Boundary Conditions for Hydrogen Schrodinger Equation

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Discussion Overview

The discussion revolves around the boundary conditions necessary for solving the hydrogen Schrödinger equation to derive energy eigenvalues and quantum numbers for the hydrogen atom. Participants explore various aspects of boundary conditions, including their implications for wave functions and normalization.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the specific boundary conditions required for the hydrogen atom's Schrödinger equation.
  • Another participant suggests that the wave function should approach zero as the radius goes to infinity, questioning if this is sufficient.
  • It is noted that while the wave function going to zero at infinity is common, it must also be square integrable, meaning the integral of the wave function squared must be finite.
  • A participant discusses the method of separation of variables, indicating that it imposes a boundary condition of time-invariance for the spatial components of the solution, but questions whether this is overly restrictive.
  • There is a clarification that normalization pertains to the finiteness of the integral of the wave function squared, rather than being a boundary condition itself.
  • Further elaboration suggests that square integrable functions can exist that do not necessarily approach zero as their argument goes to infinity.

Areas of Agreement / Disagreement

Participants express differing views on the sufficiency and nature of boundary conditions for the hydrogen atom's wave function, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

Participants highlight the dependence on the choice of coordinate systems, noting that wave functions may not be square integrable in certain coordinate systems, which could affect the boundary conditions applied.

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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 Schrödinger 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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The usual thing is to have the wave function go to zero as radius goes to infinity. Is that enough?
 
DEvens said:
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.
 
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.
 
Quantum Defect said:
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.
 
Quantum Defect said:
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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