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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Question on some basic principles of quantum theory [closed]

I asked a physicist (former head of the physics department in a university) about some of the basics of quantum theory and the double slit experiment. Here is his reply. Are any of these points ...
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2answers
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Probability in quantum physics of a wave function

I have this time-dependent wave function from solving a 3-component Schrodinger's equation, $$\psi(t)=-\frac{2}{9}(-2, 1, 2)^T+\frac{2}{9}e^{3i\omega t}(2,2,1)^T+\frac{1}{9}e^{-3i\omega t}(1, -2, 2)^T$...
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In which of Einstein's papers did he first propose interpreting square of light wave amplitude as probability of detecting photon? [closed]

In Max Born's Nobel lecture, he alludes to Einstein's proposed interpretation of EM wave amplitude (squared) as being the probability of detecting a photon: "Again an idea of Einstein’s gave me the ...
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70 views

Axiomatic Quantum Theory, and the complex numbers

In Lucien Hardy's influential paper "Quantum Theory from five reasonable axioms," Hardy states in the abstract: This work provides some insight into the reasons why quantum theory is the way it ...
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1answer
118 views

Can unitary probability theory (quantum mechanics) emerge from a lack of information about a deterministic process? [duplicate]

My initial question below appears to be unclear so I am rewording in more succinctly here. The pre-edit question remains below. There exists two types of probability theory: 1-norm (classic ...
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1answer
55 views

Inequality for quantum probability

Let $H$ be a separable Hilbert space for a quantum mechanical system then $$w (x, y) = {{\langle y \mid x\rangle\langle x \mid y \rangle} \over \langle x \mid x \rangle\langle y \mid y \rangle}$$ is ...
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1answer
39 views

Meaning of standard deviation

If two measured values of the Hubble Constant differ by 4 standard deviations, what does this mean in plain language or in probability terms?
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1answer
164 views

Must an operator that preserves probability be unitary?

One property of the unitary operator is that it preserves the norm of the state-vectors: $$ \langle \Psi | U^\dagger U | \Psi \rangle = \langle \Psi | \Psi \rangle $$ If $U$ is unitary. Is the ...
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54 views

Wavefunction of a self-adjoint operator

A self adjoint operator in general can be written as $$\mathscr{L(x)}=\frac{d}{dx}\big[ p_0(x)\frac{d}{dx}\big]+p_2(x)$$ The probability current associated can be found in the standard way $$\Psi^*\...
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Analytical solution to damped harmonic oscillator - Fokker-Planck equation

In the paper "Numerical solution of two dimensional Fokker-Planck equations" (available at: https://doi.org/10.1016/S0096-3003(97)10161-8), the authors quote an analytical solution to the damped ...
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2answers
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QM probability density function without Born's rule, invariant to wave-function phase

The QM probability density as a function of the wave function is given by Born's rule as a postulate. This leads to the probability density being invariant to the phase of the wave function. Suppose ...
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Justification behind using probability density in kinetic gas theory

Let's consider an ideal gas, of some huge number of particles. The Maxwell-Boltzmann distribution describes the probability of measuring a particle speed in a range of speeds, through integration. ...
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2answers
233 views

Richard Dawkins marble statue waving possible?

I found the following statement attributed to Richard Dawkins: "A miracle is something that happens, but which is exceedingly surprising. If a marble statue of the Virgin Mary suddenly waved its ...
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1answer
141 views

Probabilities in Quantum Mechanics: Measurement Outcomes or More?

In all treatments of quantum mechanics, the probabilistic nature of the theory enters via the Born rule for the statistical properties of the measurement outcomes of some observable. In short, this ...
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1answer
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Due to the probabilistic nature of particles, is it possible for a photon to arrive at a location slightly earlier than the speed of light?

