Question

Is there an intuitive interpretation of the shape of the angular momentum eigenstate?

Answer 1

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain physically some of the features of the eigenfunctions. In particular, he said that the reason why the spherical harmonics Y01Y10 (see the image below) is zero on the xy-plane is because the z-component of this state is zero, so intuitively the particle is not trying to rotate about the z-axis therefore it is extremely unlikely to find it around the xy-plane, while the rotation from the other two components of the angular momentum give it its lobe shape.



I thought this is quite intuitive and understandable, but when he switch to the plot of the Y02Y20 I was confused. Y02Y20 also has a zero z-component of angular momentum, and although it has a similar lobe shape like Y01Y10, it has a disk like thing on the xy-plane, which seemingly implies that the particle is rotating around the z-axis and contradicting the reasoning the professor gave for Y01Y10. I was thinking that although there is no z-axis rotation, the combined effect of the rotation about the other two axes can still lead to some sort of apparent motion on the xy-plane, but I don't think this is making any sense at all. (and after all I am not good at visualizing 3D objects...)

So what's wrong? Is there any intuitive way to understand why that disk is there (and the similar shape appear in some of the higher spherical harmonics)?



I am new to this whole subject of quantum mechanics, so please feel free to point out any mistake I made if there is any.

Thank you.

Answer 2

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain physically some of the features of the eigenfunctions. In particular, he said that the reason why the spherical harmonics Y01Y10 (see the image below) is zero on the xy-plane is because the z-component of this state is zero, so intuitively the particle is not trying to rotate about the z-axis therefore it is extremely unlikely to find it around the xy-plane, while the rotation from the other two components of the angular momentum give it its lobe shape.

I thought this is quite intuitive and understandable, but when he switch to the plot of the Y02Y20 I was confused. Y02Y20 also has a zero z-component of angular momentum, and although it has a similar lobe shape like Y01Y10, it has a disk like thing on the xy-plane, which seemingly implies that the particle is rotating around the z-axis and contradicting the reasoning the professor gave for Y01Y10. I was thinking that although there is no z-axis rotation, the combined effect of the rotation about the other two axes can still lead to some sort of apparent motion on the xy-plane, but I don't think this is making any sense at all. (and after all I am not good at visualizing 3D objects...)

So what's wrong? Is there any intuitive way to understand why that disk is there (and the similar shape appear in some of the higher spherical harmonics)?


I am new to this whole subject of quantum mechanics, so please feel free to point out any mistake I made if there is any.

Thank you