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

[philip041]

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

<br /> <br /> \frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)<br /> <br />

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

 

[tiny-tim]

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

<br /> <br /> \frac{1}{r^{2}} \frac{d}{dr} \left( r^{2} \frac{d}{dr} \right)<br /> <br />

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!

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

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)

 

[philip041]

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.

 

[tiny-tim]

product rule

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?

Hint: use the product rule to find d/dr(r²df/dr)

 

[philip041]

sweet. i also found an explanation of this in zetilli appendix b of quantum mechanics if anyone should ever search for this thread...