Due to particles not having a specific location in space unless it is observed (i.e. it is a probability wave), would this mean that a photon can appear slightly ahead of where the speed of light ...
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1answer
133 views

Probability distribution of the overlap between two random quantum states in an $n$-dimensional Hilbert space? [closed]

Let two pure states $|\Psi\rangle$ and $|\Phi\rangle$ be drawn uniformly and independently from an n-dimensional Hilbert space $\mathbb{H}^n$. (see Note). What is the probability distribution of ...
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2answers
270 views

What is difference in Dirac Notation for probability and Probability Density in Quantum Mechanics? [closed]

The Dirac Notation for wave function $$\langle\psi|\psi\rangle= \int_{-\infty}^\infty \psi^{*}\psi \,dx $$ $$\text{Probability} = \int_{-\infty}^\infty \psi^{*}\psi \,dx $$ But most often it is ...
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Understanding wave function graph

I found this graph from the internet that interprets the graphical representation of wave function.I completely understand the wave function that is depicted by blue line but i really am confused ...
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2answers
71 views

Proving the preservation for the norm of a wavefunction

How can we prove the preservation for the norm of the wave-function for a specific hamiltonian (say a spin 1/2 particle) for all times?
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How does $H^2$ effect on the probability of photon detection?

Lets consider an electromagnetic wave, between two ideal conductive plates. Maxwell's theory predicts appearance of a standing waves(of $\textbf{H}$ and $\textbf{E}$), at that nodes($\textbf{E} = 0$)...
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1answer
118 views

How much can a wave function tell us?

We can not predict the future by getting the velocity and position of particles since it’s not possible to get both of these together due to the uncertainty principle. But, according to Hawking’s book ...
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3answers
250 views

Is wave function an analogue of probability amplitude or a ket vector from Dirac notation?

The way I was introduced to wavefunctions was in form of Dirac notation: $$\psi(x)=\langle x| \psi \rangle$$ i.e. the probability amplitude of going from state $\lvert \psi \rangle$ to state $\lvert ...
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1answer
172 views

What do marginalised or marginalised error mean? Contours and posterior

I am curently working on Forecast in cosmology and I didn't grasp very well different details. Forecast allows, wiht Fisher's formalism, to compute constraints on cosmological parameters. I have 2 ...
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“Synchronization” Probability of Multiple Waves with Varying Frequencies

Update 1: I've done some digging and I think this is related to signal coherence, namely, that I'm seeing a coherence time of ~3 σ, which is consistent with the definition where Ct=1/Δv where Δv is ...
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2answers
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Random walks applied to Brownian motion

[...] There we discovered that the mean square of the distance from one end to the other of the chain of random steps, which was the intensity of the light, is the sum of the intensities of the ...
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2answers
122 views

Flipping a coin with same initial conditions

Today, in my physics class my teacher was talking about how we can never predict the outcome of a coin flip. So I thought: Will the outcome of a coin flip be the same if we do not change the initial ...
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1answer
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Experimental Physics - Defining convolution in terms of equipment resolution

So I think this video: https://www.youtube.com/watch?v=N-zd-T17uiE&t=67s By Faculty of Khan does a wonderful job in explaining what convolutions are. We basically consider two pulses $f(\tau)$ ...
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2answers
121 views

Probability density during equilibration

An ensemble of states $x$ are initially drawn from a non-equilibrium probability density $P_0(x)$. Over time they will evolve towards the equilibrium distribution $P_\mathrm{eq}(x)$. What I want to ...
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1answer
66 views

Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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1answer
189 views

Bohr's Correspondence Principle and the Born Rule

Bohr's correspondence principle and the Born rule are related right? The correspondence principle states that the behavior of systems described by the theory of quantum mechanics reproduces classical ...
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1answer
74 views

Probability of finding a particle in a superposition

In QM, is it possible to ask what the probability of finding a particle in a superposition will be? Once a particle is in a superposition, it is possible to find out the probability that it will be ...
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1answer
327 views

What is Wave Function? [duplicate]

Well, what is the meaning of wave function? What does it represent? In Schrodinger's equation, we find the value of Ψ. But what is Ψ exactly? Max Born only gave an explanation of what $Ψ^2$ (the ...
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2answers
118 views

Using Boltzmann distribution, what is the ratio of probabilities of two states?

I got the probability of state $i$ (in terms of Boltzmann distribution) as $$p_{i}=\frac{1}{Z_{i}}e^{-\epsilon _{i}/{kT}},$$ where $Z_{i}$ is the canonical partition function: $$Z_{i}=\sum_{i}e^{-\...
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2answers
155 views

What is the mathematical reasoning behind Schrodinger's equation preserving its normalization, with the evolution of time?

