Recent content by Leechie

I think I know where I'm heading now, thanks.

Thanks vela. I see how that all fits together now. I'm still a little confused with this bit though: I know how to find the largest value of ##b## using a derivative (if that's what the questions is asking), but I'm not sure I understand the bit about 'one part in a thousand'. I'm thinking...

This is how I got to (b): $$\begin{align} E_1^{(1)} & = \int_0^\infty R_{nl}^*(r) \delta \hat {\mathbf H} R_{nl}(r) r^2 dr \nonumber \\ & = \int_0^b \left( \frac {1} {a_0} \right)^{3/2} 2e^{-r/a_0} \left( - \frac {e^2} {4 \pi \varepsilon_0} \right) \left( \frac {b} {r^2} - \frac 1 r \right)...

Homework Statement Suppose there is a deviation from Coulomb's law at very small distances, with the mutual Coulomb potential energy between an electron and a proton being given by: $$V_{mod}(r)= \begin{cases} - \frac {e^2} {4 \pi \varepsilon_0} \frac {b} {r^2} & \text {for } 0 \lt r \leq b \\...

Thanks for your help John. I don't think its missing the factor 4##\pi##. The original state is given as ##\psi_{1,0,0} = R_{nl} \left( r \right) Y_{lm}\left( \theta, \phi \right) ##, and I made it that the spherical harmonics are normalized in the state: ##Y_{0,0}\left( \theta, \phi \right) =...

Homework Statement I'm trying to evaluate the following integral to calculate a first-order correction: $$\int_0^\infty R_{nl}(r)^* \delta \hat {\mathbf H} R_{nl}(r) r^2 dr$$ The problem states that ##b## is small compared to the Bohr radius ##a_o## Homework Equations I've been given...

Great. Thanks for your help!

Thanks for your reply TSny. You're right about changing the the symbol ##E## to ##V##. I think using ##E## instead of ##V## is what was confusing me when thinking about the uncertainty. I think I seen where I've gone wrong with my value for ##\left< V^2 \right>##. I've corrected it to...

Homework Statement Calculate ##\left< \frac 1 r \right>## and ##\left< \frac 1 {r^2} \right>## and the expectation value and uncertainty of the potential energy of the electron and proton for a hydrogen atom in the given state. The given state is: $$ \psi_{2,1,-1} \left( r,\theta,\phi \right)...

Thanks. I think I'm finally starting to get my head round this now.

Thanks PeroK. When I was working through that I realized it just led back to the initial spin state because of the z-basis. Is the method I used a general way of calculating a spinor in any direction ##n##?

I had a feeling there was something wrong somewhere, I'll take another look. Thanks for your help.

Homework Statement Write down a spinor that represents the spin state of the particle at any time t > 0. Use the expression to find the expectation values of ##S_x## and ##S_y## Homework Equations The particle is a spin-##\frac 1 2## particle, the gyromagnetic ratio is ##\gamma_s \lt 0##, and...

Ok, thanks for your help with this.

Thanks for the tip! I think this is where I'm getting confused. I thought the substitution ##u=\sqrt{2a}x## would mess the integral up and make it ##\frac{1}{\sqrt{2a}}\int^{\infty}_{-\infty}e^{-u^2}e^{-ikx} dx##?