# Shapes of electron orbitals

The probability distribution of the position of the electron of a hydrogen atom is related to the following polar plots

Suppose the electron is excited from the ##1s## orbital to the ##2p_x## orbital. Does it make sense to talk about the ##2p_x## orbital having a dumbbell shape pointing in the ##x## direction since the ##z## axis can be pointing in any direction? Shouldn't all orbitals be spherical when we consider there is an equal probability for the ##z## axis to point in any direction?

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blue_leaf77
Homework Helper
Does it make sense to talk about the 2px2p_x orbital having a dumbbell shape pointing in the xx direction since the zz axis can be pointing in any direction?
What are shown in the picture above are actually the cross-section of ##|Y_{lm}(\theta,\phi)|^2## in the xz plane, so when viewed as a 3D distribution, the ##2p_x## orbital does not have a dumbbell shape.

DrClaude
Mentor
Suppose the electron is excited from the ##1s## orbital to the ##2p_x## orbital. Does it make sense to talk about the ##2p_x## orbital having a dumbbell shape pointing in the ##x## direction since the ##z## axis can be pointing in any direction?
Be careful that what you posted are pictures of the the spherical harmonics, which are complex functions, while ##p_x## is a real function, linear combination of spherical harmonics, and is not an eigenfunction of ##\hat{L}_z##.

Shouldn't all orbitals be spherical when we consider there is an equal probability for the ##z## axis to point in any direction?
No, you can have a polarized sample. At equilibrium, you should have on average an isotropic distribution, but one can prepare an atomic sample in a given ##l, m_l## state. Note that the preparation aspect is going to define a laboratory z axis: the choice of axes is no longer arbitrary.

Happiness
Must the spin angular momentum of an electron have a definite direction or could it be in a superposition of being in manydifferent directions? If so, how can we prepare such a superposition?

DrClaude
Mentor
Must the spin angular momentum of an electron have a definite direction or could it be in a superposition of being in manydifferent directions? If so, how can we prepare such a superposition?
It is always in some superposition. If it is aligned along z in some coordinate system, it is in a superposition in any coordinate system where z' is not colinear with z.

Happiness
It is always in some superposition. If it is aligned along z in some coordinate system, it is in a superposition in any coordinate system where z' is not colinear with z.
So there must always exist exactly one coordinate system in which the spin is aligned along its z axis?

DrClaude
Mentor
So there must always exist exactly one coordinate system in which the spin is aligned along its z axis?
Yes. In the presence of an external field, you also have to account for spin precession, so it is possible that that coordinate system is a rotating one.

Happiness
DrDu