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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64
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Why is the application of probability in QM 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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3answers
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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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7answers
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Born rule and unitary evolution

Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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3answers
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Born's Rule, What is the Reason? [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 ...
14
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1answer
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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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6answers
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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 ...
4
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2answers
541 views

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 $|...
6
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7answers
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Is there a direct physical interpretation for the complex wavefunction?

The Schrodinger 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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3answers
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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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5answers
45k views

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 ...
11
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2answers
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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 ...
7
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3answers
388 views

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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4answers
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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 ...
11
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4answers
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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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3answers
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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 ...
18
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1answer
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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}(...
4
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1answer
1k views

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 ...
9
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2answers
768 views

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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7answers
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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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2answers
7k views

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 ...
37
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1answer
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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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3answers
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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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3answers
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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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3answers
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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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1answer
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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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2answers
975 views

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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7answers
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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 ...
10
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2answers
566 views

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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3answers
815 views

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 ...
12
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2answers
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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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3answers
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Normalize wave function with respect to time instead of space

Born's statistical interpretation of the wave function says that $|\Psi (x,t)|^2$ is the probability density of finding the particle at point $x$ at time $t$, then $$\int_{-\infty}^\infty |\Psi (x,t)|...
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2answers
950 views

How to determine the probabilities for a cuboid die?

Imagine we take a cuboid with sides $a, b$ and $c$ and throw it like a usual die. Is there a way to determine the probabilities of the different outcomes $P_{ab}, P_{bc}$ and $P_{ac}$? With $ab$, $bc$...
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3answers
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Relation between unitarity and conservation of probability

In a seminar, I heard that the unitary aspect of representations was important physically, because in quantum mechanics unitarity is closely tied to the conservation of probability. Could someone ...
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2answers
906 views

Interpretation: probability form probability amplitude (free particle)

If you compute the probability amplitude of a free 1D non-relativistic particle with mass $m$, located at position $x_0$ at time $t_0$, for beeing detected at some other point $x_N$ at time $t_N$ you ...
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3answers
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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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5answers
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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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1answer
158 views

Conditional probability between time parameter and operator in quantum mechanics?

Question and Background So I came across a question on conditional probability in quantum mechanics: How is conditional probability handled in quantum mechanics? There's an interesting comment which ...
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2answers
342 views

Are probability-preserving variations of QT with respect to the Born rule mathematically possible?

Is it possible to create (m)any theoretically workable framework(s) - that do(es) produce probabilities - by taking QM and replacing the Born(-like) rule(s) with something that is not equivalent to it ...
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2answers
648 views

Double Slit Experiment/Transition of Classical to Quantum problems in Probability Addition in “An Experiment on bullets”

The First Picture is taken out from the Book The Character of Physical Law By Richard Feynman And the second picture is from his own The Feynman Lectures on Physics. Both figures correspond to the An ...
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1answer
303 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 ...
2
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3answers
17k views

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 + ...
2
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1answer
886 views

Why electron can not be found at some node locations in the infinite potential well? [duplicate]

Consider electron in an infinite potential well, studied in quantum mechanics. Position probability density of the electron is $$ P_n(x)=\left(\frac{2}{L}\right)\sin^2\left(\frac{n\pi x}{L}\right)$$ ...
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4answers
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Probability and the many-worlds interpretation

If I toss a coin, then according to the many worlds interpretation of QM, in half those worlds I'll get a head. If I then toss again, then in a quarter I will have got two heads. And so on. There will ...
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3answers
776 views

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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1answer
316 views

Nonexistence of a Probability for the Klein-Gordon Equation

David Bohm in his wonderful monograph Quantum Theory, in Section 4.6 discusses the difficulties one encounters in trying to develop a relativistic quantum mechanics. He starts from the relation \...
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3answers
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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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3answers
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Is my baby's gender an example of Schrodinger's cat?

At some time in the conception of a child, gender is determined by the X or Y chromosomes given by the male. This is, from my perception, a random selection, like the decaying atom. Like Schrodinger'...
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4answers
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Is quantum physics truly random or does it just appear that way because of Heisenberg uncertainty principle

The behavior of an electron (and other tiny things) is said to be probabilistic because we can't say where an election will be when we measure it, but only where it will probably be. As I understand ...
4
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2answers
678 views

How to calculate the tree-level probability amplitude for the electron-positron to muon-antimuon process?

Consider the following process: $e^+ + e^- \rightarrow \mu^+ + \mu^-$. I'm trying to calculate the probability amplitude of such a process in leading order. In leading order the amplitude is given by:...
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4answers
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Can a photon be absorbed by a proton?

When incident light passes through a hydrogen gas, for example, does it have 50% chance (since it's a 1:1 ratio of protons to electrons) of getting absorbed by the proton? Any chance at all? If no, ...