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.
1,377
questions
0
votes
0
answers
39
views
Computing the probability of a measurement of energy for a free particle [closed]
$\newcommand{\ket}[1]{\left|#1\right>}$
$\newcommand{\bra}[1]{\left<#1\right|}$
$\newcommand{\braket}[2]{\left< #1 | #2 \right>}$
Problem
Consider a free particle, with no spin, in one ...
- 410
5
votes
0
answers
76
views
In statistical mechanics, why is one "allowed" to treat classical systems probabilistically?
Is the essential argument that these systems are microscopically chaotic enough that we can approximate their evolution as random (vastly simplifying calculations) and still make accurate experimental ...
- 161
0
votes
0
answers
23
views
Average probability of sampling all clusters
A system consists of $N$ nodes. The nodes are distributed into $M$ clusters such that each node belongs to a unique cluster. Each cluster $i$ has one unit of weight, $w_i = 1$. Thus the initial weight ...
- 101
0
votes
0
answers
39
views
Is there a general construction for three-outcome qutrit POVMs?
For qubits, I can consider the General POVM elements: $M_{\pm} = \frac{1}{2}(I \pm \hat{n}\cdot\overline{\sigma})$ where $\sigma $ is a vector containing the Pauli matrices and $\hat{n}$ a vector with ...
- 1
0
votes
0
answers
39
views
Probability density of electrons in double slit experiment [closed]
My professor gave some vague question in the double slit experiment setup which uses electrons instead of light, asked to found out the intensity of electrons on screen as a function of number of ...
- 1,952
12
votes
7
answers
1k
views
Why are expectation values of an observable important in QM?
I've been reading that expectation values of an observable is all what we can get and are the key quantities of the theory, but performing the same experiment many times would generate a distribution ...
- 151
1
vote
1
answer
94
views
Most probable position of finding an electron represented in cartesian and spherical coordinates
Consider the probability of finding an electron within a region:
$$P= \iiint |\psi(x,y,z)|^2 dxdydz$$
I would think that The probability of finding an electron at a single point in space would be ...
- 6,377
1
vote
0
answers
24
views
Probability of two atoms Bonding [closed]
I am trying to work out an expression for two individual atoms in a vacuum of small volume, $V$, and the probability that they will bond together. I have attempted looking at the quantum approach, in ...
- 41
1
vote
0
answers
30
views
Equivalence between CGLMP inequality and CHSH inequality
In this paper, they claim that the inequality $$I = P(A_1 = B_1) + P(B_1 = A_2 + 1) + P(A_2 = B_2) + P(B_2 = A_1) \leq 3$$ is equivalent to the CHSH inequality $$|E(A_1,B_1) - E(A_2,B_1) + E(A_2,B_2) +...
- 303
0
votes
2
answers
97
views
What is a cross section, really? [closed]
Upon looking at different resources, there is a common definition of a cross section (in the context of QFT) to be the probability that some scattering process occurs. For example, here is a ...
- 224
-1
votes
0
answers
42
views
Mean and higher moments of final position of particle subject to time dependent central force along x
I would like to find the expected value and higher moments of the x and y components of the final position of a particle moving in the xy plane, subject to a central force, centered on a positive x ...
- 11
8
votes
3
answers
1k
views
Dirac's definition of probability in quantum mechanics
I'm currently reading "The principles of quantum mechanics" by Dirac, and I'm having some trouble understanding some of his assumptions, because in the quantum mechanics course I'm following ...
- 415
1
vote
1
answer
35
views
Boundary Pressure of a system in the Grand Canonical Ensamble
Consider a system of fixed volume $V$ in equilibrium with a reservoir of both heat and particles (hence we may describe the system using the Grand Canonical Ensamble).
While I was trying to derive the ...
- 410
1
vote
1
answer
61
views
Does the master equation break down for negative times?
I'm studying stochastic dynamics and have encountered the framework of the master equation for the study of continuous time Markov processes. First, I'll state some general definitions and then say ...
1
vote
2
answers
157
views
How to know what eigenvalue corresponds to measurements of individual qubits in a multiqubit system?
I'm working through the book "Introduction to the Theory of Quantum Information Processing" by Bergou and Hillary, and I've encountered a scenario that I'm not sure how to approach. In ...
- 25
0
votes
2
answers
167
views
If the time at which a single atom decays is random, why do groups of atoms behave in predictable ways? [duplicate]
Why do groups of atoms decay at predictable rates even though a single atom’s decay point is completely unpredictable? I’m having trouble wrapping my head around this.
From my reading, it seems that ...
- 281
0
votes
2
answers
83
views
Normalization to unity, Projection Operators in QM
I have a question about something that is stated in Sakurai's MQM. It's written that if one runs a sequence of selective measurements (namely, a sequence of independent stern-gerlach apparatuses) ...
