U(x,y,z,t)*ψ(x,y,z,t)-(ħ/(2*m))*(d(adsbygoogle = window.adsbygoogle || []).push({}); ^{2}ψ(x,y,z,t)/dx^{2}+d^{2}ψ(x,y,z,t)/dy^{2}+d^{2}ψ(x,y,z,t)/dz^{2})=ħ*i*dψ(x,y,z,t)/dt

q_{proton}=-q_{e}

Schrödinger equation for electron in hydrogen atom (if we consider proton as point charge which is moving at a constant speed v_{proton}^{→}=(v_{p;x};v_{p;y};v_{p;z}).) is:

U_{e}(x,y,z,t)*ψ_{e}(x,y,z,t)-(ħ/(2*m_{e}))*(d^{2}ψ_{e}(x,y,z,t)/dx^{2}+d^{2}ψ_{e}(x,y,z,t)/dy^{2}+d^{2}ψ_{e}(x,y,z,t)/dz^{2})=ħ*i*dψ_{e}(x,y,z,t)/dt

U_{e}(x,y,z,t)=q_{e}*q_{proton}/(r_{distance from electron to proton}*ε_{0}*4*π)=-q_{e}^{2}/(((x_{p}+t*v_{p;x}-x_{e}-t*v_{e;x})^{2}+(y_{p}+t*v_{p;y}-y_{e}-t*v_{e;y})^{2}+(z_{p}+t*v_{p;z}-z_{e}-t*v_{e;z})^{2})^{(1/2)}*ε_{0}*4*π)

⇒-q_{e}^{2}/(((x_{p}+t*v_{p;x}-x_{e}-t*v_{e;x})^{2}+(y_{p}+t*v_{p;y}-y_{e}-t*v_{e;y})^{2}+(z_{p}+t*v_{p;z}-z_{e}-t*v_{e;z})^{2})^{(1/2)}*ε_{0}*4*π)*ψ(x,y,z,t)-(ħ/(2*m_{e}))*(d^{2}ψ(x,y,z,t)/dx^{2}+d^{2}ψ(x,y,z,t)/dy^{2}+d^{2}ψ(x,y,z,t)/dz^{2})=ħ*i*dψ(x,y,z,t)/dt

by solving it we can get electron wave function ψ_{e}(x,y,z,t) in hydrogen atom. Am I right?

But if we consider proton as wave like we did with electron:

U_{e}(x,y,z,t)*ψ_{e}(x,y,z,t)-(ħ/(2*m_{e}))*(d^{2}ψ_{e}(x,y,z,t)/dx^{2}+d^{2}ψ_{e}(x,y,z,t)/dy^{2}+d^{2}ψ_{e}(x,y,z,t)/dz^{2})=ħ*i*dψ_{e}(x,y,z,t)/dt

U_{p}(x,y,z,t)*ψ_{p}(x,y,z,t)-(ħ/(2*m_{p}))*(d^{2}ψ_{p}(x,y,z,t)/dx^{2}+d^{2}ψ_{p}(x,y,z,t)/dy^{2}+d^{2}ψ_{p}(x,y,z,t)/dz^{2})=ħ*i*dψ_{p}(x,y,z,t)/dt

its obvious that their potential energys are equal to each other[U_{e}(x,y,z,t)=U_{p}(x,y,z,t)], but to what potential energy function equals [U(x,y,z,t)=?]?

I know it is good approximation to consider proton as point charge .I am asking this to understand Schrödinger equation better.

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# Schrödinger equation for 2 particles

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