Ch 6: Two-Body System (Hydrogen)

We are going to bring all of these ideas together to describe the complete Hydrogen Atom (neglecting spin for now). Let’s start again with the Schrodinger Equation:

(1)   \begin{equation*} i \frac{\partial \Psi}{\partial t} +\sum _{j=1}^n \frac{1}{2m_j}\nabla _j^2\Psi -V \Psi =0 \end{equation*}

For this case, n=2, m_1=m_e=1, m_2=m_p=1836, and the Potential V = -1/|\vec{r_1}-\vec{r_2}|, so equation (1) becomes

(2)   \begin{equation*} i \frac{\partial \Psi}{\partial t}+\frac{1}{2 m_e} \nabla_1^2\Psi +\frac{1}{2 m_p} \nabla_1^2\Psi +\frac{1}{|\vec{r_1}-\vec{r_2}|} \Psi =0} \end{equation*}

Now we make a change of variables:

(3)   \begin{equation*} \vec{r}=\vec{r_1}-\vec{r_2}, \vec{R}=\frac{m_1 \vec{r_1}+m_2 \vec{r_2}}{m_1+m_2}, \mu=\frac{m_1 m_2}{m_1+m_2}, M=m_1+m_2 \end{equation*}

This will allow us to transform (2) to

(4)   \begin{equation*} i\frac{\partial \Psi }{\partial t}+\frac{1}{2M}\nabla _R{}^2\Psi +\frac{1}{2\mu }\nabla _r{}^2\Psi +\frac{1}{|r|} \Psi =0 \end{equation*}

Then by writing \Psi = \Psi(r)\Psi(R), we can separate the equation into these 2:

(5)   \begin{equation*} i\frac{\partial \psi (R)}{\partial t}+\frac{1}{2M}\nabla _R{}^2\psi (R)=0 \end{equation*}

(6)   \begin{equation*} i\frac{\partial \psi (r)}{\partial t}+\frac{1}{2\mu }\nabla _r{}^2\psi (r)+\frac{1}{|r|} \psi (r)=0 \end{equation*}

Equation (5) is identical to a free particle system. This means that ‘R’, the center of mass of the system, behaves exactly like the free particle with mass M. Meanwhile, equation (6) is the Central Potential with mass \mu, which will describe the vector joining the proton and electron. So what does this system look like? Since there are 2 particles, \Psi describes a fluid in 6-dimensional phase space. But each point in phase space represents a possible configuration of both particles. In other words, we’re looking at a fluid of pairs of particles.

The video linked below shows the system with a Gaussian “Center of Mass Distribution” and an “Electron-Proton Distribution” of \frac{1}{\sqrt{2}}(\Psi_{2,1,1}+\Psi_{3,1,1}). The green dots are electrons, the red dots are protons, and there are lines joining particles that are coupled classically. Each pair is a separate “world” in our interpretation.

Hydrogen Atom Simulation

Sources

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An essay on free content by Sam Harris

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