6
votes
3
answers
563
views
Bell's inequality for angles 120°
In 1964, John Bell first derived the original Bell inequality, $|E(a,b)-E(a,c)|\leq1+E(b,c)$. Here $a,b,c$ are three different possible spin measurement directions, and $E$ is the measured ...
4
votes
1
answer
526
views
How large a closed system does the 2nd law of thermodynamics require?
Does the second law of thermodynamics apply to a single electron in deep space? Or two electrons? Or 100? (assume the electrons are restricted in a finite area)
From which point we can start to talk ...
- 55
1
vote
2
answers
68
views
How does the state operator relate to the complete set of basis vectors?
I quantum mechanics we can represent the state of a system $\vert\psi\rangle$ in some Hilbert space as a complete set of basis vectors $\vert n\rangle$;
\begin{equation}
\vert\psi\rangle=\sum_n^Nc_n\...
- 409
-1
votes
2
answers
136
views
If you shake this sphere with sand for $n$ times ($n$ could be as large as you want), will the sand come back to its original arrangement?
"According to the Poincaré theorem, the sand will eventually come back to its original arrangement after a large number of shakes. But based on intuitive thinking, if you shake it well, the sand ...
- 218
0
votes
1
answer
102
views
What is probability? [closed]
I am looking for a definition of probability, specifically of the kind that corresponds to the frequency of repeatable phenomena rather than the epistemic kind.
I have the following idea of what ...
- 165
0
votes
0
answers
39
views
Which statistical methods are better than basic random scans for studying and exploring the parameter space of extensions of the standard model?
Given a lagrangian of a new model beyond the standard model, and given a set of constraints, say the oblique parameters for instance and the decay of the higgs boson and some signal strengths and ...
- 59
0
votes
1
answer
59
views
Probability and Thermodynamics: Analysing feasibility of 50 benches problem
Question:
I have a thought experiment that involves dropping 50 benches from a height through an open roof and examining the probability of them spontaneously arranging themselves into neat rows and ...
2
votes
1
answer
77
views
What if probability current is independent of position and time? [closed]
I worked an example in Griffiths to find probability current and then determine the direction of the flow of probability. As you can see the $J$ is constant value and independent of time and position $...
3
votes
0
answers
41
views
Confusion Regarding the Propagator [duplicate]
To my understanding, the expression $$G^+=\theta(t_f-t_i)\langle x_f|\mathcal{\hat U}(t_f,t_i)|x_i\rangle$$ represents the probability amplitude that a particle starting at position $x_i$ at time $t_i$...
- 31
1
vote
1
answer
94
views
A question about Markov chains
I'm considering a Markov process with a continuous state space. Let $V(x)$ be a differentiable function, $\Delta t$ a fixed time step, and, at every step, set
$$x_{n+1}= x_n-\alpha \frac{dV}{dx}\Big|_{...
- 219
0
votes
0
answers
36
views
Is anything truly stochastic? [duplicate]
If everything in the universe happens according to rules, thermodynamic or otherwise, then how would anything (or any choice) ever be stochastic?
Multiple choices might be probable, but in any instant ...
2
votes
2
answers
147
views
Boltzmann distribution derivation from Lagrange Multipliers
Context
In the derivation of the Boltzmann factor and the canonical partition function based essentially on Lagrange multipliers presented here, the equalities,
\begin{align*}
p_j &= \frac{1}{Z} e^...
- 1,990
2
votes
1
answer
95
views
Why must the propagator exponent be imaginary?
In response to asmaier's question, qmechanic showed why the propagator must be $\exp(cS)$. That made perfect sense. But can it also be shown that $c$ is imaginary? I believe it follows from ...
- 21
0
votes
0
answers
28
views
Fokker-Planck: Uncertainty Propagation
I am interested in propagating the probability density function along the sampled trajectories having parametric uncertainty $a$. However I discovered that the fokker-planck equation for that system ...
1
vote
1
answer
131
views
Is the Boltzmann distribution 'memoryless'? What is the physical interpretation?
The chance of occupying state with energy $E_i$ is given by the Boltzmann distribution:
$$ P(E_i)= \frac{1}{Z} \exp \left( \frac{-E_i}{kT} \right) $$
The exponential distribution is a common ...
- 1,944
5
votes
3
answers
718
views
How does indeterminism lead to deterministic laws?
Philosophers and many scientists seem to distinguish between the macro and micro world a lot. Things in the micro world seem to be indeterministic, atleast through the standard interpretation of QM.
...
- 281
0
votes
1
answer
110
views
Free particle probability to go from $a$ to $b$ [duplicate]
Feynman and Hibbs write that the probability
for a particle to go from $a$ to $b$ is
\begin{equation*}
P(b,a)=|K(b,a)|^2
\end{equation*}
The kernel for a free particle is given as
\begin{equation*}
K(...
