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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I've read that $\langle a | b\rangle$ is a probability amplitude but $\langle a | a\rangle$ is a probability. Why the inconsistency?

I'm studying elementary quantum mechanics, and I've read that $\langle a \vert b \rangle$ is the probability amplitude of a transition from state $a$ to state $b$. Thus, $|\langle a | b \rangle|^2$ ...
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Expression of stress-energy tensor with density function

I try to get the following expression defining the stress-energy tensor in General Relativity ($\big< \big>$ means average) : $$T^{\mu\nu} = \int \dfrac{\text{d}^3p}{E} F(\vec{x},\vec{p})p^{\mu}...
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Quantum mechanics calculations - must they always output a rational number?

If space is quantized, and particles are quantized, then the chance of a particle showing up in a specific unit of space must be a rational fraction? Could a quantum mechanics calculation ever output ...
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168 views

Transition Probabilities and Amplitudes of atomic levels

I am trying to understand the physical meaning behind transition amplitudes and transition probabilities. By my current understanding if $A$ is the transition amplitude then the corresponding ...
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Normalization of Probability distribution [closed]

I need to Know. Is it a condition that Probability density is bounded between 0 and 1?
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246 views

Number of microstates

I need some help with this problem: Consider a system of $N$ distinguishable particles. Each of these particles can be in a state with energy $\epsilon$ or $-\epsilon$. Consider that the states ...
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97 views

Percolation in Bethe Lattice with Alternating Neighboor Numbers

Suppose we have a Bethe Lattice with alternating numbers of neighbours,starting from 3 and then 4 and so on. So the zeroth point has 3 neighbours, each of those 3 first neighbours has 4 neighbours (3 ...
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261 views

Does a wave function never reach zero probability density?

$$ψ=e^{iκx}$$ Since the wave function is an exponential equation, is there no point with zero probability density of finding a given particle? Does that justify quantum tunneling?
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Are the diagonals of the density operator always the probabilities of finding the system in various states?

I found lecture notes online that said "Diagonal density matrix elements are the probabilities of finding system in various states." My quantum mechanics textbook doesn't say anything about this, it ...
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Can the density operator be used to describe a continuous probability distribution, analogously to a classical probability distribution function?

I've been reading about Liouville's equation and am now trying to understand Von Neumann's equation, which seems to be more or less the quantum mechanical version. Both cases involve an ensemble of ...
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58 views

Generalisation of the measurement postulate in quantum mechanics

Given an observable that has a partially discrete and partially continuous spectrum of eigenvalues associated to it with the order of the spectrum's degeneracy being greater than 1, how would you ...
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Experimentally determined probability has an “error

Feynman , in the paragraph 6-3 of the first Volume of this lectures, writes that an “experimentally determined” probability has an “error,” and writes ( We have only 2 events) $$P(H)=\frac{N_{H}}{N}(+...
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Can someone explain the concept of 'Negative Probabilities' in an intuitive manner? [duplicate]

Can someone explain the concept of Negative Probabilities in an intuitive manner? I can't seem to understand this concept. I hope someone can explain this concept in an intuitive manner.
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How to find this probability with the microcanonical ensemble?

When dealing with isolated systems we are dealing with the microcanonical ensemble. In that case, we suppose that each individual microstate has the same probability. So if $\Omega(E)$ is the number ...
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Are Ross-Littlewood-sequences or negative probabilities possible in physics?

There are claims that the Ross-Littlewood paradox could be simulated in physics. See https://stats.stackexchange.com/q/315502/ and in particular Paul's answer there. Also a solution of the Einstein-...
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1answer
35 views

Why are wave packets constructed so that the maximum probability occurs at K0 (the average wave number)?

The definition of a wave packet I have been given is that it is "a superposition of many plane waves, with wave numbers grouped around an average value $k_0$". I was told that, for a particle we want,...
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207 views

Topological entropy in Markov chains

Given a finite Markov chain, how do I find the topological entropy $h_T$? Furthermore, I should compare it with the Shannon entropy $h_S$ and show that $h_T\leq h_S$. Is this a general fact? This ...
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How deterministic nature of our world emerges?