I am currently in high-school, currently working on a physics research on the normalization of the Schrodinger's equation. I was quite interested on how we can mathematically explain preservation of ...
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4answers
214 views

Does $j=\rho v$ hold in quantum mechanics?

Let's consider the current of probablity $\vec{J}(\vec{x},t)$ associated to a particle of mass $m$ with wave function $\psi(\vec{x},t)$, given by $$\vec{J}(\vec{x},t)=\frac{i\hbar}{2m}(\psi \nabla\...
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64 views

Negative probability distribution function for Dirac equation

People say that the probability density function of the continuity equation for the Dirac equation is definite positive. I wanted to see it myself and followed the same path as Bjorken & Drell's ...
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List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
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1answer
44 views

What parameters determine the probability of virtual photon emission/absorption?

Suppose an electron is producing an electric field by emission of virtual photons and interacting with other particles. What parameters determine the probability that it will emit at least one virtual ...
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2answers
82 views

Some quantum-mechanical questions [closed]

I have recently started studying quantum mechanics, and here are some things that are really confusing me. Particle in a box: Supposedly, the square of the magnitude of the normalized wave function ...
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Method to build a polyhedral die with given probabilities [closed]

Let's define a die as a polyhedron that, if rolled over a perfect horizontal plane, ends up being in a physically stable unambiguous state labelled $n$. The die has $N$ states. Each state $n$ has a ...
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1answer
501 views

Is the Born rule indeed wrong?

This is a question about the validity of a preprint, arXiv:quant-ph/0509089, which claims that the "Copenhagen Interpretation of QM is incorrect" (same title, authored by Guang-Liang Li and Victor O.K....
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1answer
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Interpretation of the wave function in newtonian spacetime

A Newtonian spacetime is a quintuple $(M, \mathcal{O}, \mathcal{A}, \nabla, t)$ where $(M, \mathcal{O}, \mathcal{A}, \nabla)$ is a 4 dimensional differentiable manifold with a torsion free connection, ...
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Do different liquids have different distributions of kinetic energy of their particles, and does this influence their vapor pressure significantly?

This is a bit of a cross-over between a physics and a chemistry question. When we say a liquid has temperature $T$ we make a statement about the mean kinetic energy of a particle in that liquid. That ...
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2answers
158 views

Relation between the propagator and probability for the infinite well

This may be an easy question, but I am really confused about it. For the infinite square well, the (time-dependent) energy eigenfunctions are (inside the well):$$\psi_n(x,t) = \sqrt{2/L}\:e^{-iE_nt/\...
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3answers
103 views

Measurement of a State Not in the Eigenbasis of the Operator

Suppose I have a two dimensional Hilbert space $\{ |0 \rangle,|1\rangle \}$ with these states being orthonormal. Now suppose I have the Hamiltonian $H=|1\rangle \langle 0|+|0\rangle \langle 1| .$ It ...
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Is the “probabilistic nature of quantum mechanics” and quantum randomness the same?

Digital Physics are a branch of hypotheses about the fundamental physics of our universe. They basically describe the universe as an analogy to a computer and defend that everything in the universe is ...
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1answer
129 views

Probability of finding hydrogen atom in its ground state given an initial state

So I came across this question that asked what is the probability of a hydrogen atom which is prepared in an initial state $\Psi (\vec{r},t)$ to be in the ground state $\psi_{100}(\vec{r}) =2exp(-r)Y_{...
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1answer
134 views

Transmission coefficient and transmission probability

Are transmission coefficient and transmission probability the same terms? If not, could you please explain how they are related to each other?
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Error in histogram measurements

I ran into the following statement here and here but I believe it's more general. Let's suppose we're running a simulation of a system and we are interested in the distribution of a quantity (say $M$...
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1answer
52 views

Lack of intuition for distribution function in micro and macro state description

I am a mathematician who is trying to understand statistical mechanics / thermodynamics. I need a hint wrt the interpretation / meaning of the distribution function. Currently I seem to have a basic ...

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