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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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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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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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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
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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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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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Stochastic version of the Kirchoff circuit law

I assume this question could be written in a non-technical jargon, but I will try to be as simple as possible. The Kirchoff circuits law assert that the sum of inward and outward currents at a node ...
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31 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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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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446 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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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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1answer
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Probability of successive measurements

Suppose I have states $|1\rangle$ and $|2\rangle$, and my system is in a quantum mixed state \begin{equation} c \left( |1\rangle + \sqrt{3} |2\rangle \right). \end{equation} In a first measurement I ...
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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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68 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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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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38 views

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
34 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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1answer
57 views

How to evaluate the probability when a particle is detected?

Everyone knows the standard probability interpretation of the quantum mechanics. For example, the wave function of some particle at some time $t$ is $\psi (x,t)$. Therefore, if the particle is ...
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Probabilities in non-stationary states

I'm confusing myself. Let's represent some state in the eigenbasis for Hydrogen: $$|\psi\rangle = \sum_{n,l,m}|n,l,m\rangle\langle n,l,m|\psi\rangle.$$ Now denote the initial state by $\psi(t=0)\...
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Converting a discrete statistical energy distribution to a continuous version

The probability of finding a particle at a particular energy level when energy is considered discrete is according to Boltzmann: $$P(E_j) = \frac{g_j\cdot e^{-\beta E_j}}{\sum_{j=1}^\infty g_j \cdot ...
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Derive properties of fluids using Monte Carlo method on brownian motion

Given a particle inside a fluid, it's known that its movement will be unpredictable due to the random collisions with the particles of the fluid. However, the distance from the origin of motion will ...
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Probability In a Nuclear Transmutation Question

How much cobalt will form into zinc when you add a mole of the set (3 protons, 3 electrons and 5 neutrons) to the cobalt? I cannot figure out how to find the probability of creating a certain ...
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What does conservation of probability mean in Classical Mechanics and why is it true?

In the context of the Liouville equation, regularly the conservation of probability is invoked. (Of course, the overall probability is always conserved but this is a truism and not what is meant here. ...
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Doubt about the probabilistic nature of quantum stuff and the field theory

To the quantum field theory, is it like there's "two layers of reality", one in which things are just probabilities waves that collapses into the quantum fields or is the quantum field and its waves ...
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1answer
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Is possible to obtain a master equation from Rabi's coupled differential equation?

Starting from a differential equation for $c_i$ such that $c_i c_i^*$ is the probability of being at state $i$, I want to obtain a master equation for $c_i c_i^*\equiv p_i$. Consider a two-state ...
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Calculating the probability of one particle being in a certain state in two-particle system

Let's say I have the two-particle state $$|\psi\rangle=\frac{|H\rangle_a|H\rangle_b+|V\rangle_a|V\rangle_b}{\sqrt{2}}$$ where $H$ is horizontally polarized and $V$ is vertically polarized. And I ...
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2answers
99 views

Is this decoherence?

I have a very basic understanting of decoherence (i.e. I,ve read the Wikipedia page), but I was recently reading Heisenberg's The Physical Principles of the Quantum Theory and I came across a thought ...
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49 views

How does the expectation value of a partial derivative of a random variable make mathematical sense in QM?

In probability theory, I'm familiar with the definition of the expectation value of a random variable $X \colon \Omega \rightarrow \mathbb{R}$ being: $$ \langle X\rangle= \int_{-\infty}^{\infty} x ...
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Bell inequality proof

I am trying to understand how the correlation function in John Bell's paper on EPR is derived for a spin singlet state $|{\Psi}\rangle$. This is defined to be $$ \langle{\Psi}|\left(\bf{\sigma}\cdot\...
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Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?

Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
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Electron density in DFT (*ρ*(r)) and probability density (wave function squared)

Are the electron density in density functional theory, ρ(r), and probability density, defined as wave function squared, the same quantities?
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E. T. Jaynes' subjectivism vs measurement of distributions

In his paper, E. T. Jaynes argues that entropy is a measure of our ignorance about a system. As such, the probability distribution of states $\{p_k\}$ has to be chosen in the most unbiased way, thus ...
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If the probability that an alpha will deflect is $1/10000$, for $n$ layers, is the probability is only $1/10000n$?

I have attached a picture of an extract I read on Wikipedia (also in the AQA A-Level Physics specification and textbook). It says that 1/10000 alpha particles deflected in the alpha particle ...
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Radiative Transport Equation scattering phase density probability term question

The development of Radiative Transport Equation has a contribution term called the scattering phase density probability function. It account to scattering events from photons from a different solid ...
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115 views

Finding the expression for probability density (the Klein Gordon equation)

Source: Quantum Field Theory for the Gifted Amateur by Tom Lancaster, Stephen J. Blundell. I am struggling to understand the logical step from the outline of the 'proof' in the footnote, to the fact ...
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Can Quantum Mechanical Potential have a Probability Distribution

I am currently in my second semester of undergraduate quantum mechanics. We have recently starting discussing two particle systems, usually in relation to spin interactions. In all of our calculations,...
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81 views

Understanding the calculation of expectation value

The expectation value (in sense of discrete probability) can be thought of as $$ \left<a\right>=\frac{1}{N}\sum\limits^{N}{Â }\psi $$ where $N$ is the number of experiments. As the number of ...
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Can an electron of an atom be found anywhere? Does it need energy to happen? [closed]

According to quantum mechanics it should be possible. But can it happens when it has so small probability to occur? also if it can happens that means that energy must be provided in order to the ...
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Notation in a question on probabilities and particle counting

I'm working through Stephen Barnett's book on quantum information and have come across the following question (1.5, for anyone keeping track at home) A particle counter records counts with an ...
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1answer
55 views

Probability of WaveFunction [closed]

A particle is confined in a one dimensional box of length $a$. What is the probability of finding the particle at $x = a/4$? I know that the wave function is written as $$y= A\sin([(\pi x)/a]$$...
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1answer
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Dependence of wave function with time, especially probability density function. And Continuity equation

I was learning Basic Quantum mechanics. I cam across the fluid equation in QM, which suggests $\Psi^*\Psi$ is probability density function. Consider the two statements below Probability will change ...
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Probability of transition defined as the the ratio between reflected and incident fluxes

I'm reading a paper (Rapp, 1968) that treats quantum mechanically the problem of a particle $A$ "hitting" an harmonic oscillator made of two particles, $B$ and $C$: $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,...
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Unit of Spectral Emissive Power

I understand Spectral Emissive Power as the total amount of energy carried by photons having the same wavelength (energy), and it has the unit of (W/m2.um). I could not grasp the physical ...
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Why quantum map must be hermitian?

Quantum maps transform a density matrix into another one, Assume we are in the Hilbert space :$ H_A $ the quantum map on the density matrix $\rho_A$ living in $H_A$ is : $\mathcal{L}_A$ Why $\mathcal{...
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Number of Isotopes created with decay <-> Chain Yield

The chain yield (or fission yield) states how many isotopes with a certain mass $A$ are created with the decay of $^{235}$U. But how do we know the fractions of specific Isotopes that are created ...
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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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Understanding the radial distribution function [duplicate]

I am confused why the maximum of the radial distribution function for 2p orbital is closer to the nucleus than that for 2s orbital. Doesnt this mean that there is a higher chance of finding 2p orbital ...