fluidistic said:
the assumptions are that there is a lot of symmetry (at boundary condition level, initial condition and domain of the equation)
You are mixing up several different kinds of symmetry. That's not a good idea; you need to keep them separate.
The first kind of symmetry is invariance of the equation itself under some group of transformations. As I have already pointed out in previous post, if an equation has this kind of symmetry, the full set of its solutions will also have the symmetry (in the sense that they will form a group under the symmetry transformations); but not all individual solutions, by themselves, will have the symmetry. This is true for any equation, not just the Schrodinger Equation; it's true for the heat equation, for example, if you consider the full set of
all possible solutions.
Imposing a symmetry on boundary conditions or initial conditions changes things because it
restricts the set of solutions that you are considering. For example, the set of solutions of the heat equation if a spherically symmetric boundary condition is imposed is
not the same as the set of
all solutions of the heat equation, with no constraint imposed on boundary conditions.
In your original comparison between the heat equation and the Schrodinger Equation, you were imposing spherically symmetric boundary conditions on the heat equation, but
not on the Schrodinger Equation; for the Schrodinger Equation, you were only assuming spherical symmetry for the equation itself (i.e., invariance under SO(3)). So of course you're going to see individual
solutions of the Schrodinger Equation that are not spherically symmetric, even though you're not seeing such solutions of the heat equation; you restricted the available solutions to the heat equation to rule out the ones that weren't spherically symmetric, by imposing boundary conditions.
You do this again in your latest post; see below.
fluidistic said:
for Schrodinger's equation, regarding the H-atom in vacuum problem, each eigenfunction (except the ground state) depend on the angle(s).
Wrong. There are an infinite number of spherically symmetric solutions for the H-atom--all of the ##s## orbitals. The ground state is just the lowest energy ##s## orbital (1s); it's not the only one.
fluidistic said:
This differs from eigenfunctions of the heat equation, in a symmetric problem where the domain is a sphere.
"The domain is a sphere", if I'm understanding correctly what you mean by that, is a boundary condition. You're not imposing any corresponding boundary condition on the Schrodinger Equation, so of course, as above, you're going to find individual solutions of the Schrodinger Equation that aren't spherically symmetric, whereas you won't for the heat equation because you've explicitly eliminated them by imposing a boundary condition.
If you eliminate that boundary condition from the heat equation, you
will find solutions that aren't spherically symmetric, even if you maintain the spherical symmetry (invariance under SO(3)) of the equation itself. And similarly for the other PDEs you mention.