Quantum mechanics shows that nature is non-deterministic. But the world we see around us seems deterministic. Take for an example: harmonic oscillator when $n$ becomes very large the probability ...
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What are the assumptions underlying the master equation?

In Reichl, 2016; pg405 the author gives a derivation of the master equation - which I will outline below in my own words: We start with:$$P(n,t+\Delta t)=\sum_m^M P(m,t)P(n,t+\Delta t\mid m,t)\...
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Markov Chain expressed in Density Matrix formalism [closed]

Suppose we have two states of a system where I tell you that there is a probability $p_1$ of being in state $1$, and probability $p_2$ of being in state $2$. The total state can be written as a vector ...
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Proving that Shannon entropy is maximal for the uniform distribution using convexity

I need to show that $-\sum_i{p_i \log{p_i}}$ is maximal iff $p_i=p_j$ for all $i\neq j$ using the convexity inequality: $\phi (\frac{\sum{a_i}}{N})\leq \frac{\sum{\phi (a_i)}}{N}$ I tried expanding ...
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2answers
28 views

probability distribution of dependent random variables

If we have dependent random variables, then what how is the distribution the pdf look like? Can it be a normal distribution? For example, additive white Gaussian noise (AWGN) has a normal distribution,...
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140 views

Finding amplitude of probability

The mathematical structure of quantum mechanics, follows almost inevitably from the concept of a probability amplitude. For James Binney and David Skinner: "With every value in the spectrum of a ...
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What determines the determinism of observables?

It is well known that there exists certain class of physical observables like momentum and position which are common to both classical and quantum mechanics, and has different 'kinds of predictability'...
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95 views

What is the physical basis of Born's interpretations?

Did anyone has any idea how Born came up with the probabilistic interpretation of quantum mechanics. It is by all means very bizarre. And then it leads to the idea of copenhagen interpretation. Also ...
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The possiblity of one more raindrops drop the ground exact same time? [closed]

Think there is a simple rain and we have got a very very sensitive clock. Is it possible to at least two raindrops hit to ground exact in same time? I mean the date time when they hit the ground, not ...
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Does the partition function define probability of being in multiple states?

The partition function is defined as a sum over all microstates $j$ as: $Z=\sum_{j}exp(-\beta E_j)$ or $Z=\int_{-\infty}^{\infty} exp(-\beta E)dE$ if the states are continuous. We can use $Z$ to ...
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133 views

Entropy and fluctuations near the equilibrium

I'm very confused by the account on fluctuations near the equilibrium in chapter 12 of Landau's book on statistical physics. To be brief, the kernel of my doubt is that he states that if you have a ...
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1answer
61 views

Statistical error with large number of particles in weak measurements

Consider a measurement process. If $\Delta \pi$ and $\Delta x_n$ is the uncertainty in momentum and position of the measuring device. Aharonov, Albert, et al. ask us to consider the opposite limit: ...
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271 views

What's the relationship between probability amplitudes and amplitudes of a wave?

Amplitudes or probability amplitudes are the complex coefficients of the linear combination of states which represent other quantum physical states. The amplitude of a wave can be interpreted as a "...
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1answer
45 views

Particle ensemble performing shm, calculate amplitude pdf

Consider the shm for a single particle. Then the particle's position is given by (assume zero initial phase): $$x = a \times \sin(\omega t)$$ The infinitesimal probability of finding a particle ...
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243 views

Why does time evolution preserve the norm of a wavefunction?

I saw an awesome derivation of Schrodinger's equation on Wikipedia. Part of it relies on: Since time-evolution must preserve the norm of the wave-function, it follows that $U(t', t)$ must be a ...
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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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Measurement and probability

In the $\left\{\left|j_1j_2;jm\right\rangle=\left|ls;jm\right\rangle\right\}$ basis, I've got the state ket $$\left|\alpha\right\rangle=\frac{1}{2\sqrt{2}}\left(\sqrt{\frac{2}{3}}\left|11;20\right\...
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1answer
151 views

Probabilities with density matrices

Given two mixed states $\rho$ and a $\sigma$, does it make sense to say that the probability of $\rho$ being in the state $\sigma$ is given by $Tr(\rho \sigma)$? It seems to me that the answer must ...
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Postulates of inner product

In quantum mechanics, two fundamental properties of inner products (J.J Sakurai) Chapter 1.2, are: $\langle \alpha|\beta\rangle = \langle \beta|\alpha\rangle^*$ $\langle \alpha|\alpha\rangle \ge 0$ ...
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What's the distribution of scattering angles for hard spheres with random impact parameter?

