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 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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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 ...
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4answers
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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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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 ...
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3answers
480 views

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 ...
2
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3answers
631 views

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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6answers
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Probability amplitude in Layman's Terms

I am basically a Computer Programmer, but Physics has always fascinated and often baffled me. I have tried to understand probability density in Quantum Mechanics for many many years. What I ...
6
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4answers
677 views

What is the difference between + and - signs in superpositions of quantum states?

What is the difference between states $$ \frac1{\sqrt{2}} |11\rangle+\frac1{\sqrt{2}} |00\rangle $$ and $$ \frac1{\sqrt{2}} |11\rangle- \frac1{\sqrt{2}} |00\rangle~? $$ They will all eventually ...
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Would one actually find their doppelgänger in a “Googolplex Universe”?

Related: Infinite universe - Jumping to pointless conclusions I've recently become a fan of Numberphile, and today I happened to watch their video regarding Googol and Googolplex. In the video, ...
7
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1answer
236 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 ...
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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 ...
6
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3answers
765 views

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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3answers
653 views

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 ...
4
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3answers
795 views

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 ...
3
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3answers
608 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)= ...
3
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4answers
656 views

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 ...
3
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2answers
279 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 ...
3
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1answer
691 views

Transmission and reflection

What is the transmission amplitude of a wavefunction $\phi(x)=e^{ikx}(\tanh x -ik)$? I would have thought that it is $\tanh x -ik$ since this is the factor associated with the forward travelling ...
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1answer
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Can binary sequences generated from ergodic maps be chaotic?

Briefly, the way symbols are generated is: Consider a one-dimensional chaotic map $T: [0,1]→[0,1]$ and a time series $\{x_n\}_{n=1}^N$ generated with this map. Define a threshold $A$ and a ...
0
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2answers
112 views

Probability from classical physics compared with quantum mechanics [closed]

$|\psi\rangle$=$\frac{1}{\sqrt5}|\uparrow_z\rangle+\frac{2}{\sqrt5}|\downarrow_z\rangle$ a)What is the probability of obtaining $+\frac{\hbar}{2}$ for $S_x$? b)If after obtaining ...
0
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3answers
715 views

Why is $|\Psi|^2$ the probability density?

I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for $|\Psi|^2$ being the probability density of finding an electron. They've just taken it for granted ...
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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 ...
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6answers
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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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4answers
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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 ...
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3answers
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Determinism, classical probabilities, and/or quantum mechanics?

[I]f you want a universe with certain very generic properties, you seem forced to one of three choices: (1) determinism, (2) classical probabilities, or (3) quantum mechanics. [My emphasis.] ...
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2answers
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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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2answers
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Why is quantum mechanics based on probability theory? [duplicate]

What makes us formulate quantum mechanics based on probability theory? Isn't the real quantum world based on unknown laws to us? Is it possible that results of an experiment will be measurable in ...
3
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0answers
362 views

Moments of a Distribution via Laplace Transforms and Wick Rotations [closed]

On a mathematical level, the statistical mechanical partition function is just a Laplace transform of the microcanonical probability distribution, i.e. it's moment generating function. Understanding ...
9
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1answer
549 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 $ab$, ...
6
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3answers
589 views

Could quantum mechanics work without the Born rule?

Slightly inspired by this question about the historical origins of the Born rule, I wondered whether quantum mechanics could still work without the Born rule. I realize it's one of the most ...
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1answer
882 views

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 ...
0
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5answers
489 views

Why do we need a wave function?

Assuming our only aim is to solve double slit experiment (or other problems that can be mapped into that). Knowing that electron does some strange thing not expected of a particle, we need a function ...
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2answers
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Bell polytopes with nontrivial symmetries

Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, v_i}$, possibly in a nondeterministic manner. We are interested in joint ...
7
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1answer
155 views

Is there a mathematical basis for Born rule?

Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the ...
7
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2answers
435 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 ...
4
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3answers
222 views

How do probabilities emerge in the many-worlds interpretation?

My understanding is that at each quantized unit of time that a split occurs, every possible recombination of particles occurs in the 'objective' universe. If this is the case, what relevance to ...
2
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3answers
360 views

Probability distribution in phase space and Liouville's theorem?

We can define a probability distribution over phase space (say 1D) $\rho(x,p)$ such that, for example, $$\langle x\rangle = \int x \rho(x,p) dxdp$$ etc. It can be shown here that such a distribution ...
2
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3answers
413 views

How to show that probability and statistics are very important in Quantum Mechanics?

I'm doing a research for my stats class in high school and I chose quantum mechanics as my subject. I narrowed down to electron localization in an atom and radial probability distribution. However, I ...
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0answers
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Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
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2answers
141 views

If the universe is spatially infinite, and if something (A) is possible, does that mean the thing (A) happens?

Assuming that the universe is spatially infinite, extending outwards in all directions without end, and is consistent with what we have thus far observed, and predict to find (and predict to be out ...
12
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1answer
554 views

The measure problem in the anthropic principle

The anthropic principle is based upon Bayesian reasoning applied to the ensemble of universes, or parts thereof, conditioned upon the existence of conscious observers. That still leaves us with the ...
9
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3answers
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What is the relationship between Maxwell–Boltzmann statistics and the grand canonical ensemble?

In the grand canonical ensemble one derives the expectation value $\langle \hat n_r\rangle^{\pm}$ for fermions and bosons of sort $r$: $$ \langle \hat n_r\rangle^{\pm} \ \propto \ ...
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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 ...
3
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2answers
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How do I figure out the probability of finding a particle between two barriers?

Given a delta function $\alpha\delta(x+a)$ and an infinite energy potential barrier at $[0,\infty)$, calculate the scattered state, calculate the probability of reflection as a function of ...
2
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0answers
328 views

Probability and probability amplitude [duplicate]

What made scientists believe that we should calculate probability $P$ as the $P = \left|\psi\right|^2$ in quantum mechanics? Was it the double slit experiment? How? Is there anywhere in the ...
2
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1answer
194 views

Can someone explain probability flux in the tunneling boundary condition of Vilenkin?

This is what's leading to the notion of a quantum universe tunneling from nothing into existence, right? The idea is that probability flux flows out of superspace (configuration space) at ...
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1answer
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Probability current

Conservation of probability: Suppose a wavefunction has ${\partial \mathbb P \over \partial t} = -t f(x,t)$ and ${\partial j \over \partial x} = i f(x,t)$. How does it follow that ${\partial \mathbb P ...
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0answers
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What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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3answers
283 views

Intuition/derivation behind the probability current definition

The definition is: $${\bf{j}} = \frac{\hbar}{2mi} (\psi^* \nabla \psi - \psi \nabla \psi^*)$$ However: Where ever I have looked, the above "pops out of nowhere". I was wondering how can I obtain ...