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## Main Question or Discussion Point

I am currently undergoing a project for my undergraduate course in Quantum Physics on electron tunneling in the [tex]H_2^+[/tex] ion. Essentially, I am trying to model this situation in Fortran based on the variables that the user supplies (For example, the energy levels of the atoms, the separation distance, etc..). An illustration of my goal is examined at http://www.chem1.com/acad/webtut/bonding/TunnelBond.html" [Broken]. At first glance it seems this situation boils down to nothing more than a fancy double well potential problem - however upon beginning to draw out the wavefunctions for the various regions involved I find that I may have been mistaken.

For normal potential-well situations, we have nice and neat wavefunctions that take the form (for instance):

[tex]\Psi_(x)=Ae^{Kx}+B_Ie^{-Kx}[/tex]

where

[tex]K^2=\frac{2mE}{\hbar^2}[/tex]

HOWEVER, since I am concerning myself with a Hydrogenic molecule - the wave function takes the form:

[tex]\Psi_{nlm}(r,\theta,\phi)=\left(\sqrt{\left(\frac{2}{na_o}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}}\right)e^{\frac{-r}{na_0}}\left(\frac{2r}{na_0}\right)^lL_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)Y_l^{m_l}(\theta,\phi)[/tex]

So, before I can get anywhere I need to modify this wavefunction to properly describe my system. But, as you will notice (as I surly did), that the hydrogen wavefunction does not explicitly include a value for the energy, as the previous wavefunction does. This leads me to belive that I may have to delve into various time-dependent perturbation methods of the Schrodinger equation - which my introductory coursebook offers little insight into. Since this seems like it would be a fairly common computational endevour, and the community here are extremely experienced, I was wondering if anybody could offer some insight as to how to get me started on this project (literature, specific topics of relevance, etc..). Any help would be greatly appreciated. Thanks all

IHateMayonnaise

For normal potential-well situations, we have nice and neat wavefunctions that take the form (for instance):

[tex]\Psi_(x)=Ae^{Kx}+B_Ie^{-Kx}[/tex]

where

[tex]K^2=\frac{2mE}{\hbar^2}[/tex]

HOWEVER, since I am concerning myself with a Hydrogenic molecule - the wave function takes the form:

[tex]\Psi_{nlm}(r,\theta,\phi)=\left(\sqrt{\left(\frac{2}{na_o}\right)^3\frac{(n-l-1)!}{2n[(n+l)!]^3}}\right)e^{\frac{-r}{na_0}}\left(\frac{2r}{na_0}\right)^lL_{n-l-1}^{2l+1}\left(\frac{2r}{na_0}\right)Y_l^{m_l}(\theta,\phi)[/tex]

So, before I can get anywhere I need to modify this wavefunction to properly describe my system. But, as you will notice (as I surly did), that the hydrogen wavefunction does not explicitly include a value for the energy, as the previous wavefunction does. This leads me to belive that I may have to delve into various time-dependent perturbation methods of the Schrodinger equation - which my introductory coursebook offers little insight into. Since this seems like it would be a fairly common computational endevour, and the community here are extremely experienced, I was wondering if anybody could offer some insight as to how to get me started on this project (literature, specific topics of relevance, etc..). Any help would be greatly appreciated. Thanks all

IHateMayonnaise

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