Boundary Conditions for Hydrogen Schrodinger Equation

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If I am trying to derive the energy eigenvalues and quantum numbers for the hydrogen atom (basic hydrogen-1), I obviously need to solve the hydrogen Schrodinger equation and account for some boundary conditions. However, no website ever gives me the boundary conditions. What would be the boundary conditions for the hydrogen atom?
 

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  • #2
DEvens
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The usual thing is to have the wave function go to zero as radius goes to infinity. Is that enough?
 
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Quantum Defect
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The usual thing is to have the wave function go to zero as radius goes to infinity. Is that enough?
Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.
 
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The usual method of separation of variables imposes a 'boundary' condition that the spatial components of the solution are independent of time - i.e. time-invariant.
This may be excessive, as the requirement that the final wave function be square-integrable permits wave functions that are periodic with finite integrability.
The other boundary conditions also imposed by the method are that the components of the wave function in polar coordinates are orthogonal and normal.
The polar coordinates themselves are part of this, in that in other coordinate systems (rectilinear, cylindrical) the wave functions are NOTsquare-integrable.
 
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DEvens
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Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.
That is normalization, not boundary condition.
 
  • #6
dextercioby
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Integral of Psi*Psi needs to be finite (square integrable). You can have a function go to zero as r->inf. that would not be square integrable.
And you can have square integrable functions which don't go to 0 when their argument goes to infinity.
 

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