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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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Quantum Mechanical Current Normalisation

Consider an electron leaving a metal. The quantum-mechanical current operator, is given (Landau and Lifshitz, 1974) by $$ j_x\left[\psi_{\mathrm{f}}\right]=\frac{\hbar i}{2 m}\left(\psi_{\mathrm{f}} \...
Tomi's user avatar
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What is the physical meaning of the normalization of the propagator in quantum mechanics?

Suppose we have a quantum field theory (QFT) for a scalar field $\phi$ with vacuum state $|\Omega\rangle$. Then, in units where $\hbar = 1$, we postulate that the vacuum expectation value (VEV) of any ...
zeroknowledgeprover's user avatar
4 votes
2 answers
917 views

What Does Feynman Mean When He Says Amplitude and Probabilities?

In Feynman lectures on gravitation section 1.4, he tries to debate over whether one should quantize the gravitation or not. He provides a two-slit diffraction experiment with a gravity detector, which ...
Ting-Kai Hsu's user avatar
1 vote
4 answers
115 views

Quantum: Which improbable macroscopic events are possible?

Basically, the title. Web search had not found pages in top results with similar QA. E.g. I understand nuclear blast can just end at any time because random chain-reaction has probability of not ...
Martian2020's user avatar
-3 votes
1 answer
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Probabilistic behavior of quantum mechanics [closed]

In a hypothetical scenario, if I were to measure the quantum spin of an electron and it showed "up," and then I traveled back in time without changing the initial conditions, would measuring ...
Vishnu's user avatar
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3 answers
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Is the zero vector necessary to do quantum mechanics?

Textbook quantum mechanics describes systems as Hilbert spaces $\mathcal{H}$, states as unit vectors $\psi \in \mathcal{H}$, and observables as operators $O: \mathcal{H} \to \mathcal{H}$. Ultimately, ...
Silly Goose's user avatar
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5 votes
2 answers
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Physical meaning of each term of the square modulus of a wave function

The expression below is the square modulus of the wave function of a harmonic potential ($V=\frac{1}{2}m\omega^2 x^2$) in which it's stated that the probability of finding the particle in the $\psi_0$ ...
zzzzzzzzz's user avatar
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1 answer
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Factorization on increments in Markov chain

I'm trying to show the following property for a Markov chain: $$\left<[x(t+\tau)-x(t)][x(t'+\tau)-x(t')]\right> =\left<x(t+\tau)-x(t)\right>\left<x(t'+\tau)-x(t')\right> $$ Where $t\...
SSh2402's user avatar
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1 vote
3 answers
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Boltzmann distributions on atomic orbitals: infinite degeneracy?

The (unnormalized) Boltzmann probability distribution of states as a function of energy and temperature is given by $$P(\epsilon_i) \propto g_i\exp\left(\frac{-\epsilon_i}{k_BT}\right)$$ with $P(\...
ChangedMyName's user avatar
0 votes
2 answers
109 views

What is the connection between moments in probability theory and the moment of inertia?

My question arises as the moment of inertia (MOI) has been described as a second moment. In my understanding if the MOI is indeed a second moment of a distribution of mass, this suggests the MOI could ...
Luke K's user avatar
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Fermi's Golden Rule: Interpreting the Dirac Delta in Transition Probabilities [duplicate]

I am trying to understand an aspect of Fermi's golden rule in the case of a constant perturbation, $V$. The formula for the transition probability from an initial state $i$ to a final state $f$ is ...
SimoBartz's user avatar
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8 votes
1 answer
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Is there a name for the number of '9's in numbers such as 0.999 (where it would be 3)?

I am doing an optics simulation involving transmission and reflection coefficients very close to 1, such as 0.999. While I was an undergraduate student, a professor mentioned that, in certain fields, ...
jcuk's user avatar
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0 answers
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How to deal with the divergence in tree-level diagrams? where the propagator momentum is on-shell

Only consider the interaction term between electron Higgs and $Z$ boson $$ \mathcal{L}_{h ff}=-\frac{Y_f v}{\sqrt{2}} \bar{\psi} \psi\left(1+\frac{h}{v}\right) =-m_f \bar{\psi} \psi\left(1+\frac{h}{...
MW L's user avatar
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10 votes
6 answers
3k views

How are quantum systems different from dice?

