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1answer
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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 ...
5
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2answers
166 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 ...
3
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2answers
318 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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3answers
1k views

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 \ ...
8
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1answer
316 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 ...
2
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1answer
317 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 ...
3
votes
1answer
235 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
418 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 ...
2
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1answer
549 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 ...
0
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2answers
139 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.
1
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1answer
204 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. ...
2
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2answers
263 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
735 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 ...
3
votes
1answer
120 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 ...
4
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3answers
360 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
296 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 ...
1
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1answer
323 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
849 views

Born's Rule, What is the Reason?

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
795 views

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} ...
4
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3answers
160 views

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 $\overline{w}$
5
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2answers
961 views

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
480 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 ...
2
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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
569 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 ...
0
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1answer
342 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
3k 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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2answers
404 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 ...
0
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1answer
119 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 ...
8
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2answers
160 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. ...
4
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2answers
466 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 ...
2
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1answer
95 views

Statistical sum of physical quantities in a quantum system

Let $C = A + B$ (statistical sum, so $\mathbb{E}[C] = \mathbb{E}[A] + \mathbb{E}[B]$), and let $p(A = a) = 1$. Are the following true? $\mathbb{E}[C^2] = a^2 + 2a\mathbb{E}[B] + \mathbb{E}[B^2]$ ...
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2answers
109 views

Independent systems and Lagrangians

Definition 1: The notion of independent systems has a precise meaning in probabilities. It states that the (joint) probability or finding the system ($S_1S_2$) in the configuration ($C_1C_2$) is ...
3
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2answers
317 views

Probability of position in linear shm?

The problem that got me thinking goes like this:- Find $dp/dx$ where $p$ is the probability of finding a body at a random instant of time undergoing linear shm according to $x=a\sin(\omega t)$. ...
8
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3answers
989 views

Determinism, classical probabilities, and/or quantum mechanics?

[I]f you want a universe with certain very generic properties, you seem forced to one of three choices: (1) determinism, (2) classical probabilities, or (3) quantum mechanics. [My emphasis.] ...
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4answers
233 views

Does entropy alter the probability of independent events?

So I have taken an introductory level quantum physics and am currently taking an introductory level probability class. Then this simple scenario came up: Given a fair coin that has been tossed 100 ...
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2answers
167 views

Probabilistic vs Statistical interpretation of Double Slit experiment

Why is it assumed that the results seen in the double slit experiment are probabilistic and not just a statistical result of some unknown variable or set of variables within the system.
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6answers
5k views

Probability amplitude in Layman's Terms

I am basically a Computer Programmer, but Physics has always fascinated and often baffled me. I have tried to understand probability density in Quantum Mechanics for many many years. What I ...
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3answers
854 views

Operators explaination and momentum operator in QM

I know and understand why equation below holds. But i am new to operator thing in QM and would need some explaination on this. $$\langle x \rangle = \int\limits_{-\infty}^\infty |\Psi|^2 x \, ...
5
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1answer
513 views

't Hooft for laypersons

I have looked at some of 't Hooft's recent papers and, unfortunately, they are well beyond my current level of comprehension. The same holds for the discussions that took place on this website. (See, ...
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3answers
338 views

Is “entanglement” unique to quantum systems?

My text shows (sections 0.2 and 0.3) that the joint "state space" of a system composed of two subsystems with $k$ and $l$ "bits of information", respectively, requires $kl$ bits to fully describe it. ...
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0answers
78 views

Equivalence of simple formulations of qubit entanglement

I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a ...
6
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3answers
489 views

Could quantum mechanics work without the Born rule?

Slightly inspired by this question about the historical origins of the Born rule, I wondered whether quantum mechanics could still work without the Born rule. I realize it's one of the most ...
0
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3answers
967 views

Normalisation factor $\psi_0$ for wave function $\psi = \psi_0 \sin(kx-\omega t)$

I know that if I integrate probabilitlity $|\psi|^2$ over a whole volume $V$ I am supposed to get 1. This equation describes this. $$\int \limits^{}_{V} \left|\psi \right|^2 \, \textrm{d} V = 1\\$$ ...
6
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2answers
970 views

Amplitude of Probability amplitude. Which one is it?

QM begins with a Born's rule which states that probability $P$ is equal to a modulus square of probability amplitude $\psi$: $$P = \left|\psi\right|^2.$$ If I write down a wave function like this ...
2
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0answers
301 views

Probability and probability amplitude [duplicate]

What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ...
2
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2answers
221 views

Why does the amplitude of a ripple tells us that it is a particle?

The quote below is from Matt Strassler's blog: a particle is a ripple with many crests and troughs; its amplitude, relative to its overall length, is what tells you that it is a single ...
5
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3answers
386 views

Probability and probability amplitude

The equation: $$P = |A|^2$$ appears in many books and lectures, where $P$ is a "probability" and $A$ is an "amplitude" or "probability amplitude". What led physicists to believe that the square of ...
6
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1answer
160 views

Is there an equivalent of a Galton box for a converging probability?

This is a question about probability. The Galton box (or quincunx) uses the physical process of shot moving down a pin-board, to demonstrate central limit theorem, eg: So I am interested in events ...
3
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2answers
255 views

Are probability-preserving variations of QT with respect to the Born rule mathematically possible?

Is it possible to create (m)any theoretically workable framework(s) - that do(es) produce probabilities - by taking QM and replacing the Born(-like) rule(s) with something that is not equivalent to it ...