1. ### I Normalization and the probability amplitude

Given two probability amplitude wavefunctions, one in position space ##\psi(r,k)## and one in wavenumber space ##\phi(r,k)##, where ##r## and ##k## are Fourier conjugates, how is it possible for the modulus squared, i.e., probability density, of BOTH wavefunctions to be normalized? It seems...
2. K

### Show that the Hydrogen wave functions are normalized

Homework Statement Show that the (1,0,0) and (2,0,0) wave functions are properly normalized. We know that: Ψ(1,0,0) = (2/(a0^(3/2))*e^(-r/a0)*(1/sqrt(2))*(1/sqrt(2*pi)) where: R(r) = (2/(a0^(3/2))*e^(-r/a0) Θ(θ) = (1/sqrt(2)) Φ(φ) = (1/sqrt(2*pi)) Homework Equations (1) ∫|Ψ|^2 dx = 1 (2)...
3. ### A Probability amplitude

1. Given a Markov state density function: ## P((\textbf{r}_{n}| \textbf{r}_{n-1})) ## ##P## describes the probability of transitioning from a state at ## \textbf{r}_{n-1}## to a state at ##\textbf{r}_{n} ##. If ## \textbf{r}_{n-1} = \textbf{r}_{n}##, then ##P## describes the probability of...
4. ### Probability Density in an infinite 1D square well

Homework Statement The wave function of a particle of mass m confined in an infinite one-dimensional square well of width L = 0.23 nm, is: ψ(x) = (2/L)1/2 sin(3πx/L) for 0 < x < L ψ(x) = 0 everywhere else. The energy of the particle in this state is E = 63.974 eV. 1) What is the rest energy...
5. ### Using Noether's Theorem find a continuity equation for KG

Homework Statement Consider the Klein-Gordon equation ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0##. Using Noether's theorem, find a continuity equation of the form ##\partial_\mu j^{\mu}=0##. Homework Equations ##(\partial_\mu \partial^{\mu}+m^2)\varphi(x)=0## The Attempt at a Solution...
6. ### I Sum of Probability Density Function > 1?

I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modelled by a probability density function: $$f(x)=2x$$ Obviously the...
7. ### What probability density is used in Brownian motion?

Homework Statement I have a free Brownian particle and its coordinate is given as a function of time: And its first moment, or mean, is given as But what kind of probability density was used to calculate this first moment? Homework Equations I know that the first moment is calculated...
8. ### Plotting the Probability Density of the Coulomb Wave Function

Hey there - I think I have an issue with my 3D density plots of the probability density of the Coulomb wave function. The reason I think something is going wrong is because my plots of |ψ(n=2, l=1, m=-1)|² and |ψ(2, 1, 1)|² are identical, while I would expect them to have the same shape but be...
9. ### Probability current density in E.M. field

Derive the probability current density for a particle in an electromagnetic field. (I previously posted this on StackExchange. Please pardon, but I have been spending a lot of time on this and if anyone knows exactly what the subtle trick involved is, I would really appreciate it.)...
10. ### Probability density function of simple Mass-Spring system.

Homework Statement We know that after long run of simple mass-spring system, there should be a probability of finding the mass at certain points between -A and A.. Obviously in probability of finding the particle near A or -A is higher than finding the particle at 0, because the speed is the...
11. ### The Probability Density of X^2?

Here is a question about probability density. I am trying to work it out using a different method from the method on the textbook. But I get a different answer unfortunately. Can anyone help me out? Question: Let X be uniformly distributed random variable in the internal [ 0, 1]. Find the...
12. ### Calculating log liklihood: Zero value of likelihood function

Hello, I am analysing hydrology data and curve fitting to check the best probability distribution among 8 candidate distribution. (2 and 3 parameter distributions) The selection is based on the lowest AIC value. While doing my calculation in excel, how is it suggested to treat very low (approx...
13. ### Step Potential with incident and reflected waves

Homework Statement A woman is walking along a road. She has a mass of 52 kg and is walking at 1 m/s. (a) She is not paying careful attention and is walking straight towards the wall of a nearby building. Assume that the wall is infinitely hard and that she can be described as a plane wave (a...
14. ### Help Please -- cumulative distribution function, probability density

Homework Statement For the next probability function: f(x)=x/4 for 0<x<2 Homework Equations a) Get the probability function b) Get the cumulative distribution function The Attempt at a Solution I don´t know if the problem is well written, and for that I'm lost with the first question...
15. ### Quantum physics - probability density,

Homework Statement consider a particle at an interval ##[-L/2, L/2]##, described by the wave function ## \psi (x,t)= \frac{1}{\sqrt{L}}e^{i(kx-wt)}## a) Calculate the probability density ##\rho (x,t) ## and the current density ## j(x,t)## of the particle b) How can you express ## j(x,t)## as a...
16. ### Quantum harmonic oscillator tunneling puzzle

My problem is described in the animation that I posted on Youtube: For the sake of convenience I am copying here the text that follows the animation: I have made this animation in order to present my little puzzle with the quantum harmonic oscillator. Think about a classical oscillator, a...
17. ### Skewed Generalized Gaussian Distribution

I am looking for more information (e.g., reference, the CDF and descriptive stats) about a four-parameter skewed generalized Gaussian (SGG) distribution. I have come across the PDF for this distribution, but with no reference and not a lot of other information. Here is a snippet... On...