- 176
1
vote
1
answer
134
views
Is the Stirling approximation fundamental to the derivation of the Boltzmann distribution?
In the derivations of the Boltzmann distribution I've encountered, the Stirling approximation seems to play a pivotal role. From these derivations, I've gleaned that the Stirling formula might be ...
- 89
4
votes
0
answers
111
views
Is the path integral emergent?
I have recently read a couple of papers on lattice QCD and found that there is a well-established connection between Boltzmann distribution and the path integral in QFT (disclaimer: I am not a huge ...
0
votes
0
answers
30
views
Gaussian approximation integrated over a unit square, written in terms of error function?
So I am new here, and I apologize for my last question being badly formatted with too much extra information making it just painful to read. I am struggling with getting down a 2D Gaussian estimation ...
- 1
0
votes
1
answer
34
views
What is the cross-section impact probability for an extended object moving trough a field of randomly moving particles?
In trying to get an estimate of the probability for an earth orbiting (LEO) satellite to collide with small debris particles.
So I need to understand what is the impact probability for an orbiting ...
- 143
0
votes
1
answer
63
views
Can Quantum Heisenburg Spin Model be regarded as a random walk?
Can Quantum Heisenburg Spin Model be regarded as a random walk ?
We know that the 1D classical Ising Model can be regarded as a 1D random walk. For example, in the simplest case of the 1D Ising model, ...
- 89
4
votes
1
answer
553
views
Total cross-section for Bhabha scattering
The Bhabha scattering differential cross section is given by
$$\frac{d\sigma}{d\Omega}=\frac{\alpha}{2s}\left(\frac{3+\cos^{2}\theta}{1-\cos\theta}\right)^{2}$$
where $\theta$ denotes the angle of the ...
- 1,697
1
vote
2
answers
88
views
What is the interpretation of these hydrogen probability density diagrams?
In the diagram above, what is the interpretation of all of the individual renders? Does the hydrogen atom continuously change between these states? For example, will $(n, l, m_l)$ become $(2, 0, 0)$ ...
- 87
4
votes
3
answers
955
views
Is unitary time evolution the same as obeying the Schrödinger equation?
In this question, the answer says that unitary time evolution means that probability is conserved. Is this the same as saying that a system obeys the Schrödinger equation?
- 1,254
11
votes
2
answers
889
views
Distributions "more singular than a Dirac delta" must have negativity
I am looking at properties of the Glauber P-functions, which are distributions (in the sense of a dirac delta) on the complex plane, normalized so that $\int_{\mathbb{C}} d^2 \alpha P(\alpha) = 1$. On ...
- 187
1
vote
1
answer
62
views
Expectation Value Definition [closed]
I think I might just not be thinking things completely through here but
I've seen the expectation value of a Hermitian operator written in many sources as both $$\langle A \rangle=\frac{\langle\psi|A| ...
- 133
2
votes
3
answers
27
views
Probability distributions of bounded measurement results?
Say that I am measuring a length very inaccurately. For instance, I might measure $1m$ with an uncertainty of $50cm$. If I model the probability distribution as a Gaussian with $\sigma=50cm$, I run ...
- 23
1
vote
1
answer
75
views
Quantum physics: Probability of first state using dirac notation
I am struggling with this question.
Let a two-dimensional Hilbert space H has orthonormal basis vectors |A⟩ and |B⟩. Consider a vector
in H ⊗ H:
$ |v⟩ = α_1 |AA⟩ + α_2 |AB⟩ + α_3 |BA⟩ + α_4 |BB⟩$ .
...
- 13
1
vote
1
answer
81
views
Probability density function (pdf) of a Wiener process
I am working through a book right now in which there is a short introduction to Brownian motion and Wiener processes. I assume it is not treated nearly as rigorous as in mathematics but still more of ...
- 133
1
vote
0
answers
31
views
Probability of measuring velocity
In a one dimensional problem in quantum mechanics, does the probability of measuring momentum greater than $p_0$ correspond to the probability of measuring velocity greater than $v_0= \frac{p_0}{m}$?
- 11
1
vote
1
answer
178
views
Radioactive decay: calculating probability of decay from half life/decay constant
Let's say due to a nuclear reaction a radionuclide of half-life $T_{1/2}$ was created. I am trying to find out what will be the probability of that radionuclide undergoing radioactive decay within ...
- 13
0
votes
2
answers
50
views
Is a qubit just a bit that hasn't been measured or observed?
A qubit is a superposition of a 0 and 1, apparently meaning that as long as the qubit isn't observed it is both 0 and 1, or a probability of being both. If this is the case, why don't regular bits ...
