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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.

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If I sliced the universe in half, would the slice go through a star?

This question is based on a discussion with a 10-year old. So if it is not clear how to interpret certain details, imagine how a 10-year old would interpret them. This 10-year old does not know about ...
60
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8answers
5k views

Why is the application of probability in QM fundamentally different from application of probability in other areas?

Why is the application of probability in quantum mechanics (QM) fundamentally different from its application in other areas? QM applies probability according to the same probability axioms as in other ...
36
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1answer
2k 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 right?...
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5answers
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Why do all the atoms of a radioactive substance not decay at the same time?

Why does the substance decay at a rate which is proportional to the amount of the substance at that moment? As all atoms are in hurry to become a stable atom and as their decay do not depend on any ...
33
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6answers
8k views

What is the probability for an electron of an atom on Earth to lie outside the galaxy?

In this youtube video it is claimed that electrons orbit their atom's nucleus not in well-known fixed orbits, but within "clouds of probability", i.e., spaces around the nucleus where they can lie ...
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5answers
37k views

What is a wave function in simple language?

In my textbook it is given that 'The wave function describes the position and state of the electron and its square gives the probability density of electrons.' Can someone give me a very simple ...
24
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2answers
238 views

Bell polytopes with nontrivial symmetries

Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, v_i}$, possibly in a nondeterministic manner. We are interested in joint ...
20
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5answers
563 views

Connections and applications of SLE in physics

In probability theory, the Schramm–Loewner evolution, also known as stochastic Loewner evolution or SLE, is a conformally invariant stochastic process. It is a family of random planar curves that are ...
19
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7answers
2k views

Why is a Hermitian operator a “quantum random variable”?

To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
19
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7answers
6k views

Why was quantum mechanics regarded as a non-deterministic theory?

It seems to be a wide impression that quantum mechanics is not deterministic, e.g. the world is quantum-mechanical and not deterministic. I have a basic question about quantum mechanics itself. A ...
17
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2answers
2k views

What is the Continuity Equation in QM?

I have an exercise for my homework that mentions the "continuity equation". Don't tell me how to solve it please, just tell me what the continuity equation is. I tried googling it but I couldn't find ...
16
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6answers
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Probability amplitude in Layman's Terms

What I understood is that probability amplitude is the square root of the probability of finding an electron around a nucleus, but the square root of the probability does not mean anything in the ...
16
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1answer
801 views

How does QFT predict the probability density to find a particle at x?

In quantum mechanics, the probability density of a particle's position is $$\rho(x)=|\langle x|\psi\rangle|^2$$ What is the corresponding expression in QFT to predict this distribution? Since $\rho(x)...
16
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2answers
565 views

Nonexistence of a Probability for Real Wave Equations

David Bohm in Section (4.5) of his wonderful monograph Quantum Theory gives an argument to show that in order to build a physically meaningful theory of quantum phenomena, the wave function $\psi$ ...
15
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7answers
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Born rule and unitary evolution

Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
15
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1answer
249 views

Monte Carlo integration over space of quantum states

I am currently facing the problem of calculating integrals that take the general form $\int_{R} P(\sigma)d\sigma$ where $P(\sigma)$ is a probability density over the space of mixed quantum states, $...
14
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3answers
729 views

Derive Poisson distribution from probability per time of event

Suppose we have a probability per time $\lambda$ that something (e.g. nuclear decay, random walk takes a step, etc.) happens. It is a known result that the probability that $n$ events happen in a time ...
14
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3answers
334 views

Is there a condition of quantum mechanics that forbids Lorentzian distributions?

Imagine a particular potential that allows a superposition of eigenstates such that in space basis the probability density $|\psi(x)|^2$ is a lorentzian (Cauchy) distribution. The properties of the ...
14
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1answer
2k views

What are the odds of finding an electron a long way from an atom? As in metres away? [closed]

Pretty self-explanatory, heard that there is a chance that you might find an electron on the other side of the universe, just wanted to know.
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5answers
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Why do coherent states have Poisson number distribution?

In quantum mechanics, a coherent state of a quantum harmonic oscillator (QHO) is an eigenstate of the lowering operator. Expanding in the number basis, we find that the number of photons in a ...
14
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0answers
247 views

Renormalisation and the Fisher-Rao metric

The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be though of as a flux in a space of possible Hamiltonians for a ...
13
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5answers
3k views

Earth still exists - does this fact tell us anything about LHC safety?

When LHC was about to be launched there were many fears that it would destroy the world. To counter them scientists tried to carefully examine all possibilities and concluded that there is nothing ...
13
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4answers
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What is the fundamental probabilistic interpretation of Quantum Fields?

In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the ...
12
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2answers
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Is there actually a 0 probability of finding an electron in an orbital node?