I've had this question for a while: Is a state space $\mathcal{H}$ for a quantum system just a sample space in a probability space? The question arises because i can't really tell a difference between ...
Simón Flavio Ibañez's user avatar
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1 answer
66 views

Equation for probability of quantum tunneling

I am looking at fusion reactions in stars and came across how particles will bypass the Coulomb barrier through quantum tunneling. I was wondering if there is an equation for the probability of a ...
Waev's user avatar
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2 answers
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Questioning the Probability Expression for Neutrino Oscillation in Griffiths' "Introduction to Elementary Particles"

In Griffiths' book, Introduction to Elementary Particles (Griffiths, D. (2020). John Wiley & Sons, p. 390), the author defines the pure electron and muon neutrino states as: $$|ν_{e}\rangle=-\sinθ|...
Okba's user avatar
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0 answers
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How are uncertainties of the quantum state/wavefunctions themselves modeled?

This question might be confusing so let me try to clarify this carefully. The wavefunction is a tool that allows us to calculate probability distributions that model uncertainties. Thus makes sense. ...
Maximal Ideal's user avatar
1 vote
3 answers
111 views

Why a probability distribution in RHS in deriving Bell's Inequality?

Why is there typically an integral over a probability distribution in the RHS of a derivation of Bell's inequality $$|P(\boldsymbol{a}, \boldsymbol{b}) - P(\boldsymbol{a}, \boldsymbol{c})| \leq \int{p(...
No infinity's user avatar
3 votes
1 answer
75 views

Why the transition probability in the master equation approach just the rate$*dt$ for a simple birth process?

I am modeling a process of an exponentially growing population of cells as $\frac{dn}{dt}=\lambda n$. To account for the intrinsic noise in the birth process of these cells, I write down the ...
M. Z.'s user avatar
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0 answers
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The role of probability in the many-worlds interpretation [duplicate]

A quantum system can transition to one of two states, with probabilities 30% and 70%. The many worlds interpretation says that the universe splits into two, one for each state. If so, what do the 30% ...
Maurice Mizrahi's user avatar
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1 answer
51 views

How is the inner product of two quantum states related to their associated Bloch vectors?

I have a doubt about how two equivalent ways of calculating the inner product between two states seem to not be actually equivalent, as they should. In particular, I'm interested in the case where the ...
Sebastián Torres's user avatar
2 votes
1 answer
41 views

Fermi-Dirac Distribution for Multiple Species

If I have a system containing two types of fermions, what is the probability of a state of energy $E$ being occupied? Is it just the sum of two standard Fermi probabilities for each type of fermion?
S.T. Zweig's user avatar
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1 answer
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A question about time evolution of position distributions

If I have two probability distributions $P$ at $t$ and $P’$ at $t’$ separated by some time interval. Then, can I describe the transform between the two distributions as $$P’(x) = \int P(a) D(a, x-a, t’...
Adam Kabbeke's user avatar
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1 answer
27 views

How to go from probability distribution to transitions probability distribution?

For the past few days I have been studying Advanced statistical mechanics. I am studying a Wiener process in general. Such a process is a non-stationaty time-independent Gaussian process. The ...
luki luk's user avatar
-4 votes
3 answers
176 views

How would one calculate the actual probability of a macro event given what we know of physics?

Suppose I wanted to calculate the “true” probability of me tossing a coin tomorrow and it landing on heads. Now, even though we often say that this is 50%, correct me if I’m wrong, but this can’t be ...
user avatar
0 votes
1 answer
81 views

Statistical independence of $x,y,z$ dimensions for Maxwell velocity distribution function

I have been looking into the derivation of the Maxwell speed distribution function as for instance given in https://en.wikipedia.org/wiki/Maxwell%E2%80%93Boltzmann_distribution#...
Thomas's user avatar
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-2 votes
1 answer
145 views

Why do the Schrödinger and Dirac equations contain the mass?

