57-71

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Periods 1-4 (H-Kr) are modeled here via Slater’s rules; periods 5-7 (greyed out) aren’t yet.

Schrodinger's Equation

Time-Independent

\frac{\hbar^2}{2m} \nabla^2\Psi + U\Psi = E\Psi

Separation of Variables

In spherical coordinates, the Laplacian expands as:

\nabla^2 = \frac{1}{r^2}\frac{\partial}{\partial r} \bigg( r^2\frac{\partial}{\partial r} \bigg) + \frac{1}{r^2\sin\theta}\frac{\partial}{\partial \theta} \bigg( \sin\theta\frac{\partial}{\partial \theta} \bigg) + \frac{1}{r^2 \sin^2\theta}\frac{\partial^2}{\partial \phi^2}

Assuming $\Psi(r,\theta,\phi)=R(r)Y(\theta,\phi)$ and dividing by $RY$ , the equation separates into independent radial and angular parts:

\frac{1}{R}\frac{d}{dr} \bigg[ r^2\frac{dR}{dr} \bigg] - \frac{2mr^2}{\hbar^2}\bigg(U(r)-E\bigg) = -\frac{1}{Y} \bigg[ \frac{1}{\sin\theta}\frac{\partial}{\partial\theta} \bigg( \sin\theta\frac{\partial Y}{\partial\theta} \bigg) + \frac{1}{\sin^2\theta}\frac{\partial^2 Y}{\partial \phi^2} \bigg]

Each side equals the same separation constant $\ell(\ell+1)$ , giving the factored wavefunction:

\Psi_{n,\ell,m}(r, \theta, \phi) = R_{n,\ell}(r)\, Y_{\ell,m}(\theta, \phi)

Radial part

R_{n,\ell}(\rho) = \rho^\ell e^{-\rho/2} L_{n-\ell-1}^{2\ell+1}\bigg(\frac{2r}{na}\bigg)

Angular part

Y_\ell^m(\theta, \phi) = \sqrt{\frac{(2\ell+1)(\ell-m)!}{4\pi(\ell+m)!}} e^{im\phi} P_{\ell}^{m}(\cos\theta)

where

\rho = \frac{2Z}{na_0}r \quad\quad c_{j+1} = \frac{j+\ell+1-n}{(j+1)(j+2\ell+2)} c_j

P_{\ell}^{m}(x) \equiv (-1)^m (1-x^2)^{m/2} \bigg(\frac{d}{dx}\bigg)^{m} P_{\ell}(x) \quad\quad P_{\ell}(x) \equiv \frac{1}{2^\ell \ell!} \bigg(\frac{d}{dx}\bigg)^{\ell} (x^2-1)^\ell

*Associated Legendre Polynomials $\left(P_{\ell}^{m}\right)$

The Hydrogen Atom

General Wavefunction

\Psi_{n,\ell,m}(r, \theta, \phi) = \sqrt{\bigg(\frac{2}{na}\bigg)^3 \frac{(n-\ell-1)!}{2n(n+\ell)!}} e^{-r/na} \bigg(\frac{2r}{na}\bigg)^\ell \bigg[L_{n-\ell-1}^{2\ell+1}\bigg(\frac{2r}{na}\bigg)\bigg] Y_{\ell,m}(\theta,\phi)

where

L_{q}^{p}(x) \equiv (-1)^p \bigg(\frac{d}{dx}\bigg)^p L_{p+q}(x) \quad\quad L_{q}(x) \equiv \frac{e^x}{q!} \bigg(\frac{d}{dx}\bigg)^q (e^{-x} x^q)

*Associated Laguerre Polynomials $\left(L_{q}^{p}\right)$

Select a State

n: l: m:

Wavefunction

\psi_{210}(r,\theta,\phi)=\cos\theta\left(\frac{2r}{2a_0}\right)e^{-\frac{r}{2a_0}}\sqrt{\frac{1}{32a_0^3\pi}}

Radial Equation

R_{21}(r)=\left(\frac{2r}{2a_0}\right)e^{-\frac{r}{2a_0}}\sqrt{\frac{1}{24a_0^3}}

Angular Equation

Y_{10}(\theta,\phi)=\cos\theta\sqrt{\frac{3}{4\pi}}

H (Z=1) — 2p

Z_eff (Slater) = 1.000
Approx. orbital energy = -3.40 eV

n:2

l:1

m:0

Probability: 50%

Show lower orbital(1s)

Show Phase

Phase (arg ψ):

−π−π/20+π/2+π

Grid: 48³

10 a₀ (5.29 Å)

Radial

R_21(r)

Angular

Y_10(theta, phi=0) core