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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Notation for Propability Amplitudes

I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude ...
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1answer
254 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
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Are frequency and likelihood the same across the multiverse?

My probability text distinguishes between two interpretations of probability values: the frequency of occurrence "as percentage of success in a moderately large number of similar situations" (coin ...
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4answers
654 views

Probability and the many-worlds interpretation

If I toss a coin, then according to the many worlds interpretation of QM, in half those worlds I'll get a head. If I then toss again, then in a quarter I will have got two heads. And so on. There will ...
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Basic introductory quantum mechanics question [closed]

Given that $\psi = \frac{1}{\sqrt{32 \pi a_{0}^{3}}}(2-\frac{r}{a_0})exp(\frac{-r}{2a_0})$ is a wavefunction of the hydrogen atom, write down the probability density for r and calculate the ratio ...
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328 views

Expectation values in QFT?

What is the meaning of different expectation values in QFT? For instance: $$\langle 0|{\cal O}(0)|q,s\rangle$$ or $$\langle 0|{\cal O}(0)|0\rangle$$ with ${\cal O}$ being some operator and ...
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2answers
257 views

Physics Standard Deviation

I am a physics enthusiast and I have a question: Why is it meaningless to express the '$\pm$' (standard deviation) value as a percentage?
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76 views

What is the probability that all the air ends up in the upper right corner of the room and we suffocate

Since someone commented this on this question(What is the probability of ice in boiling water?), I would like to ask what is the probability that all the air ends up in the upper right corner of the ...
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1answer
1k views

Probability of finding n particles in a volume v

I'm trying to calculate the probability of finding $n$ particles in a certain volume $v$ in a system with a total of $N$ particles and total volume of $V$. My problem is that I've tried two approaches ...
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2answers
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Particle in a 1-D box and the correspondence principle

Consider the particle in a 1-d box, we know very well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density ...
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2answers
161 views

Probability of having energy $E$ when $E$ is bounded

For a canonical ensemble the probability of a system to have energy $E$ is $P(E)=e^{-\beta E}$. For that we consider the that the system can have any energy between $0$ to $\infty$. What will be the ...
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4answers
200 views

What aspect of quantum mechanics forces probabilities to be (conventionally, at least) central?

I understand how to compute probability distributions and expected values and such from quantum states, but a lot of treatments of QM make it look like this is what the wavefunction is essentially ...
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3answers
411 views

How to show that probability and statistics are very important in Quantum Mechanics?

I'm doing a research for my stats class in high school and I chose quantum mechanics as my subject. I narrowed down to electron localization in an atom and radial probability distribution. However, I ...
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435 views

Interpretation: probability form probability amplitude (free particle)

If you compute the probability amplitude of a free 1D non-relativistic particle with mass $m$, located at position $x_0$ at time $t_0$, for beeing detected at some other point $x_N$ at time $t_N$ you ...
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3answers
627 views

Normalization of basis vectors with a continuous index?

I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
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422 views

Physical intuition for independence of components of velocity in derivation of Maxwell–Boltzmann distribution

Maxwell derived the shape of the probability distribution of velocity of gas particles by starting with just two assumptions. These are: The probability distribution is rotation invariant. The ...
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1answer
693 views

Spinors and Probabilities of Electron-Positron Pair

Question: An electron and positron are moving in opposite directions, and are in the spin singlet state. Two Stern-Gerlach machines are orientated in some ...
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140 views

How to use The Schwarzchild Metric formula to get distribution representing “free-fall”

Given formula: How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should ...
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1answer
685 views

Probability in Quantum Mechanics: General

How do I find the most probable value of position of a (non-Gaussian) wave function? Is it the same value as the expectation value of the position? And is it true that the most probable value of ...
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2answers
183 views

What information is lost in the symmetrization necessary to derive the BBGKY hierarchy?

The book on Kinetic theory I'm reading derives the BBGKY hierarchy after introducing the reduced distribution functions $f_s(q^1,p_1,q^2,p_2,\dots,q^s,p_s):=\int\ \rho\ \ \mathrm d q^{s+1} \mathrm d ...
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2answers
388 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
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What is the relationship between Maxwell–Boltzmann statistics and the grand canonical ensemble?

In the grand canonical ensemble one derives the expectation value $\langle \hat n_r\rangle^{\pm}$ for fermions and bosons of sort $r$: $$ \langle \hat n_r\rangle^{\pm} \ \propto \ ...
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1answer
432 views

Quantum version of the Galton Board

If classical particles fall through a Galton Board they pile up in the limit of large numbers like a normal distribution, see e.g. http://mathworld.wolfram.com/GaltonBoard.html What kind of ...
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2answers
443 views

Probability current vs. direction of wave function

I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi ...
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1answer
273 views

Percolation and number of phases in the 2D Ising model

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive. After a long time I came back to try to understand an article on the Ising model. ...
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1answer
905 views

Confusion about the probability cloud

What is the meaning of the electron probability cloud? I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading ...
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1answer
640 views

Why is classical physics not valid for a harmonic oscillator in its lowest energy state? [closed]

I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum ...
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2answers
207 views

Moment of Inertia [closed]

Let $f(x) = \frac{1}{L}$ be a probability function, where $L$ is constant. Find the mean and variance. Discuss your results by making a connection to the moment of inertia definition.
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1answer
381 views

