3
votes
3answers
79 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 ...
0
votes
0answers
32 views

Definition of transmission and reflection probability

This is a basic question, but it does not seem to be well defined anywhere. Generally, two terms are mixed somewhat randomly: transmission PROBABILITY and transmission coefficient. So to be clear, ...
0
votes
1answer
36 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 ...
2
votes
1answer
111 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 ...
6
votes
5answers
502 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 ...
2
votes
2answers
94 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 ...
3
votes
5answers
234 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 ...
-1
votes
3answers
199 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: ...
1
vote
1answer
33 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 ...
2
votes
1answer
109 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 ...
0
votes
0answers
62 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 ...
0
votes
2answers
46 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 ...
0
votes
0answers
26 views

Stochastic process corresponding to Schrödinger evolution

In probability theory, the Fokker-Planck equations governs the trajectory of a sample path of a stochastic process -- say the heat equation in the case of a Wiener process. Consider a Gaussian ...
3
votes
1answer
42 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 ...
1
vote
0answers
77 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 ...
2
votes
5answers
647 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 ...
19
votes
7answers
1k views

Why is the application of probability in QM fundamentally different from application of probability in other areas?

Why is application of probability in QM fundamentally different than application of probability in other areas? Quantum mechanics applies probability according to the same probability theory that ...
2
votes
0answers
85 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 ...
1
vote
3answers
140 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 ...
0
votes
0answers
42 views

Measurement of the amplitude

I want to ask how do we actually measure the probability amplitude that appears in Schrödinger equation. From what I read in quantum mechanics textbooks, it appears that after the measurement, the ...
1
vote
1answer
26 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 ...
1
vote
0answers
41 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 ...
1
vote
0answers
51 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 ...
7
votes
2answers
369 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 ...
1
vote
1answer
100 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 ...
0
votes
0answers
34 views

How do I overlap this two-particle symmetric wavefunction?

Suppose we have a symmetric wavefunction that composed of a two-particle system: $$ \psi_s = \frac{1}{\sqrt 2} \left(|u,A\rangle|v,B\rangle + |v,A\rangle|u,B\rangle\right)$$ where $u_{(x)}$ and ...
0
votes
3answers
240 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 ...
2
votes
2answers
447 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 ...
2
votes
2answers
120 views

Free-particle solution to Schrödinger Equation

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 ...
8
votes
2answers
901 views

What is probability current in quantum mechanics?

What is probability current in quantum mechanics? Why define such a thing? I mean the meaning of probability current. I know the formula for it but I just don't get the idea of a flow of probability ...
2
votes
2answers
158 views

Does quantum randomness predicate an infinite number of realities?

I am a layman when it comes to physics and especially quantum mechanics. I have seen many documentaries on the subject, and often in these productions there is a physicist featured explaining the ...
0
votes
1answer
64 views

Regarding derivation of Probability Current

The question for the full derivation of Probability Conservation -> Probability Current was already asked here: Probability current. I apologize for not retyping it out, but it's already beautifully ...
3
votes
1answer
107 views

Why do we use $\psi$ instead of a straightforward probability?

What is the advantage/purpose of using $\psi$ for wavefunctions and getting the probability with $|\psi|^2$ as opposed to just defining and using the probability function?
2
votes
2answers
136 views

Probability density of electron orbital

Why the probability of density is higher in the area that is closer to the nucleus? I'm a high school student. I don't know much about wave functions.
0
votes
0answers
59 views

Notation for Propability Amplitudes

I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude ...
2
votes
1answer
110 views

Three dimensional wave packets in momentum space

I am given the 3D wave packet: $$\psi(x,y,z)=N\,\exp\left(\frac{-(x^2+y^2+2z^2)}{2a^2}\right).$$ I was asked to find N (easy enough). Then I was asked the probability that we measure $z$ greater than ...
1
vote
4answers
210 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 ...
1
vote
0answers
119 views

Basic introductory quantum mechanics question [closed]

Given that $\psi = \frac{1}{\sqrt{32 \pi a_{0}^{3}}}(2-\frac{r}{a_0})exp(\frac{-r}{2a_0})$ is a wavefunction of the hydrogen atom, write down the probability density for r and calculate the ratio ...
1
vote
1answer
138 views

Expectation values in QFT?

What is the meaning of different expectation values in QFT? For instance: $$\langle 0|{\cal O}(0)|q,s\rangle$$ or $$\langle 0|{\cal O}(0)|0\rangle$$ with ${\cal O}$ being some operator and ...
6
votes
2answers
616 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 ...
2
votes
1answer
112 views

What aspect of quantum mechanics forces probabilities to be (conventionally, at least) central?

I understand how to compute probability distributions and expected values and such from quantum states, but a lot of treatments of QM make it look like this is what the wavefunction is essentially ...
2
votes
3answers
214 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 ...
6
votes
2answers
248 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 ...
1
vote
1answer
293 views

Spinors and Probabilities of Electron-Positron Pair

Question: An electron and positron are moving in opposite directions, and are in the spin singlet state. Two Stern-Gerlach machines are orientated in some ...
0
votes
1answer
195 views

Probability in Quantum Mechanics: General

How do I find the most probable value of position of a (non-Gaussian) wave function? Is it the same value as the expectation value of the position? And is it true that the most probable value of ...
3
votes
2answers
288 views

Formulation and probability of a wave-function [closed]

I have got this problem where I have been given the following wave function: $$\Psi = 0\quad\text{if}~|x| > a\quad\text{and}\quad A(a^2-x^2)\quad \text{if} \quad |x|< a$$ Now the first question ...
7
votes
1answer
213 views

Quantum version of the Galton Board

If classical particles fall through a Galton Board they pile up in the limit of large numbers like a normal distribution, see e.g. http://mathworld.wolfram.com/GaltonBoard.html What kind of ...
2
votes
1answer
234 views

Probability current vs. direction of wave function

I did an exercise for my Quantum-Mechanics Lecture: Let $\hbar$=2m=1. A particle in 1 dimension has $j(x)=2\ Im(\overline{\psi} (x) \ \psi'(x))$ and it's to show that there are superpositions $\psi ...
1
vote
1answer
239 views

Confusion about the probability cloud

What is the meaning of the electron probability cloud? I understood it to mean that the electron has a probability to be found in a certain postion before measurement, but now after reading ...
2
votes
1answer
384 views

Why is classical physics not valid for a harmonic oscillator in its lowest energy state? [closed]

I am reading Born's interpretation of wave function in quantum physics by Eisberg & Resnick and I am not able to understand this description about comparison between the classical and quantum ...