Preferred direction about hydrogen atom

1. Mar 16, 2013

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.

2. Mar 16, 2013

tiny-tim

Hi ENDLESSYOU!

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

Each of the (eg) px py and pz 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, pk, for some direction k.

All directions are treated equally.

3. Mar 16, 2013

Staff: Mentor

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.

4. Mar 16, 2013

ENDLESSYOU

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?

Last edited: Mar 16, 2013
5. Mar 16, 2013

ENDLESSYOU

Thanks for your interpretation in a mathematical way!

6. Mar 20, 2013

tiny-tim

if there's a field, they will line up either with or opposite to it
it's physical … the angular momentum can be transferred to another body
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