Inner product of two spherical functions

In summary, the conversation discusses the inner product between two functions in spherical coordinates and the appropriate volume element to use. The solution involves using the Jacobian and converting the integral to spherical coordinates.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

The inner product between two functions f(x) and g(x) is defined as:

[tex]
<f | g> = \int f^*(x)g(x) dx,
[/tex]

where the * denotes the complex conjugate. Now if my functions f and g are functions of r, theta and phi (i.e. they are written in spherical coordinates), is the volume element then given as [itex]drd\theta d\phi[/itex] or [itex]r^2\sin \theta drd\theta d\phi[/itex]?

The Attempt at a Solution



I personally think the last, because originally we have <f(x,y,z) | g(x,y,z)>, which we convert to spherical coordinates, so we need the Jacobian. Am I correct?

Thanks in advance.
 
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  • #2
Putting [tex]h(x,y,z) = f^\ast(x,y,z)g(x,y,z)[/tex], we have

[tex]\int\int\int h(x,y,z) dz \, dy\, dx = \int\int\int h(r\cos{\theta}\sin{\phi}, r\sin{\theta}\sin{\phi}, r\cos{\phi}) r^2{\underbrace{\sin{\phi}}_{\text{not $\theta$}} dr \, d\theta\, d\phi[/tex]

with appropriate limits, as per usual. That is all your question appears to boil down to, Niles.
 
  • #3
Then I was correct. Thanks for replying!
 

Related to Inner product of two spherical functions

What is the concept of the inner product of two spherical functions?

The inner product of two spherical functions is a mathematical operation that measures the similarity or correlation between two spherical functions. It is a generalization of the dot product in Euclidean space and is used to analyze functions that are defined on a sphere.

What is the formula for calculating the inner product of two spherical functions?

The formula for calculating the inner product of two spherical functions is given by:



where and are the two spherical functions and and are the spherical coordinates.

What is the significance of the inner product of two spherical functions?

The inner product of two spherical functions is used to determine the orthogonality and completeness of a set of spherical functions. It is also used in various applications such as signal processing, image processing, and quantum mechanics.

How is the inner product of two spherical functions related to the spherical harmonics?

The inner product of two spherical functions is closely related to the spherical harmonics, which are a set of orthogonal functions on a sphere. The inner product of any two spherical harmonics is equal to 0 if they have different degrees or orders, and 1 if they have the same degree and order. This property makes spherical harmonics useful in expanding functions on a sphere.

Can the inner product of two spherical functions be negative?

Yes, the inner product of two spherical functions can be negative. This indicates that the two functions are not in phase or are out of phase with each other. The magnitude of the inner product represents the strength of correlation between the two functions, while the sign represents the phase relationship.

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