- #1

hokhani

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- TL;DR Summary
- The non spherical orbitals of spherical Hamiltonian

Oritals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?

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- Thread starter hokhani
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- #1

hokhani

- 473

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- TL;DR Summary
- The non spherical orbitals of spherical Hamiltonian

Oritals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?

- #2

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Summary::The non spherical orbitals of spherical Hamiltonian

Oritals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?

You can compare this to the case of a free particle in 1D. The Hamiltonian operator

##\displaystyle\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}##

is invariant in any coordinate translation ##x\mapsto x+\Delta x##, but the energy eigenstates are of form

##\displaystyle\psi (x) = Ae^{ikx} + Be^{-ikx}##

which are not translation invariant unless ##k=0## and ##\psi## is a constant function.

- #3

PeterDonis

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How is this possible?

Because the symmetry of the Hamiltonian shows up in the symmetry of the entire set of solutions, not in every individual solution. The entire set of solutions has spherical symmetry, but not all individual solutions do.

- #4

Nugatory

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That's true of any central force problem, even classically. For example the Sun's gravitational field is (too good approximation) spherically symmetrical but the solar system obviously is not.Orbitals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?

- #5

A. Neumaier

Science Advisor

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The reason is that the named orbitals commonly drawn are basis-dependent. It is like drawings of a circle from different (basis-dependent) perspectives, which produces unsymmetric ellipses.Orbitals, other than s-orbitals, don't have spherical symmetry while the atomic Hamiltonian does have spherical symmetry. How is this possible?

The general orbital is quite arbitrary, and the set of all orbitals has spherical symmetry.

- #6

DrClaude

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$$

\sum_{m=-l}^l \left| Y_{l,m} (\theta, \phi) \right|^2

$$

is spherically symmetric.

- #7

hokhani

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There is no preferred z-direction and the shape of orbitals helps to determine the dynamics of electrons.

- #8

DrClaude

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What is the source of atoms?

There is no preferred z-direction and the shape of orbitals helps to determine the dynamics of electrons.

If the source is a thermal one, then there will be an equal probability of the atom being in all states of the same energy, as is the case for states that only differ in the magnetic quantum number. In that case, the electronic distribution is isotropic, see my post #6 above.

If the atom is in a specific state, as in your example above, then there was some state preparation (or selection) and there is no requirement for an isotropic distribution, since the preparation process can break the symmetry. In that case, the atom is said to be polarized, and there

- #9

HomogenousCow

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