events. Then I think if we fix a time interval $T$ then we ...
1
vote
1answer
93 views

How does one make sense of a delta function of a scalar field?

Disclaimer: Originally posted on math SE, but thought that it was better in physics SE, so deleted my post on math SE and posted here. In the classic review summary of stochastic quantization here, ...
2
votes
2answers
70 views

Why don't both equivalent forms of this delta function give the correct answer?

I am a bit confused on a basic problem involving a Dirac delta function being integrated over in a multiple integral. The original problem is to find the probability distribution in position-momentum ...
0
votes
0answers
23 views

Construct bivariate symmetric (polynomial) Hilbert-Schmidt two-qubit volume functions over the unit square with certain properties

Construct bivariate symmetric polynomials (two-qubit volume functions) f(r,R) = f(R,r) >= 0 over [0,1]^2, with f(1,R) = f(r,1)=0, such that the univariate marginal (integrating over r or R) ...