#### olgerm

Gold Member

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Can you say whether I understood these things correctly?

to get condition on wavefunction ##\Psi## for a system that consists of 2 electrons(without taking spin into account) and helium nuclei I can solve schrödinger equation: ##i*\frac{\partial \Psi}{\partial t}=-\frac{q_e*q_e}{\sqrt{(x_{electron1}-x_{electron2})^2+(y_{electron1}-y_{electron2})^2-(z_{electron1}-z_{electron2})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron2}-x_{proton})^2+(y_{electron2}-y_{proton})^2+(z_{electron2}-z_{proton})^2}}-\\

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron2}^2}+\frac{\partial^2 \Psi}{\partial y_{electron2}^2}+\frac{\partial^2 \Psi}{\partial z_{electron2}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2}))##

probability of particles being in corresponding positions is ##P(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})=|\Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})|^2##

to form helium atom wavefunction must be stationary aka not change in time. By solving time-independent schrdinger equation I find wavefunctions that satisfy normal schrödinger equation and so not change in time (##\frac{\partial |\Psi|}{\partial t}=0##).

##\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron2}^2}+\frac{\partial^2 \Psi}{\partial y_{electron2}^2}+\frac{\partial^2 \Psi}{\partial z_{electron2}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2})

-\frac{q_e*q_e}{\sqrt{(x_{electron1}-x_{electron2})^2+(y_{electron1}-y_{electron2})^2-(z_{electron1}-z_{electron2})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron2}-x_{proton})^2+(y_{electron2}-y_{proton})^2+(z_{electron2}-z_{proton})^2}})=E*\Psi(r)##

to have hydrogen atom I have 1 electron and 1 proton and wavefunction must be stationary:

##-\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2})

-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}=E*\Psi(r)##

and the solution is: ##\psi _{n,l,m}(r,\theta ,\phi )= (\frac{2^{l+1}*r^l*m_e^{l+3/2}*m_p^{l+3/2}*q_e^{2*l+3}*((n-l -1)!)^{1/2}} {n^{1+l}*(m_e+m_p)^{l+3/2}*((n+l )!)^{1/2}})*e^{-\frac{r*m_e*m_p*q_e^2}{n*(m_e+m_p)}}*L_{n-l -1}^{2l +1}(\frac{2*r*m_e*m_p*q_e^2}{n*(m_e+m_p)})*Y_l^m(\theta ,\phi )##(got it from Wikipedia)

since ##P(A,B,C,D,E,F) \neq \int_{-\infty}^{\infty}(dx_1*\int_{-\infty}^{\infty}(dx_2*\int_{-\infty}^{\infty}(dx_3*P(A,B,C,x_1,x_2,x_3))))*\int_{-\infty}^{\infty}(dx_1*\int_{-\infty}^{\infty}(dx_2*\int_{-\infty}^{\infty}(dx_3*P(x_1,x_2,x_3,D,E,F))))## positions of electron and proton in hydrogen atom are entagled.

To get in y-direction inertia of proton I us inertia operator ##-i*\frac{\partial}{\partial y}## :

##-i*\frac{\partial}{\partial y_{proton}} \Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})=\Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},p_{y\ proton},z_{proton})##

I used units where ##\hbar=1## and ##k_{Coulomb}=1##

to get condition on wavefunction ##\Psi## for a system that consists of 2 electrons(without taking spin into account) and helium nuclei I can solve schrödinger equation: ##i*\frac{\partial \Psi}{\partial t}=-\frac{q_e*q_e}{\sqrt{(x_{electron1}-x_{electron2})^2+(y_{electron1}-y_{electron2})^2-(z_{electron1}-z_{electron2})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron2}-x_{proton})^2+(y_{electron2}-y_{proton})^2+(z_{electron2}-z_{proton})^2}}-\\

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron2}^2}+\frac{\partial^2 \Psi}{\partial y_{electron2}^2}+\frac{\partial^2 \Psi}{\partial z_{electron2}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2}))##

probability of particles being in corresponding positions is ##P(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})=|\Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})|^2##

to form helium atom wavefunction must be stationary aka not change in time. By solving time-independent schrdinger equation I find wavefunctions that satisfy normal schrödinger equation and so not change in time (##\frac{\partial |\Psi|}{\partial t}=0##).

##\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron2}^2}+\frac{\partial^2 \Psi}{\partial y_{electron2}^2}+\frac{\partial^2 \Psi}{\partial z_{electron2}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2})

-\frac{q_e*q_e}{\sqrt{(x_{electron1}-x_{electron2})^2+(y_{electron1}-y_{electron2})^2-(z_{electron1}-z_{electron2})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}-\frac{q_e*q_p}{\sqrt{(x_{electron2}-x_{proton})^2+(y_{electron2}-y_{proton})^2+(z_{electron2}-z_{proton})^2}})=E*\Psi(r)##

to have hydrogen atom I have 1 electron and 1 proton and wavefunction must be stationary:

##-\frac{1}{2*m_{electron}}* (\frac{\partial^2 \Psi}{\partial x_{electron1}^2}+\frac{\partial^2 \Psi}{\partial y_{electron1}^2}+\frac{\partial^2 \Psi}{\partial z_{electron1}^2})-

\frac{1}{2*m_{proton}}* (\frac{\partial^2 \Psi}{\partial x_{proton}^2}+\frac{\partial^2 \Psi}{\partial y_{proton}^2}+\frac{\partial^2 \Psi}{\partial z_{proton}^2})

-\frac{q_e*q_p}{\sqrt{(x_{electron1}-x_{proton})^2+(y_{electron1}-y_{proton})^2+(z_{electron1}-z_{proton})^2}}=E*\Psi(r)##

and the solution is: ##\psi _{n,l,m}(r,\theta ,\phi )= (\frac{2^{l+1}*r^l*m_e^{l+3/2}*m_p^{l+3/2}*q_e^{2*l+3}*((n-l -1)!)^{1/2}} {n^{1+l}*(m_e+m_p)^{l+3/2}*((n+l )!)^{1/2}})*e^{-\frac{r*m_e*m_p*q_e^2}{n*(m_e+m_p)}}*L_{n-l -1}^{2l +1}(\frac{2*r*m_e*m_p*q_e^2}{n*(m_e+m_p)})*Y_l^m(\theta ,\phi )##(got it from Wikipedia)

since ##P(A,B,C,D,E,F) \neq \int_{-\infty}^{\infty}(dx_1*\int_{-\infty}^{\infty}(dx_2*\int_{-\infty}^{\infty}(dx_3*P(A,B,C,x_1,x_2,x_3))))*\int_{-\infty}^{\infty}(dx_1*\int_{-\infty}^{\infty}(dx_2*\int_{-\infty}^{\infty}(dx_3*P(x_1,x_2,x_3,D,E,F))))## positions of electron and proton in hydrogen atom are entagled.

To get in y-direction inertia of proton I us inertia operator ##-i*\frac{\partial}{\partial y}## :

##-i*\frac{\partial}{\partial y_{proton}} \Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},y_{proton},z_{proton})=\Psi(t,x_{electron1},y_{electron1},z_{electron1},x_{electron2},y_{electron2},z_{electron2},x_{proton},p_{y\ proton},z_{proton})##

I used units where ##\hbar=1## and ##k_{Coulomb}=1##

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