I have recently read that an orbital node in an atom is a region where there is a 0 chance of finding an electron. However, I have also read that there is an above 0 chance of finding an electron ...
12
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2answers
460 views

Random Walk Randomly Reflected

Hi I am not specialist in probability so I will not be surprised if the answer for this question is just a simple consequence of well known results from the random walk theory. In this case, I will ...
12
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2answers
361 views

Probability measure implies quantum mechanics?

The article "Quantum Logic and Probability Theory," by Wilce, has the following in section 1.4: 1.4 The Reconstruction of QM From the single premise that the “experimental propositions” ...
12
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2answers
249 views

Probabilistic Intuition behind connected correlations and 1PI vertex function

In the context of statistical field or quantum field theory, one encounters so called generating function(al) for connected correlations, aka the following function(al): $$ W(J) = \ln (Z(J))$$ $$ Z(...
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3answers
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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 ...
12
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1answer
632 views

The measure problem in the anthropic principle

The anthropic principle is based upon Bayesian reasoning applied to the ensemble of universes, or parts thereof, conditioned upon the existence of conscious observers. That still leaves us with the ...
11
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4answers
2k views

How is quantum mechanics consistent with statistical mechanics?

Let's say we have an harmonic oscillator (at Temperature $T$) in a superposition of state 1 and 2: $$\Psi = \frac{\phi_1+\phi_2}{\sqrt{2}}$$ where each $\phi_i$ has energy $E_i \, .$ The probability ...
11
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4answers
626 views

Connection between Hamiltonian version of the least action principle and probability amplitude in the Schrödinger equation

If I'm not mistaken, Schrödinger was influenced to look at wave equations because of de Broglie's assertion about particles having a wavelength. He started with the Hamiltonian equation which is ...
11
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4answers
1k views

How can I intuitively understand the Boltzmann factor?

It is known that for a system at thermal equilibrium described by the canonical ensemble, the probability of being in a state of energy $E$ at temperature $T$ is given by the Boltzmann distribution: $$...
10
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2answers
5k views

What is probability current in quantum mechanics?

What is probability current in quantum mechanics? Why define such a thing? I mean the meaning of probability current. I know the formula for it but I just don't get the idea of a flow of probability ...
10
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2answers
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Itô or Stratonovich calculus: which one is more relevant from the point of view of physics?

Langevin equation provides an example of a physical model which involves a differential equation with a stochastic term. Now, I wonder, how should one treat this? When I studied stochastic processes, ...
10
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1answer
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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 ...
10
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2answers
500 views

Motivation for Wigner phase space distribution

Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula $$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
10
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2answers
6k views

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 ...
10
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2answers
827 views

How to determine the probabilities for a cuboid die?

Imagine we take a cuboid with sides $a, b$ and $c$ and throw it like a usual die. Is there a way to determine the probabilities of the different outcomes $P_{ab}, P_{bc}$ and $P_{ac}$? With $ab$, $bc$...
10
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1answer
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Interpretation of the Random Schrödinger Equation

I should preface this by admitting that my physics background is rather weak so I beg you to keep that in mind in your responses. I work in mathematics (specifically probability theory) and a paper ...
10
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4answers
666 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. ...
10
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2answers
919 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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3answers
1k views

Form of Schrödinger equation for the probability density

Is it possible to formulate the Schrödinger equation (SE) in terms of a differential equation involving only the probability density instead of the wave function? If not, why not? We can take the ...
9
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2answers
708 views

Is it really impossible to calculate in advance the result of throwing dice?

Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics.
9
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3answers
3k views

Relation between unitarity and conservation of probability

In a seminar, I heard that the unitary aspect of representations was important physically, because in quantum mechanics unitarity is closely tied to the conservation of probability. Could someone ...
9
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3answers
4k views

How to work out the relation between the “mean relative speed” and the “mean speed”?

I'm a freshman and am taking the general physics course. I just learned intro thermodynamics. One problem that really puzzles me is the calculation of "collision mean-free path", where calculating the ...
9
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2answers
1k views

Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?

Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix? Does a density matrix always have a a basis of eigenvectors?  
9
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7answers
2k views

Mathematically possible vs physically probable outcomes

A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with ...
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5answers
15k views

Would one actually find their doppelgänger in a “Googolplex Universe”?

Related: Infinite universe - Jumping to pointless conclusions I've recently become a fan of Numberphile, and today I happened to watch their video regarding Googol and Googolplex. In the video, I ...
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3answers
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Infinite universe - Jumping to pointless conclusions

I watched an episode of thee BBC Horizon series titled 'To infinity and beyond'. In this program a number of respected physicists and mathematicians were talking about the nature of infinity and an ...
9
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1answer
674 views

Can't the Negative Probabilities of Klein-Gordon Equation be Avoided?

I came across these notes of Dyson on Relativistic Quantum Mechanics. There on p. 3, he mentions that the issue with the Klein-Gordon equation is that the only way to relate $\psi$ with a probability ...