Vectors in spherical coordinates

[eoghan]

Hi! I'm studying the selection rules and the spectrum of one-electron atoms. In the textbook it is said: "It is convenient to introduce the spherical components of the vector $$\epsilon$$ which are given in terms of its Cartesian components by:

$$\epsilon_1=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)$$
$$\epsilon_0=\epsilon_z$$
$$\epsilon_-1=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)$$

Can you please explain me these expressions?
I thought that
$$\epsilon_1=sin\theta cos\phi$$
$$\epsilon_2=sin\theta sin\phi$$
$$\epsilon_3=cos\theta$$

so I can't understand the expressions given in the textbookP.s. $$\epsilon$$ is the polarization vector, so it's a unit vector

 

[kuruman]

Hi! I'm studying the selection rules and the spectrum of one-electron atoms. In the textbook it is said: "It is convenient to introduce the spherical components of the vector $$\epsilon$$ which are given in terms of its Cartesian components by:

$$\epsilon_1=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)$$
$$\epsilon_0=\epsilon_z$$
$$\epsilon_-1=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)$$

Can you please explain me these expressions?
I thought that
$$\epsilon_1=sin\theta cos\phi$$
$$\epsilon_2=sin\theta sin\phi$$
$$\epsilon_3=cos\theta$$

so I can't understand the expressions given in the textbook


P.s. $$\epsilon$$ is the polarization vector, so it's a unit vector

It looks like you are confused about notation and I don't blame you. Sometimes subscripts 1,2,3 are used respectively for x,y,z and sometimes not. The confusion arises when you consult different sources using differing notations. Let me recast the unit vectors as follows:

$$\epsilon_+=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)$$
$$\epsilon_0=\epsilon_z$$
$$\epsilon_-=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)$$

where

$$\epsilon_x=sin\theta cos\phi$$
$$\epsilon_y=sin\theta sin\phi$$
$$\epsilon_z=cos\theta$$

This should keep the meanings of the subscripts clear for you.

 

[eoghan]

Thank you!