Schrödinger Wave Equation for Hydrogen Atom

Getting from the Schrödinger Wave Equation, to a better model than the Bohr Model.

The Potential energy of the electron-proton system is electrostatic.

$$V(r)=-\frac{e^2}{4\pi {ϵ}_{0}r}$$

Negative means potential is attractive

Transform the Time-dependent Equation into spherical coordinates. Also use reduced mass in the equation, because it gives a more accurate result for free.

Assume $\psi (r,\theta ,\varphi )=R(r)f(\theta )g(\varphi )$, so it is a separable function. This makes the above equation simplify a lot!

Azimuthal-angle equation

$$g_{\varphi \varphi }=-m_l^{2}g$$ $$\text{Solution: }g(\varphi )=\exp (\pm i{m}_l\varphi )$$

Sines and cosines satisfy this equations, but its convenient to choose the complex exponential.

Quantizing $m_l$

Because $\varphi$ and $\varphi +2\pi$ are the same angle, you can set g of each equal and you solve for this:

$$\exp (\pm 2\pi i{m}_l\varphi )=1$$

$m_l$ must be an integer for this to be true.

Polar Equation for H

$$\frac{1}{\sin \theta }\frac{d}{d\theta }\left(\sin \theta \frac{\partial f}{\partial \theta }\right)+\left[l(l+1)-\frac{m_l^2}{{\sin }^2\theta }\right]f=0$$

See Polar-Angle Equation

Radial Equation for H

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\frac{2\mu }{{\hslash }^2}\left[E+\frac{e^2}{4\pi {ϵ}_{0}r}-\frac{{\hslash }^2}{2\mu }\frac{l(l+1)}{r^2}\right]R=0$$

See Radial Wave Equation

Complete Hydrogen Wave Functions

$${\Psi }_{nl{m}_l}(r,\theta ,\varphi ,t)=R_{nl}(r)f_{l{m}_l}(\theta )\exp (i({\omega }_{n}t\pm m_l\varphi ))$$ $$\text{where }{\omega }_n=\frac{E_n}{\hslash },\text{and }n,l,m_l\text{ are constrained}$$

The result is a wave; states have some angular momentum proportional to $m_l$. However, such motion can only be inferred indirectly, so we opt for the spatial wave-function instead, which is a product of the radial wave function $R_{nl}$ and the spherical harmonics $Y_{lm}$:

$${\Psi }_{nl{m}_l}(r,\theta ,\varphi )=R_{nl}(r)Y_{l{m}_l}(\theta ,\varphi )$$ $$\text{where }n,l,m_l\text{ are constrained}$$

See also

Quantum Number