# Tagged Questions

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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### Do doomsday arguments influence doomsday hypotheses?

The doomsday argument supposes that in the absence of any other knowledge, if we know the age of something now, we may assume that we are seeing it in the middle of its lifetime and then calculate our ...
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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 self-information given by $-\log(p(m))$? Shannon derived a measure of information content called the self-information or "surprisal" of a message $m$: $$I(m) = \log \left( \frac{1}{p(... 1answer 263 views ### Which position and momentum distributions arise from some wave function? Consider a particle in one dimension with wave function \psi(x). The probability density function describing how likely it is to find it in a given position is given by f(x)=\left|\psi(x)\right|^2.... 1answer 57 views ### Probability to be in a particular state If I have a wavefunction \psi = \sum_{n=0}^{\infty} a_n e^{i \phi_n} | n \rangle and (|n \rangle) is a set of orthonormal functions. Is it correct that the probability to be in a state |k\rangle ... 1answer 163 views ### Was Max Born the first to notice a connection between quantum mechanics and randomness? Max Born introduced the Born Rule in a paper from 1926. But was this really the first time that a connection between quantum mechanics and randomness was noticed? Today, quantum mechanics and ... 0answers 47 views ### Books on collision probability and collision processes Are there any books specifically on collision processes between atoms and molecules and collision probability? I would like to get an overview of the factors that determine collision probability ... 0answers 67 views ### probability of sequence for given rate constants lets consider a copolymer, C_{r,s}^A containing r number of A monomers and s number of B monomers with A at the reactive end of the polymer. The equilibrium constant for A-A, A-B, B-A, and B-B bonds ... 3answers 367 views ### Probability and lifetime What is the relation between the lifetime of a particle and the probability of decaying that particle? Here it says that the probability of survival is exponential if the decay process is a Poisson ... 5answers 3k views ### Differences between probability density and expectation value of position The expression \int | \Psi\left(x\right)|^2dx gives the probability of finding a particle at a given position. If wave function gives the probabilities of positions, why do we calculate "... 2answers 819 views ### Differences between wave function and set of orthonormal wave functions? I'm reading a QM book. It first says for wave function: "The state of a physical system (or particle) is completely specified by an entity associated with it called a wave function, Ψ , that in ... 5answers 578 views ### About the definition of expectation value in quantum mechanics In quantum mechanics, the expectation value of a observable A is defined as$$\int\Psi^*\hat A\Psi$$But in probability theory the expectation is a property of a random variable, with respect to a ... 0answers 82 views ### Help with deriving an asymptotic expression Note: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a ... 1answer 57 views ### How many state configurations are possible for N particles in completely different states? How many state configurations are possible for N particles in completely different states? I cannot remember if the total number of state configurations for N particles in completely different ... 5answers 2k views ### Proof of Liouville's theorem: Relation between phase space volume and probability distribution function I understand the proof of Liouville's theorem to the point where we conclude that Hamiltonian flow in phase-space is volume preserving as we flow in the phase space. Meaning the total derivative of ... 4answers 10k 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/... 1answer 51 views ### Why does the probability of obtaining a value of a measurement follow from Dirac's general assumption? In Dirac's The Principle of Quantum Mechanics he makes the general assumption that "if the measurement of the observable \xi for the system in the state corresponding to |x\rangle is made a large ... 0answers 123 views ### Probability and the propagator Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to ... 1answer 145 views ### From where randomness comes from and why it exists? [closed] I recently began to study statistics and probability and I have two questions: Where does randomness come from? What is the source of randomness? Why does the randomness exist? Is it possible to ... 0answers 105 views ### Quantum mechanics and probability I've done an intro course on QM and I'm now hoping to understand exactly how to use probability theory rigorously in solving problems. My question is: How do I do the same thing, or the closest thing ... 2answers 56 views ### Modeling Quantum Aspects with Probability This is a question I've had a while about quantum theory. Many times when I look at books and equations about this subject matter I see that the use many concepts in probability. (Correct me if Im ... 1answer 143 views ### How does determinism manifest out of QFT? Classical electrodynamics is deterministic. QED is indeterministic, or probabilistically random. Yet they agree with each other? What am I missing? 1answer 300 views ### Probability of measuring momentum [closed] Suppose we have this wavefunction:$$ \psi = A \left( cos(kx) + cos (2kx) \right) $$I have to find the possible results of measurement of momentum and their probabilities. Attempt For a momentum ... 0answers 365 views ### Moments of a Distribution via Laplace Transforms and Wick Rotations [closed] On a mathematical level, the statistical mechanical partition function is just a Laplace transform of the microcanonical probability distribution, i.e. it's moment generating function. Understanding ... 7answers 1k views ### Mathematically possible vs physically probable outcomes A good buddy of mine and I have had a friendly debate about the origins of the current state of our universe (namely; Earth and life on Earth) and have fundamentally disagreed in our stances with ... 