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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Why is the application of probability in Quantum Mechanics fundamentally different from application of probability in other areas?
Why is the application of probability in Quantum Mechanics (QM) fundamentally different from its application in other areas? QM applies probability according to the same probability axioms as in other ...
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Normalization of basis vectors with a continuous index?
I have an infinite basis which associates with each point, $x$, on the $x$-axis, a basis vector $|x\rangle$ such that the matrix of $|x\rangle$ is full of zeroes and a one by the $x^{\mathrm{th}}$ ...
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Is the Born rule a fundamental postulate of quantum mechanics?
Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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Normalizing Propagators (Path Integrals)
In the context of quantum mechanics via path integrals the normalization of the propagator as
$$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$
is incorrect. But why?
It gives the correct pre-...
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Where does the Born rule come from? [duplicate]
As far as I've read online, there isn't a good explanation for the Born Rule. Is this the case? Why does taking the square of the wave function give you the Probability? Naturally it removes negatives ...
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Normalization of the path integral
When one defines the path integral propagator, there is the need to normalize the propagator (since it would give you a probability density). There are two formulas which are used.
1) Original (v1+v2)...
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Is there a direct physical interpretation for the complex wavefunction?
The Schrödinger equation in non-relativistic quantum mechanics yields the time-evolution of the so-called wavefunction corresponding to the system concerned under the action of the associated ...
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Probability, quantum physics, and why (can't it/does it) apply to macroscale events?
Quantum physics dictates that there are probabilities that determine the outcome of an event, ie: the probability of a quark passing through a wall is X, due to the size of the quark in comparison to ...
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Probability amplitude in Layman's Terms
What I understood is that probability amplitude is the square root of the probability of finding an electron around a nucleus, but the square root of the probability does not mean anything in the ...
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Is there actually a 0 probability of finding an electron in an orbital node?
I have recently read that an orbital node in an atom is a region where there is a 0 chance of finding an electron.
However, I have also read that there is an above 0 chance of finding an electron ...
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Why, in spin sums, we sum over final spin states and average over initial states?
I am reading Halzen's book about quarks and leptons and on page 120 he talks about spin sums.
He says that in order to calculate the amplitude between unpolarized states we have to sum over FINAL ...
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Why is $\langle x| x' \rangle=\delta(x-x')$? [duplicate]
I've tried to find any solution or proof for $$\langle x| x' \rangle=\delta(x-x'),$$ but I only came to this post: Wave function and Dirac bra-ket notation
So I got the information, that the vector $|...
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How close does a particle-antiparticle pair need to be for annihilation to happen?
I've most often seen the statement that the annihilation of a particle and its antiparticle occurs when they 'collide' with one another. So in other words when they get very close to one another right?...
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Why can't the Uncertainty Principle be broken for individual measurements if it is a statistical law?
The Heisenberg Uncertainty Principle is derived for two operators $\hat A$ and $\hat B$ as
$$\Delta \hat A\ \Delta \hat B \geq \dfrac{1}{2}|\langle[\hat A, \hat B] \rangle|$$
where $\Delta$ denotes ...
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How do we choose the standard probability current?
In quantum mechanics, the probability current is defined as
$$\mathbf{J} \propto \text{Im}(\psi^* \nabla \psi)$$
and satisfies the continuity equation
$$\nabla \cdot \mathbf{J} = - \frac{\partial \...
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What is a wave function in simple language?
In my textbook it is given that
'The wave function describes the position and state of the electron and its square gives the probability density of electrons.'
Can someone give me a very simple ...
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What is the fundamental probabilistic interpretation of Quantum Fields?
In quantum mechanics, particles are described by wave functions, which describe probability amplitudes. In quantum field theory, particles are described by excitations of quantum fields. What is the ...
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Can 1 kilogram of radioactive material with half life of 5 years just decay in the next minute?
I wondered this since my teacher told us about half life of radioactive materials back in school. It seems intuitive to me to think this way, but I wonder if there's a deeper explanation which proves ...
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How is conditional probability handled in quantum mechanics?
In ordinary probability theory the conditional probability/likelihood is defined in terms of the joint and marginal likelihoods. Specifically, if the joint probability of two variables is $\mathcal{L}(...
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How can I intuitively understand the Boltzmann factor?
It is known that for a system at thermal equilibrium described by the canonical ensemble, the probability of being in a state of energy $E$ at temperature $T$ is given by the Boltzmann distribution:
$$...
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Particle in a 1-D box and the correspondence principle
Consider the particle in a 1-d box, we know very well the solutions of it. I'd like to see how the correspondence principle will work out in this case, if we consider position probability density ...
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Is it really impossible to calculate in advance the result of throwing dice?
Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics.
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Infinite universe - Jumping to pointless conclusions
I watched an episode of thee BBC Horizon series titled 'To infinity and beyond'.
In this program a number of respected physicists and mathematicians were talking about the nature of infinity and an ...
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Can a particle pass through a nodal point where its wave function is zero?
Let's consider an infinite square well. In the first exited state there is a node at the middle of the well (i.e. wave function and thus probability of finding the particle is zero there).
If I ...
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How does QFT predict the probability density to find a particle at x?
