Phy5646/Born-Oppenheimer Approximation

Consider the problem of two protons and one electron.

As for the two protons, we consider the two bodies problem as one body problem.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\frac{P^2}{2M}+\frac{p^2}{2m}+V\left(\overset{\rightharpoonup }{R},\overset{\rightharpoonup }{r}\right)}

The wave function is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi _{total}=\left|n_{electron}\left(\overset{\rightharpoonup }{R}\right)>\right|\psi _{proton}\left(\overset{\rightharpoonup }{R}\right)>}

First step:

Consider R is fixed, to solve the schrodinger equation of electron.

Second step:

Seek an solution of H as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\Psi >=\int \Psi \left(\overset{\rightharpoonup }{R}\right)\right|\overset{\rightharpoonup }{R}>|n\left(\overset{\rightharpoonup }{R}\right)>d^3R}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}_{eff} = \frac{1}{2m}(\overrightarrow{P} - \overrightarrow{A}^{n})^2 + \Phi^{(n)} }

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | n(R) \rangle } is the eigenstate of the fast degree of freedom, the following quantities are defined:

The Berry Vector Potential: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow{A}^{(n)} = i\hbar \langle n(R) | \overrightarrow{\nabla}_R |n(R) \rangle }

The Berry Scalar Potential: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi^{(n)} = \frac{\hbar^2}{2m} [ \langle \overrightarrow{\nabla}_R n(R) | \overrightarrow{\nabla}_R n(R) \rangle - \langle \overrightarrow{\nabla}_R n(R) | n(R) \rangle \langle n(R) | \overrightarrow{\nabla}_R n(R) \rangle ]}