1answer 136 views ### Change of variables in calculating the integral of multivariable differential entropy I have already asked this question in math.SX but here might be more proper. So I decided to put a copy here and delete the one which is not the one that got an answer: I know that for one ... 2answers 572 views ### Is it really impossible to calculate in advance the result of throwing dice? Is it really impossible to calculate in advance the result of throwing dice? After all, the physics of dice throwing is in the world of classical mechanics, rather than quantum mechanics. 0answers 1k views ### tritium beta decay - probability of being in 1s state Hydrogen-like wavefunctions have the form:$$R_{10} = \left(\frac{Z}{a_0}\right)^{\frac{3}{2}} 2\space e^{-\frac{Zr}{a_0}}Y_{00} = \frac{1}{\sqrt {4\pi}} $$where a_0 = \frac{4\pi \epsilon_0 \... 5answers 793 views ### Is there any operator behind probability, in quantum mechanics? In Quantum mechanics, the probability of finding a particle at position x is given by |\psi(x)|^2, where \psi is the wave function. Wonder what is the operator which gives this probability? Is ... 8answers 3k views ### Why is the application of probability in QM fundamentally different from application of probability in other areas? Why is the application of probability in quantum mechanics (QM) fundamentally different from its application in other areas? QM applies probability according to the same probability axioms as in other ... 0answers 511 views ### Probability density of Klein-Gordon equation This may, perhaps, stir some healthy debate; at least I am having some "fun" thinking about it, hopefully I can solicit some outside views too. It is often regarded that the Klein-Gordon equation ... 0answers 96 views ### Reference for stochastic processes which helps moving from a basic level to a measure theory one I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ... 3answers 284 views ### Intuition/derivation behind the probability current definition The definition is:$${\bf{j}} = \frac{\hbar}{2mi} (\psi^* \nabla \psi - \psi \nabla \psi^*)$$However: Where ever I have looked, the above "pops out of nowhere". I was wondering how can I obtain ... 1answer 253 views ### Reflection Probability for Different Potentials - Quantum Mechanics My question is above. Firstly, I don't actually know whether it is true or not (!). Secondly, if I were to try to prove it, then I have very little idea how to. The potential steps that I have always ... 1answer 130 views ### Deriving probability distributions from the Wigner distribution I know that I can calculate the probability distributions of x and p from the Wigner quasiprobability distribution, and I can calculate the probability distributions of other operators by ... 0answers 159 views ### Quantum Rigid Rotor Perturbation As the title says, I have a rigid rotor with a perturbation given below$$H=\frac{L^2}{2I}-\alpha B L_z.$$So I know that the eigenvalues of H will be \ell(\ell+1)/2I -\alpha B m where m is our ... 2answers 670 views ### Interpreting the Partition Function and Free Energy Mathematically Given that The partition function in statistical mechanics tells us the number of quantum states of a system that are thermally accessible at a given temperature http://vallance.chem.ox.... 6answers 1k views ### Why was quantum mechanics regarded as a non-deterministic theory? It seems to be a wide impression that quantum mechanics is not deterministic, e.g. the world is quantum-mechanical and not deterministic. I have a basic question about quantum mechanics itself. A ... 1answer 498 views ### What do matrices in the Gaussian orthogonal ensemble look like? I've been reading a fair amount about quantum chaos, and random matrix theory comes up a lot. I get that they're looking at the distribution of eigenvalues from an ensemble of random matrices, but I ... 1answer 176 views ### Why don't we need to normalize wavefunction to find probability distribution? Consider an unormalized wavefunction of a rotor at t = 0, a combination of n=0 and n=2 states:$$\psi(\phi) = 3 - 2 \cos (2\phi).$$Find the probability distribution in angle. The ... 3answers 364 views ### Probability distribution in phase space and Liouville's theorem? We can define a probability distribution over phase space (say 1D) \rho(x,p) such that, for example,$$\langle x\rangle = \int x \rho(x,p) dxdp$$etc. It can be shown here that such a distribution ... 0answers 150 views ### Ising model. What is large fluctuations of magnetization? My background is in mathematics. I have studied the Ising model in \mathbb{Z}^2. The main model of statistical mechanics. Yesterday, I was reading the preliminary notes of the book Statistical ... 1answer 168 views ### Is there a known equation for evolution of classical particle probability density? Suppose we have some very imprecise knowledge of classical particle's coordinates and momentum: what we can only tell is the probability density to find it in some point of phase space. This is (... 3answers 1k views ### What is a linear probability density function? In the following question, what is meant by linear probability density function? Is it a uniformly distributed variable or triangularly distributed? Thanks in advance. The kinetic energy of any ... 3answers 742 views ### Why is |\Psi|^2 the probability density? I am starting with Quantum Mechanics, learning online. I can't seem to find the reason for |\Psi|^2 being the probability density of finding an electron. They've just taken it for granted everywhere.... 1answer 4k views ### Kolmogorov-Smirnov test vs Chi-squared test What is the difference between the Kolmogorov-Smirnov test and the Chi-squared test? When should we use one instead of the other? I was reading this article, and I got confused a lot. It is hard to ... 2answers 4k views ### Calculating the most probable radius for an electron of a hydrogen atom in the ground state This link describes a method for determining the most probable radius of an electron for a Hydrogen atom in the ground state. It states that : The radial probability density for the hydrogen ... 2answers 96 views ### Calculating the Probability Current of a Travelling Wave Calculate the probability current density vector \vec{j} for the wave function :$$\psi = Ae^{-i(wt-kx)}.$$From my very poor and beginner's understanding of probability density current it is :$$\...
The free particle solution in stationary state (with definite energy) to the Schrödinger equation is $$\psi(x,t) =Ae^{i(kx-\omega t)} + Be^{-i(kx+\omega t)}$$ Since the energy is definite, and ...