In quantum mechanics, the probability density of a particle's position is $$\rho(x)=|\langle x|\psi\rangle|^2$$
What is the corresponding expression in QFT to predict this distribution? Since $\rho(x)...
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Why is a Hermitian operator a "quantum random variable"?
To me, as a stupid mathematician, a random variable is a measurable function from some probability space $(\Omega, \sigma, \mu)$ to $(\Bbb{R}, B(\Bbb{R}))$. This makes sense. You have outcomes, events,...
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Derive Poisson distribution from probability per time of event
Suppose we have a probability per time $\lambda$ that something (e.g. nuclear decay, random walk takes a step, etc.) happens.
It is a known result that the probability that $n$ events happen in a time ...
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Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?
Is there a clear and intuitive meaning to the eigenvectors and eigenvalues of a density matrix?
Does a density matrix always have a a basis of eigenvectors?
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Can't the Negative Probabilities of Klein-Gordon Equation be Avoided?
I came across these notes of Dyson on Relativistic Quantum Mechanics. There on p. 3, he mentions that the issue with the Klein-Gordon equation is that the only way to relate $\psi$ with a probability ...
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What is the unit (dimension) of the 3-dimensional position space wavefunction $\Psi$ of an electron?
I googled for the above question, and I got the answer to be $$[\Psi]~=~L^{-\frac{3}{2}}.$$
Can anyone give an easy explanation for this?
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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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Form of Schrödinger equation for the probability density
Is it possible to formulate the Schrödinger equation (SE) in terms of a differential equation involving only the probability density instead of the wave function? If not, why not?
We can take the ...
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When are all microstates equally probable?
So I am taking my introductory statistical mechanics course, and this concept is something that I can not wrap my head around. My professor said that all microstates are equally probable and this is ...
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Is it true that quantum mechanics technically allows anything to happen?
Maybe this is a silly question (I think it is), but it's a question I'm arguing with some of my friends for a long time.
The ultimate question is: Is everything (in our Universe) possible ?
I've ...
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Are negativity of the Wigner function and quantum behaviour equivalent?
I've read the following question: Negative probabilities in quantum physics
and I'm not sure I understand all the details about my actual question. I think mine is more direct.
It is known that the ...
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Fermi's golden rule and infinite probablity?
I am slightly confused about the application of Fermi's golden rule. Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of:
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Probability and probability amplitude
The equation:
$$P = |A|^2$$
appears in many books and lectures, where $P$ is a "probability" and $A$ is an "amplitude" or "probability amplitude". What led physicists to believe that the square of ...
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Calculating the probability of a given energy
Given a normalised wavefunction say $$\psi(x) = A\sin(n\pi x),$$ (where $A$ is a normalisation constant) I can calculate the probability of finding the particle being between a position $x$ and $x + ...
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Dirac Delta Function and Position [duplicate]
How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator $\hat{x}$? In math, why does $\langle x’|x\rangle = \delta(x’-x)$?
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An example of a quantum system for which Wigner function transitions to negative values
I want to check my understanding of the Wigner transform and try to understand why and how exactly the probabilistic interpretation drops down as the function goes to zero and then to negative values
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Physical intepretation of nodes in quantum mechanics
I am taking my second course in QM, and my head is starting to spin as it probably should.
But I would very much like to clear up my head about a few details regarding the wave function. As I know it ...
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Wavefunction, probability and impossible events
A friend of mine asked me a question, which I considered trivial at first, but after a while gave rise to some doubts.
For instance, we have a potential well in 1 dimension defined by
$$
V(x)=
\begin{...
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Value of momentum of a particle in 1D box prepared in a particular state?
What is the value of momentum of particle in 1D box in state $\sin(10\pi x/a)$?
My understanding
Standing waves representing particle in 1D box is not an momentum eigenstate so if we measure the ...
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Determine the state $|\psi \rangle$
The question is:
The angular momentum components of an atom prepared in the state $|\psi\rangle$ are measured and the following experimental probabilities are obtained:
\begin{equation} P(+\hat{z}) = ...
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Proof of normalization constant of wave function to be independent of time
I am trying to prove that the normalization constant is independent of time. If we have fixed it for a particular time then it will remain constant for all time.
Suppose $\psi(x,t)$ is a wavefunction.
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Probability and double slit
if a beam of identical particles at random distances from each other (or exactly 1/2 lambda between each other) travelling with the same v towards a double sllit do not interfere with each others wave ...
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Why do coherent states have Poisson number distribution?
In quantum mechanics, a coherent state of a quantum harmonic oscillator (QHO) is an eigenstate of the lowering operator. Expanding in the number basis, we find that the number of photons in a ...
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Why was quantum mechanics regarded as a non-deterministic theory?
It seems to be a wide impression that quantum mechanics is not deterministic, e.g. the world is quantum-mechanical and not deterministic.
I have a basic question about quantum mechanics itself. A ...
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Nonexistence of a Probability for Real Wave Equations
David Bohm in Section (4.5) of his wonderful monograph Quantum Theory gives an argument to show that in order to build a physically meaningful theory of quantum phenomena, the wave function $\psi$ ...
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Motivation for Wigner phase space distribution
Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula
$$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\...
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