Classical mechanics from Quantum mechanics

I'm looking at a way to prove that one recovers, under ad hoc assumptions, classical mechanics from quantum theory. Usually, we can find in textbooks that the propagator $$K(x,x_0;t)=\langle x|e^{-i ...
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1answer
259 views

Quantum randomness and brownian motion in biological systems, e.g., fertilization

I am looking for examples of physical indeterminacy impacting the macroscopic world. By physical indeterminacy, I mean physical sources of randomness such as quantum indeterminacy or brownian motion. ...
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2answers
316 views

How to understand wavefunction in quantum mechanics in math

I am reading some introduction on quantum mechanics. I don't understand all but I get the point that the wavefunction tells some probability aspects. In one book, they show one example of the ...
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2answers
1k views

Quantum mechanics potential barrier problem [duplicate]

While reviewing some quantum mechanics, I cam across a very interesting situation. For a potential barrier, if a particle has an energy $E$ less than the potential barrier $V_0$, it is possible to ...
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1answer
133 views

Probability density of detection of collinearly emitted photons in two detectors

Update: As proposed by @dmckee, I added equation numbers and improved the display of some equations. The answer by @Trimok inspired me to look at coordinate systems which are not specific to the ...
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3answers
651 views

Are negativity of the Wigner function and quantum behaviour equivalent?

I've read the following question: Negative probabilities in quantum physics and I'm not sure I understand all the details about my actual question. I think mine is more direct. It is known that the ...
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2answers
329 views

Probabilities in statistical mechanics

I am reviewing some concepts in statistical mechanics and am becoming confused with how to calculate probabilities when a system has $N$ non-interacting particles. For instance, let's say we have ...
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2answers
2k views

The probability of finding the electron in the H-atom

In the book Arthur Beiser - Concepts of modern physics [page 213] author separates the variables in the polar Schrödinger equation assuming: $$\psi_{nlm}=R(r)\Phi(\phi)\Theta(\theta)$$ then there a ...
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1answer
385 views

Fermi's golden rule and Probabilities in QM

In Fermi's golden rule $$P_{ab}(t)=2\pi t/\hbar \left|\langle\psi_b|V|\psi_a\rangle\right|^2 \delta(E_f-E_i)$$ for transition probability from state $a$ to $b$, how can the probability grow with ...
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3answers
1k views

Born's Rule, What is the Reason? [duplicate]

As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives ...
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2answers
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What is the probability density function over time for a 1-D random walk on a line with boundaries?

If a single particle sits on an infinite line and undergoes a 1-D random walk, the probability density of its spatio-temporal evolution is captured by a 1-D gaussian distribution. \begin{align} ...
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4answers
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Is there a phenomenon where physicists are only interested in the standard deviation of the quantity to be measured?

or a phenomenon where we can only measure the standard deviation ($\sigma_w$) of a variable $w$ and not the mean $\overline{w}$
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2answers
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Why is quantum mechanics based on probability theory? [duplicate]

What makes us formulate quantum mechanics based on probability theory? Isn't the real quantum world based on unknown laws to us? Is it possible that results of an experiment will be measurable in ...
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1answer
658 views

What is probability to find electron at certain distance from nucleus

Given for example, Hydrogen electron in ground state. What is probability to find that electron at certain distance (not interval of distances) from center of nucleus, for example at radial coordinate ...
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1answer
1k views

Probability for harmonic oscillator outside the classical region

I'm having some trouble finding an expression for the probability to find the particle outside the classical area in the harmonic oscillator. I have a wavefunction defined as: $\psi \left( x,\,t ...
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3answers
837 views

Classical/Quantum Coin Toss

I am having a brainfreeze moment and have confused myself, help appreciated! Classical Coin: Heads OR tails. Quantum Coin: Superposition Heads AND Tails. Classical Mechanics: Deterministic (in ...
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1answer
507 views

spontaneous disintegration of an unstable particle

Suppose one wants to describe an unstable particle that spontaneously disintegrates with a life time say "tau". In that case the total probability of finding the particle is not constant. But should ...
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2answers
4k views

Physical interpretation of normalization of wave fuctions

Does normalization of wave function mean that we are getting our state vector to unit length? If that's the case what does it mean physically? Also is the underlying vector space finite dimensional? ...
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3answers
496 views

Fermi-Dirac Statistics

In Fermi-Dirac statistics the probability of being in a certain energy state is $$f(E) = \left[1 + \exp\left(\frac{E-E_F}{k T}\right)\right]^{-1}$$ In the area that I'm looking at the texts always ...
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1answer
123 views

Is this hypo-theoretical model of future prediction feasible? [closed]

First let me state that I am not, nor ever have I been, a physics student. I am working on an idea for a book I'm writing. This is a thought experiment that posits the existence of a computer system ...
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177 views

How to design a deliberately biased coin?

For demonstrating basic probability concepts, it would be nice to have a coin-like object that lands heads/tails not in 50/50% ratio, but biased in a way that can be revealed in a short experiment. ...
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2answers
605 views

How do you come up with a POVM?

This is a made-up example, just to understand a concept. If changing the probability values aids your explanation, that's fine by me. Say you have a physical quantity $E$ that can take values 1, 2, 3 ...