Vectors in spherical coordinates

In summary, the expressions given in the textbook are for describing the spherical components of the vector \epsilon.
  • #1
eoghan
207
7
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 [tex]\epsilon[/tex] which are given in terms of its Cartesian components by:

[tex]\epsilon_1=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)[/tex]
[tex]\epsilon_0=\epsilon_z[/tex]
[tex]\epsilon_-1=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)[/tex]

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

so I can't understand the expressions given in the textbookP.s. [tex]\epsilon[/tex] is the polarization vector, so it's a unit vector
 
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  • #2
eoghan said:
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 [tex]\epsilon[/tex] which are given in terms of its Cartesian components by:

[tex]\epsilon_1=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)[/tex]
[tex]\epsilon_0=\epsilon_z[/tex]
[tex]\epsilon_-1=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)[/tex]

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

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


P.s. [tex]\epsilon[/tex] 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:

[tex]\epsilon_+=-\frac{1}{\sqrt2}(\epsilon_x+i\epsilon_y)[/tex]
[tex]\epsilon_0=\epsilon_z[/tex]
[tex]\epsilon_-=-\frac{1}{\sqrt2}(\epsilon_x-i\epsilon_y)[/tex]

where

[tex]\epsilon_x=sin\theta cos\phi[/tex]
[tex]\epsilon_y=sin\theta sin\phi[/tex]
[tex]\epsilon_z=cos\theta[/tex]

This should keep the meanings of the subscripts clear for you.
 
  • #3
Thank you!
 

What are vectors in spherical coordinates?

Vectors in spherical coordinates are a way of representing a vector in three-dimensional space using spherical coordinates instead of the more commonly known Cartesian coordinates. This system is useful for solving problems involving spherical symmetry, such as those found in physics and engineering.

How are vectors expressed in spherical coordinates?

In spherical coordinates, vectors are expressed using three components: the radial distance (r), the inclination angle (θ), and the azimuthal angle (φ). These components represent the distance from the origin, the angle from the z-axis, and the angle from the x-axis, respectively.

What is the relationship between spherical and Cartesian coordinates?

Spherical coordinates can be converted to Cartesian coordinates using the following equations:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
Conversely, Cartesian coordinates can be converted to spherical coordinates using the inverse equations.

How do you add or subtract vectors in spherical coordinates?

To add or subtract vectors in spherical coordinates, you must first convert them to Cartesian coordinates, perform the operation, and then convert back to spherical coordinates. This is because the components of vectors in spherical coordinates are not directly additive.

What are some applications of vectors in spherical coordinates?

Vectors in spherical coordinates are used in various fields, including physics, engineering, and astronomy. They are particularly useful for solving problems involving spherical symmetry, such as calculating forces or velocities in a system with a central point of attraction.

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