Radial component of del^2 in spherical coordinates? (again)

In summary, the question being discussed is about finding the radial component of the Laplacian operator in spherical polar coordinates for a 3D isotropic harmonic oscillator. The lecturer has written the expression for the operator and the student is trying to understand it. The conversation also includes a link to a resource for further clarification and a hint to use the product rule in solving the problem.
  • #1
philip041
107
0
I'm doing a question on a 3D isotropic harmonic oscillator. At one point I need to find write the radial component of del^2.

Lecturer has written

[tex]

\frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)

[/tex]

where the del^2 used to be in the set of equations.

Am I correct in saying the radial part of in spherical polar coordinates is just dr. Then del^2 would be dr^2? Well I'm not?

I had a look at http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html but it didn't explain anything more for me.

PS. I asked this question earlier but internet broke, (thanks virgin media), and I don't understand the answer given... https://www.physicsforums.com/showthread.php?p=2006618#post2006618
 
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  • #2
philip041 said:
I'm doing a question on a 3D isotropic harmonic oscillator. At one point I need to find write the radial component of del^2.

Lecturer has written

[tex]

\frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)

[/tex]

where the del^2 used to be in the set of equations.

Am I correct in saying the radial part of in spherical polar coordinates is just dr. Then del^2 would be dr^2? Well I'm not?

I had a look at http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html but it didn't explain anything more for me.

Hi philip041! :smile:

(have a theta: θ and a phi: φ and a curly d: ∂ :wink:)

isotropic means dθ = dφ = 0

so in http://hyperphysics.phy-astr.gsu.edu/hbase/sphc.html at http://hyperphysics.phy-astr.gsu.edu/hbase/lapl.html#c2, the only non-zero terms in the last line are ∂2f/∂r2 + 2/r ∂f/∂r,

which is the same as 1/r2 d/dr(r2d/dr) :smile:
 
Last edited by a moderator:
  • #3
Thanks,

With ∂^2f/∂r^2 + 2/r ∂f/∂r how come it doesn't go to ∂^2/∂r^2 -1/r^2?

Sorry I don't understand how you get to that final line you wrote.
 
  • #4
product rule

philip041 said:
With ∂^2f/∂r^2 + 2/r ∂f/∂r how come it doesn't go to ∂^2/∂r^2 -1/r^2?

uhhh? how did you get -1/r^2? :confused:

Hint: use the product rule to find d/dr(r2df/dr) :smile:
 
  • #5
sweet. i also found an explanation of this in zetilli appendix b of quantum mechanics if anyone should ever search for this thread...
 

1. What is the radial component of del^2 in spherical coordinates?

The radial component of del^2 in spherical coordinates is a mathematical concept that represents the second derivative of a function with respect to the radial distance in a three-dimensional spherical coordinate system. It is used in physics and engineering to calculate the rate of change of a physical quantity in relation to the radial distance from a point in space.

2. How is the radial component of del^2 calculated in spherical coordinates?

The radial component of del^2 in spherical coordinates is calculated using the formula:

del^2 f(r,θ,φ) = (1/r^2) ∂/∂r (r^2 ∂f/∂r)

where r is the radial distance, θ is the polar angle, and φ is the azimuthal angle. This formula is derived from the del operator in spherical coordinates and represents the curvature of the function in the radial direction.

3. What is the physical significance of the radial component of del^2?

The radial component of del^2 has physical significance in various fields of science and engineering. In fluid dynamics, it represents the rate of change of fluid velocity in the radial direction, and in electromagnetism, it represents the rate of change of electric or magnetic fields in relation to the radial distance. It is also used in solving differential equations and analyzing the behavior of physical systems in spherical coordinates.

4. How does the radial component of del^2 differ from the Cartesian coordinate system?

The radial component of del^2 differs from the Cartesian coordinate system in that it takes into account the spherical nature of space. In the Cartesian system, the del operator represents the derivative of a function in all three directions, while in spherical coordinates, the del operator is modified to account for the radial, polar, and azimuthal directions. This allows for a more accurate representation of physical systems with spherical symmetry.

5. Can the radial component of del^2 be used to solve real-world problems?

Yes, the radial component of del^2 can be used to solve real-world problems in various fields such as fluid dynamics, electromagnetism, and quantum mechanics. It is a fundamental concept in mathematics and physics and has many practical applications in analyzing and understanding physical phenomena. Its use in solving differential equations makes it an essential tool for scientists and engineers in problem-solving and modeling real-world systems.

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