I know the Schrödinger equation is bascially the "quantized" Hamiltonian formalism from classical mechanics, and the Dirac equation is the special-relativistic version. But these equations ...
ldfjglfkgj's user avatar
0 votes
1 answer
50 views

Where is my mistake in using a measurement operator instead of Born’s rule to calculate the probability of detecting photons at an arbitrary angle?

As I asked in this question: https://quantumcomputing.stackexchange.com/questions/36998/how-can-i-calculate-the-measuring-probabilities-of-a-two-qubit-state-along-a-cer/37000#37000 From here I know ...
Alex1111's user avatar
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How to properly discretize and solve the Liouville equation?

Consider some dynamical system $\dot{\textbf{X}}(\textbf{x},t)=F(\textbf{X})$ where $\textbf{X}$ is discretized along a 1-dimensional spatial coordinate $\textbf{x}=(x_1,\dots,x_N)^T$. Let $\rho(\...
thespaceman's user avatar
1 vote
1 answer
74 views

Infinite potential well suddenly expanding

Problem statement: an electron is in its fundamental state in an infinite (1-dimensional) potential well, its walls being located at $x=0$ and $x=a$. Suddenly, the right wall moves from $x=a$ to $x=2a$...
Lagrangiano's user avatar
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0 votes
1 answer
80 views

Momentum probability density and its normalization

Let the (normalized) wave function $\Psi(x,y)$ represent a free particle in the XY plane. I know $|\Psi|^2$ gives me the probability density function of the particle's position, which I can then ...
Lagrangiano's user avatar
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1 vote
1 answer
76 views

Understanding $\mathrm dP_x$ in the derivation of Maxwell-Boltzmann distribution

In my physical chemistry book, it says: In the study of molecular speeds, we must consider a range of speeds. If we don’t, the probability would be zero. This probability is proportional to the range ...
Kintoke 's user avatar
1 vote
3 answers
112 views

Relation between classical probability and quantum probability formulae

Assuming superposition state $$ \Psi = C_1 \psi_1 + C_2 \psi_2 $$ ,one can write the expectation value $\langle A \rangle$ of a physical magnitude A as follows $$ \langle A\rangle = P_1 \langle A\...
Takopako's user avatar
  • 117
4 votes
0 answers
112 views

Uncertainty principle for incompatible observables whose probability distributions lack well-defined moments

The Heisenberg uncertainty principle states that the product of standard deviations (or variances) for incompatible observables has a non-zero lower bound (with a zero lower bound reserved for ...
Omid's user avatar
  • 342
-3 votes
1 answer
78 views

Does the inner product of wavefunctions really have units? [closed]

Let $\psi(x)$ and $\phi(x)$ be wavefunctions. I usually see the inner product defined as $$\int dx\, \overline{\psi(x)} \, \phi(x)$$ and interpreted, I think, as "the amplitude that state $\phi$ ...
Upasker's user avatar
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0 votes
0 answers
90 views

Axiomatic Treatment of Quantum Probability Theory

Define quantum probability theory to be an axiomatic mathematical theory which appropriately generalizes classical (Kolmogorov) probability theory to provide the precise probabilistic framework ...
Silly Goose's user avatar
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2 votes
1 answer
115 views

Measurement of electrons positions in an orbital, thought experiment

An orbital can hold upto 2 electrons. Let's take 1s orbital of helium. Now, we use probability density to depict where we can find the electrons in the orbital if we make measurements. Since two ...
LuffyYadav's user avatar
0 votes
2 answers
253 views

What particles are described by the Klein-Gordon Equation?

