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 ...
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81 votes
8 answers
7k views

Why is the application of probability in Quantum Mechanics 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 ...
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66 votes
8 answers
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Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute?

I wondered this since my teacher told us about half life of radioactive materials back in school. It seems intuitive to me to think this way, but I wonder if there's a deeper explanation which proves ...
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39 votes
1 answer
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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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36 votes
6 answers
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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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6 answers
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Why doesn't the nucleus have "nucleus-probability cloud"?

While deriving the wave function why don't we take into the account of the probability density of the nucleus? My intuition says that the nucleus is also composed of subatomic particles so it will ...
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36 votes
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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 ...
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30 votes
5 answers
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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 ...
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27 votes
1 answer
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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)...
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25 votes
4 answers
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Is the probability of an electron being somewhere zero?

So recently I've been reading "How to teach Quantum Mechanics to your Dog" by Chad Orzel. In chapter 3, he says, if I understood this right, that electrons can only exist in specific quanta - that is ...
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23 votes
5 answers
836 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 ...
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21 votes
7 answers
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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,...
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20 votes
5 answers
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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 ...
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7 answers
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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 ...
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1 answer
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How is conditional probability handled in quantum mechanics?

In ordinary probability theory the conditional probability/likelihood is defined in terms of the joint and marginal likelihoods. Specifically, if the joint probability of two variables is $\mathcal{L}(...
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18 votes
4 answers
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Can our hand pass through a table?

I recently read that "there is a $1$ in $5.2^{61}$ chance that the molecules in your hand and table would miss each other, making your hand go through it". To me, it seems completely false, ...
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7 answers
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Is the Born rule a fundamental postulate of quantum mechanics?

Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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18 votes
3 answers
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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 $\psi$ ...
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17 votes
6 answers
23k views

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 ...
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17 votes
2 answers
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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 ...
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17 votes
3 answers
545 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 ...
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17 votes
1 answer
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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 thought of as a flux in a space of possible Hamiltonians for a ...
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16 votes
3 answers
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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 ...
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15 votes
2 answers
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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?  
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1 answer
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Normalizing Propagators (Path Integrals)

In the context of quantum mechanics via path integrals the normalization of the propagator as $$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$ is incorrect. But why? It gives the correct pre-...
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15 votes
2 answers
407 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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14 votes
1 answer
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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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14 votes
2 answers
744 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|\...
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14 votes
1 answer
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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 ...
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14 votes
1 answer
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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, $...
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14 votes
3 answers
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Where does the Born rule come from? [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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13 votes
2 answers
9k views

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 ...
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13 votes
5 answers
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 ...
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13 votes
4 answers
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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 ...
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13 votes
5 answers
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Why can't the Uncertainty Principle be broken for individual measurements if it is a statistical law?

The Heisenberg Uncertainty Principle is derived for two operators $\hat A$ and $\hat B$ as $$\Delta \hat A\ \Delta \hat B \geq \dfrac{1}{2}|\langle[\hat A, \hat B] \rangle|$$ where $\Delta$ denotes ...
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13 votes
1 answer
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If frequency of photons is a continuous spectrum, wouldn't the chance of a photon having the exact right frequency to excite an electron be zero?

As far as I'm aware, the energy needed to excite an electron to a different orbital is discrete. Since the frequency of light is continuous, wouldn't it be impossible for a photon to have the exact ...
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13 votes
1 answer
22k views

Kolmogorov-Smirnov test vs Chi-squared test

What is the difference between the Kolmogorov-Smirnov test and the Chi-squared test? When should we use one instead of the other? I was reading this article, and I got confused a lot. It is hard to ...
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13 votes
2 answers
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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, ...
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13 votes
4 answers
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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: $$...
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13 votes
1 answer
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Naive question about time-dependent perturbation theory

In time-dependent perturbation theory where $H=H_0+V$ and $V$ is considered small and has no explicit time dependence, the standard text-book treatment of the leading order probability amplitude for ...
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13 votes
2 answers
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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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12 votes
3 answers
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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 ...
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12 votes
4 answers
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What is the unit (dimension) of the 3-dimensional position space wavefunction $\Psi$ of an electron?

I googled for the above question, and I got the answer to be $$[\Psi]~=~L^{-\frac{3}{2}}.$$ Can anyone give an easy explanation for this?
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12 votes
2 answers
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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 ...
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12 votes
1 answer
2k 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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12 votes
2 answers
575 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 ...
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12 votes
2 answers
548 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” ...
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12 votes
1 answer
692 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 ...
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11 votes
3 answers
3k 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 ...
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11 votes
4 answers
774 views

Why are probabilities for each micro-state equal within a micro-canonical-ensemble?

This question is about statistical mechanics: Why does it make sense to postulate, that in thermal equilibrium all micro-states with fixed U within a micro-canonical ensemble are equally probable? ...
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