Preferred direction about hydrogen atom

[ENDLESSYOU]

When solving the three dimensional Schrödinger equation, we obtain a probability distribution on θ. But it seems like the procudure produces a particular direction the z-axis. While the Coulomb field is spherical symmetric, it shouldn't exist such a preferred direction. I'm puzzled.

 

[tiny-tim]

Hi ENDLESSYOU!

Individual solutions to a symmetric equation don't have to be symmetric.

Each of the (eg) p_x p_y and p_z distributions

is a solution to the Schrödinger equation, and does of course have a preferred direction.

But any combination of them will be a new solution, p_k, for some direction k.

All directions are treated equally.

 

[jtbell]

If there are no external influences on the hydrogem atom (it's just sitting there) the "preferred direction" in spherical coordinates (the z-axis) is physically arbitrary. It could be up, to the east, to the south, etc. Any actual physical results of experiments that you calculate will be spherically symmetric.

However, when we apply a magnetic field, for example, that creates a physically preferred direction, which breaks the spherical symmetry. Then it is natural to align the z-direction along the magnetic field.

Notice the solutions of the SE with same n and l, but with different m, are not spherically symmetric. Without a magnetic field, they all have the same exact energy. With a magnetic field, they have different energies, and those energies are exact only if the z-axis is along the magnetic field direction.

 

[ENDLESSYOU]

If there are no external influences on the hydrogem atom (it's just sitting there) the "preferred direction" in spherical coordinates (the z-axis) is physically arbitrary. It could be up, to the east, to the south, etc. Any actual physical results of experiments that you calculate will be spherically symmetric.

However, when we apply a magnetic field, for example, that creates a physically preferred direction, which breaks the spherical symmetry. Then it is natural to align the z-direction along the magnetic field.

Notice the solutions of the SE with same n and l, but with different m, are not spherically symmetric. Without a magnetic field, they all have the same exact energy. With a magnetic field, they have different energies, and those energies are exact only if the z-axis is along the magnetic field direction.

Thanks! But I still have some problems.
1. If there are several free atoms, their z-axes may be in different directions?
2. This arbitary direction is physically existence or just a mathematical construct?
3. The solutions with same n and l have different probability density, but when we add them together, it becomes spherical. How to interpret this by measurement?

 

[ENDLESSYOU]

Hi ENDLESSYOU!

Individual solutions to a symmetric equation don't have to be symmetric.

Each of the (eg) p_x p_y and p_z distributions

is a solution to the Schrödinger equation, and does of course have a preferred direction.

But any combination of them will be a new solution, p_k, for some direction k.

All directions are treated equally.

Thanks for your interpretation in a mathematical way!

 

[tiny-tim]

1. If there are several free atoms, their z-axes may be in different directions?

if there's a field, they will line up either with or opposite to it

2. This arbitary direction is physically existence or just a mathematical construct?

it's physical … the angular momentum can be transferred to another body

3. The solutions with same n and l have different probability density, but when we add them together, it becomes spherical. How to interpret this by measurement?

spherical means that the distribution is random, ie we measure the direction as random … we don't know anything about the particle, it is a mixture of all possible solutions