# 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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### 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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### 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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### 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 ... 3k views

### 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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### 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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### 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 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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### 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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### 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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### 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,... 6k views

### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
1 vote
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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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### 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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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 $$V(x)= \begin{... 2 votes 3 answers 793 views ### Value of momentum of a particle in 1D box prepared in a particular state? What is the value of momentum of particle in 1D box in state \sin(10\pi x/a)? My understanding Standing waves representing particle in 1D box is not an momentum eigenstate so if we measure the ... 0 votes 3 answers 1k views ### Probability and double slit if a beam of identical particles at random distances from each other (or exactly 1/2 lambda between each other) travelling with the same v towards a double sllit do not interfere with each others wave ... 4 votes 1 answer 1k views ### Dirac Delta Function and Position [duplicate] How does one prove that the Dirac Delta distribution is the eigenfunction of the position operator \hat{x}? In math, why does \langle x’|x\rangle = \delta(x’-x)? 27 votes 1 answer 2k views ### How does QFT predict the probability density to find a particle at x? In quantum mechanics, the probability density of a particle's position is$$\rho(x)=|\langle x|\psi\rangle|^2$$What is the corresponding expression in QFT to predict this distribution? Since \rho(x)... 20 votes 7 answers 12k views ### Why was quantum mechanics regarded as a non-deterministic theory? It seems to be a wide impression that quantum mechanics is not deterministic, e.g. the world is quantum-mechanical and not deterministic. I have a basic question about quantum mechanics itself. A ... 20 votes 5 answers 3k views ### Why do coherent states have Poisson number distribution? In quantum mechanics, a coherent state of a quantum harmonic oscillator (QHO) is an eigenstate of the lowering operator. Expanding in the number basis, we find that the number of photons in a ... 2 votes 4 answers 48k views ### Differences between wavefunction, probability and probability density? I am trying to understand the differences between wavefunction, probability and probability density. There are different definitions on the internet. For example: http://inside.mines.edu/~fsarazin/... 14 votes 2 answers 744 views ### Motivation for Wigner phase space distribution Most sources say that Wigner distribution acts like a joint phase-space distribution in quantum mechanics and this is justified by the formula$$\int_{\mathbb{R}^6}w(x,p)a(x,p)dxdp= \langle \psi|\... 718 views

### 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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### 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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### Symmetry transformations on a quantum system; Definitions

We define a symmetry transformation of a system to be any transformation that, when performed, does not change the outcome of a measurement. Wigner's symmetry theorem says that any symmetry of a ...
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### 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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### 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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### 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 ...
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: ...
In some textbooks about quantum mechanics, the position-momentum uncertainty principle is treated as being valid for an individual "particle", with $\Delta x\cdot\Delta p\geq\hbar/2$ ...