Hydrogen atom — solving in spherical coordinates

2.3.12 · Physics › Modern Physics

1. Starting equation (KYA solve karte hain)

KYU reduced mass? Proton infinitely heavy nahi hai, isliye dono particles common centre of mass ke around move karte hain. $m_e$ ko $\mu =\frac{m_e{m}_p}{m_e+m_p}\approx m_e$ se replace karne par two-body problem exactly ek one-body problem ban jaata hai.

Spherical coordinates mein Laplacian (yeh ek geometry fact hai, chain rule se derive hota hai, physics se nahi): ${\nabla }^2=\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial }{\partial r}\right)+\frac{1}{r^2\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{r^2{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}.$

2. Separation of variables (KAISE split hota hai)

Substitute karo aur $\frac{2\mu r^2}{{\hslash }^{2}RY}$ se multiply karo. $r$ wali har cheez ek taraf jaati hai, angular har cheez doosri taraf:

$\underset{\text{sirf }r}{\underbrace{\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{2\mu r^2}{{\hslash }^2}(V-E)}}=\underset{\text{sirf }\theta ,\varphi }{\underbrace{-\frac{1}{Y}\left[\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial Y}{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^{2}Y}{\partial {\varphi }^2}\right]}}$

Toh hume do equations milti hain:

Angular: $\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial Y}{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^{2}Y}{\partial {\varphi }^2}=-\ell (\ell +1)Y$

Radial: $\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)-\frac{2\mu r^2}{{\hslash }^2}(V-E)R=\ell (\ell +1)R$

3. Angular part → phir se split

$\varphi$ equation: $\frac{d^2\Phi }{d{\varphi }^2}=-m_{\ell }^2\Phi \Rightarrow \Phi (\varphi )=e^{i{m}_{\ell }\varphi }.$

$\theta$ equation associated Legendre equation hai; iski solutions tab hi finite hoti hain jab $\ell$ ek non-negative integer ho aur $|m_{\ell }|\le \ell$ ho. Poori angular solutions spherical harmonics $Y_{\ell }^{m_{\ell }}(\theta ,\varphi )$ hain.

4. Radial part → energy quantization

Equation ko clean karne ke liye $u(r)=rR(r)$ define karo. Yeh ek 1D Schrödinger equation ban jaata hai: $-\frac{{\hslash }^2}{2\mu }\frac{d^{2}u}{d{r}^2}+\underset{V_{\text{eff}}}{\underbrace{\left[V(r)+\frac{{\hslash }^2\ell (\ell +1)}{2\mu r^2}\right]}}u=Eu.$

$u$ ko normalizable (decay as $r\to \infty$) aur origin par finite hone ki requirement ek aur integer force karti hai, ==principal quantum number $n$==. Solve karne par famous energy levels milte hain.

5. Worked examples

Flashcards

Connections

• Bohr model of hydrogen — same $E_n$ deta hai lekin wavefunctions ke bina
• Spherical harmonics — $Y_{\ell }^{m_{\ell }}$ angular solutions
• Angular momentum operators — $\ell ,m_{\ell }$ ka origin
• Separation of variables (PDEs)
• Reduced mass and two-body problem
• Zeeman effect — magnetic field ke saath $m_{\ell }$ degeneracy lift ho jaati hai
• Quantum numbers and the periodic table

Concept Map