The Klein-Gordon equation $$\left(\frac{\partial ^2}{\partial t^2} - |\nabla|^2 + m^2\right)\phi = 0\tag{1}$$ should describe non interacting particles without spin. So what particles in the standard ...
Noumeno's user avatar
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1 vote
1 answer
110 views

Interpretation of a probability that does not normalize to one in stat mech?

I am trying to understand the meaning of the "n-particle distribution function" as defined by the three references below([1][2][3]), primarily those by Claudio Zannoni. Setup: For a system ...
McKinley's user avatar
0 votes
2 answers
93 views

What is the relationship between an electron's wave-like and particle-like qualities? Is "Electrons are waves and particles" the whole truth? [duplicate]

Upon researching the double-slit experiment, it seems to me that electrons are somehow cloaked in wavelike behavior (not at all like my previous idea that electrons were waves and somehow were also ...
Ruchir Kavulli's user avatar
0 votes
1 answer
58 views

Quantum Chemistry Help FWHM [closed]

I am struggling with this question and cannot find any resources to help me. Even the tutors at my school don't know how to help me and my professor just tells me to reference the slides, but I have ...
Lorena Alvarado's user avatar
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0 answers
56 views

Continuum bases: why do we use dirac delta function? [duplicate]

In discrete bais, we can express a vector as $$ |\psi\rangle=\sum_{i} c_i|e_i\rangle $$ with orthonormality $$ \langle e_i|e_j\rangle=\delta_{ij}.$$ $\delta_{ij}$ is usual kronecker delta. If we ...
B. Silasan's user avatar
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0 answers
46 views

Definition of (Differential) Scattering Cross Section in QED

In QED we like to define the (differential) cross section for a scattering process as follows: $$d\sigma \ \dot= \ \frac{w_{fi}dN_f}{|j_{inc}|}\tag{1}$$ where $w_fi$ is the probability of transition ...
Noumeno's user avatar
  • 4,577
1 vote
2 answers
189 views

Derivation of mean life of a radioactive nucleus

How can the mean life of a radioactive nucleus be derived? Consider R.dt number of nuclei decaying in the time interval t and t+dt. Then, isn't the lifetime of those R.dt number of nuclei is t? But, I ...
Vinay5101's user avatar
0 votes
0 answers
72 views

Why is psi square a possibility? [duplicate]

Is psi square just an assumption? Or there is a physical reason why they defined like that? My procedure is: It is intuitive for me to think possibility is proportional to energy distribution. ...
user avatar
8 votes
2 answers
1k views

Is energy only conserved statistically in quantum mechanics?

I know that a system's energy can be measured with an energy that can be below or above the expectation value, if the system was not in an energy eigenstate, so that energy is only conserved on ...
FACald's user avatar
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0 votes
0 answers
87 views

Can the probability of finding a particle in a certain finite region be zero?

Don't worry this time isn't about doubleslit but I'll still use it for my question. Imagine an electron is emitted from the source and I shall allow a certain amount of time to lapsed so as to provide ...
user6760's user avatar
  • 13k
2 votes
0 answers
95 views

Do we have "amplitudes over configurations over spacetime" for QFT in terms of path integral?

Suppose we work in 1+1 spacetime and consider only a scalar field. In canonical quantization of QFT, a state is a density on "configuration on a time slice" (Let's forget the fact that there ...
Peter Wu's user avatar
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0 votes
1 answer
39 views

Why does computing mean free path from an air molecule's reference frame seem to greatly overestimate path length?

I was interested in seeing if I could derive the mean free path of an "air molecule" by considering the reference frame of an individual molecule as other particles moved around it randomly. ...
trevR's user avatar
  • 1
1 vote
1 answer
127 views

Calculating the average kinetic energy (expectation value) of gas particles from the Maxwell Boltzmann distribution

From what I already know, to calculate the expectation value/average from a probability distribution, you use the formula: $$ \langle x \rangle \ = \int_{-\infty}^{\infty} x f(x) \,\mathrm{d}x \tag{1}$...
user374355's user avatar

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