I am modelling the scattering of hydrogen atoms against each other. In this model, one hard sphere scatters elastically off another hard sphere, they are identical with radius $r$. They meet with ...
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Question about calculating |Wavefunction|^2

In one of my homework in Quantum mechanics, I was asked to find $|Ψ(x,t)|^2$, where \begin{align} Ψ(x,t)&=1/\sqrt{10}[3ψ_1(x)e^{-iE_1t/ħ}-ψ_3(x)e^{-iE_3t/ħ}]\, ,\\ &=1/\sqrt{10}[3\sqrt{2/a} \...
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1answer
135 views

Normal Distribution vs Poisson Distribution

I am not sure that I understand when to use Normal Distribution and when to use Poisson distribution! For example, in RF communication the channel noise is mainly modeled as Normal Gaussian ...
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Probability of measuring eigenvalue of non-normalised eigenstate

This came up while working on a question about measuring the angular momentum of a particle in a superposition of angular momentum eigenstates: Given that: $$\langle\theta,\phi|\psi\rangle \propto \...
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1answer
133 views

In statistical mechanics, why do we consider number of states of a system in energy interval?

In statistical mechanics,when we go for calculating the no. of accessible micro states of a system, I notice that we always calculate the no. of micro states of that system in some energy interval say ...
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1answer
51 views

Free particle in one dimension, gaussian distribution question [closed]

Okay, so the question im struggling with is: Given a free particle in one dimension(with $H=\frac{p^2}{2m}$) in a state $p$ at $x=0$ with uncertainty $\sigma$(Gaussian Distribution) determine the ...
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1answer
165 views

Defining Kraus operators with normal distribution

I am interested in defining Kraus operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of ...
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1answer
52 views

Probability densities at the LHC

Let $\sigma$ be a cross-section for the collision of two protons as given by $$ \sigma = \intop_0^1 \mathrm{d}x_1 \intop_0^1 \mathrm{d}x_2 \, \sum_{a,b}f_a(x_1,Q^2) f_b(x_2,Q^2) \frac{1}{2\hat{s}} \...
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143 views

Probability of a specific energy state

We consider the normalized wave function: $$\psi(x,t) = \sqrt{\frac{2}{3}}\psi_0(x)\exp\left(\frac{-iE_0t}{\hbar}\right) + \sqrt{\frac{1}{3}}\psi_1(x)\exp\left(\frac{-iE_1t}{\hbar}\right) $$ To ...
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How is the measure for the wavefunction determined in quantum mechanics?

Given some quantum mechanical system described by a lagrangian ${\cal L}=\frac{1}{2}\dot{q}^2-V(q)$, I can imagine solving for the wavefunction $\Psi[q]$ and then using this to compute expectation ...
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Movement of a random walk in the limit (a particle in diffusion)

I asked this question in Math Exchange and MathOverflow and obtained no answer. This question may lack of mathematical rigorous, but I would like to understand why this type of reasoning is sometimes ...
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1answer
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Stochastic dynamics of rotation intergral over $d\hat n$?

I am looking into the stochastic dynamics of rotation in which we describe the orientation with a unit vector $\hat n$. If we let $p(\hat n',t)$ denote the probability that $\hat n=\hat n'$ at time $t$...
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211 views

Path integral kernel dimensions and normalizing factor

I am currently reading Quantum Mechanics and Path Integrals by Feynman and Hibbs. Working on problem 3.1 made me wonder why the 1D free particle kernel: $$ K_0(b,a) = \sqrt\frac{m}{2\pi i \hbar(t_a - ...
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Probability of photoelectric effect for gamma rays [duplicate]

Why is the probability of the photoelectric effect occurring higher for low energy gamma rays? I'd like a physical answer not just equations.