Recent content by Dazzabaijan

Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...

So following your advice of converting my wavepacket wavefunction into spherical polar coordinates, I get $$ \begin{equation*} \begin{split} \psi(\textbf{r},0) &= \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0 \cdot...

Just used Mathematica for the first time and had to figure out how to do integration. So doing the whole integral of the scalar product gave me this, not sure if it's right or what to make of it. Any opinions would be appreciated!

Just in case that you're still interested in helping me, all of the equations above were followed closely from Chapter 10.1, 10.9, and 10.10 of the book "The Picture Book of Quantum Mechanics 4th edition", so it might be useful if you have a look. Meanwhile I'm trying all the approach I could lol

Which also sounds right because partial waves decomposition works well with a central potential, which is exactly the case with this problem because the potential in the Hamiltonian is central in the Hamiltonian of the SE that I had to solve earlier.

It seems that the Legendre functions are related to the Spherical Harmonics by $$P_l(\cos\theta) = \sqrt{\frac{4\pi}{2l+1}} Y_{l,0}(\theta, \phi)$$ when m = 0, so the decomposition of stationary harmonic plane wave into partial waves can now be expressed as $$ e^{\textbf{k}\cdot\textbf{r}} =...

My supervisor did his master's thesis on it too but he did it on a Quantum Harmonic Oscillator. Most of the stuff can be found from the book "The Picture Book of Quantum Mechanics" but they didn't show how to find the expansion coefficients. This project is waaay above my knowledge of QM which...

Yes how did you know?Ultimately I'm trying to investigate the correspondence principle of the hydrogen atom, so in the limit of high quantum number how it'd would collapse from a quantum mechanical regime to a classical one.

I'm having trouble with trying to find the expansion coefficients of a superposition of a Gaussian wave packet. First I'm decomposing a Gaussian wave packet $$\psi(\textbf{r},0) = \frac{1}{(2\pi)^{3/4}\sigma^{3/2}}\text{exp}\left[ -\frac{(\textbf{r} - \textbf{r}_0)^2}{4\sigma^2} + i\textbf{k}_0...