Laplacian in Polar Cooridinates

In summary, the conversation is about using the del or laplacian operator on a function in polar coordinates. The laplacian operator is defined as the sum of the second derivatives with respect to each coordinate. The potential for the problem is described as V = infinity outside the 3D box. The conversation also mentions using spherical coordinates and the ease of computing in cartesian coordinates.
  • #1
QuantumMech
16
0
I need to take the [itex]\nabla^2[/itex] of [itex]x^2+y^2+z^2[/itex]. This is how far I got

[tex]
\begin{gather*}
\nabla^2 = \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + \frac{1}{r^2}(\frac{1}{sin^2\theta}\frac{d^2}{d\Phi^2} + \frac{1}{sin\theta}\frac{d}{d\theta} sin\theta\frac{d}{d\theta})\\
\nabla^2(r^2sin^2\theta cos^2\Phi + r^2sin^2\theta sin^2\Phi + r^2cos^2\theta = \frac{1}{sin\theta} \frac{d}{d\theta}(sin\theta \frac{d}{d\theta}) + \frac{1}{sin^2\theta} \frac{d^2}{d\Phi^2})
\end{gather*}
[/tex]


Also, can degeneracy occur with n not in order? Like for a part. in 3D box can degeneracy occur for [tex]\Psi_{1,3,5}[/tex] [tex]\Psi_{5,3,1}[/tex] or do the n have to be next each other like [tex]\Psi_{1,2,1}[/tex] [tex]\Psi_{2,1,1}[/tex]?
 
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  • #2
Is that a spherical box?And how does the potential look like...?

Daniel.
 
  • #3
Im not sure. I just need to use the del operator on [tex]x^2+y^2+z^2[/tex].
 
  • #5
I mean del squared or laplacian.

Oh, for the 2nd question: a 3D box with V = infinity outside box.
 
Last edited:
  • #6
It's simple.

[tex] \nabla^{2}=\Delta=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}+\frac{\partial^{2}}{\partial z^{2}} [/tex]

Use it to differentiate what u had to ([itex] x^{2}+y^{2}+z^{2} [/itex]).

Daniel.
 
  • #7
Oh, but I mean using polar coordinates for [itex]x^2+y^2+z^2 = r [/itex].

Thanks for the the p chem help dextercioby.
 
  • #8
Nope,i think you mean spherical coordinates and

[tex] x^{2}+y^{2}+z^{2}=r^{2} [/tex]

Daniel.
 
  • #9
And one more thing:please take my advice and compute that in cartesian coordinates...It's easier.Maths should be made as easy as possible,here's an opportunity

Daniel.
 
  • #10
That's why I was so confused with the first post, why were we straying away from cartesian when the Laplacian operator is so easily used on the described fct?
 

What is the Laplacian in polar coordinates?

The Laplacian in polar coordinates is a mathematical operator that determines the rate of change of a function with respect to its spatial coordinates in a polar coordinate system. It is commonly denoted as ∇² or ∆ and is used in various fields such as physics, engineering, and mathematics.

How is the Laplacian in polar coordinates calculated?

The Laplacian in polar coordinates can be calculated using the formula ∇²f = 1/r ∂/∂r (r ∂f/∂r) + 1/(r² sin θ) ∂/∂θ (sin θ ∂f/∂θ) where r is the radial coordinate and θ is the angular coordinate. This formula is derived from the Cartesian form of the Laplacian operator and takes into account the unique geometry of a polar coordinate system.

What is the significance of the Laplacian in polar coordinates?

The Laplacian in polar coordinates is an important mathematical tool in many scientific and engineering applications. It is used to solve differential equations that describe physical phenomena in a polar coordinate system, such as heat flow, fluid flow, and electromagnetic fields. It also plays a crucial role in the study of harmonic functions and Laplace's equation.

What are the advantages of using the Laplacian in polar coordinates?

One advantage of using the Laplacian in polar coordinates is that it simplifies certain types of problems that have circular or rotational symmetry. This reduces the number of variables and makes the problem easier to solve. Additionally, the Laplacian in polar coordinates is used in conjunction with other coordinate systems to describe complex systems and phenomena.

Are there any real-world applications of the Laplacian in polar coordinates?

Yes, there are many real-world applications of the Laplacian in polar coordinates. Some examples include predicting the temperature distribution in a circular metal plate, calculating the electric field around a point charge, and analyzing the flow of air around a spinning propeller. It is also used in image processing and computer graphics to smooth out noise and enhance edges in images.

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