# Laplacian in spherical coordinates

• BRN
In summary: The author solves a Schrodinger equation for nuclear systems in spherical coordinates and finds that there is an equation independent of angular coordinates. He then calculates the total energy of the system using different equations. One is using the Laplace operator and the other is using the triangledown.
BRN

## Homework Statement

Hello at all!

I have to calculate total energy for a nucleons system by equation:

##E_{tot}=\frac{1}{2}\sum_j(t_{jj}+\epsilon_j)##

with ##\epsilon_j## eigenvalues and:

##t_{jj}=\int \psi_j^*(\frac{\hbar^2}{2m}\triangledown^2)\psi_j dr##

My question is: if I'm in spherical coordinates, should I use Laplacian defined by:

##\triangledown^2=\frac{1}{r}\frac{\partial^2 }{\partial r^2}r-\frac{l^2}{r^2}##

or

##\triangledown^2=\frac{1}{r}\frac{\partial^2 }{\partial r^2}r## ?

## The Attempt at a Solution

At the moment I try to use

##\triangledown^2=\frac{1}{r}\frac{\partial^2 }{\partial r^2}r##

I'm not sure it's the right solution...

Could someone help me?

Thanks!

The Laplace operator in spherical coordinates is
$$\nabla^2 = \frac{1}{r^2}\partial_r r^2 \partial_r + \frac{1}{r^2\sin(\theta)}\partial_\theta \sin(\theta) \partial_\theta + \frac{1}{r^2\sin^2(\theta)}\partial_\varphi^2.$$
If it is acting on an eigenfunction of the angular momentum operator then the angular part can be replaced by ##-\ell(\ell+1)/r^2##. It is not clear from your post whether this is the case or not. Please give the full problem statement exactly as given as per forum guidelines.

I have not a exercise text to post because it is not an exercise, but part of a work.
I try to explain better what I need to do.
I have created an Schrodinger equation solver for nuclear systems in spherical coordinates. In this case, spherical symmetry leads to an equation independent of angular coordinates and my Schrodinger equation depends only on radial coordinate:

##[-\frac{\hbar^2}{2m}\triangledown^2+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)]\psi(r)=E\psi (r)##

Now, I need to calculate a total energy of the system by equations:

##E_{tot}=\frac{1}{2}\sum_j(t_{jj}+\epsilon_j)##

##t_{jj}=\int \psi_j^*(\frac{\hbar^2}{2m}\triangledown^2)\psi_j dr##

I'm not sure if I have to use Laplacian definition, in spherical coordinates, as:

##\triangledown^2=\frac{1}{r}\frac{\partial^2 }{\partial r^2}r-\frac{l^2}{r^2}##

or as:

##\triangledown^2=\frac{1}{r}\frac{\partial^2 }{\partial r^2}r##

I hope I explained myself better.

Thaks!

## 1. What is the Laplacian in spherical coordinates?

The Laplacian in spherical coordinates, also known as the spherical Laplacian, is a mathematical operator used to describe the curvature and shape of a three-dimensional space.

## 2. How is the Laplacian in spherical coordinates calculated?

The Laplacian in spherical coordinates is calculated by taking the second derivative of a function with respect to the radial distance, theta angle, and phi angle.

## 3. What is the significance of the Laplacian in spherical coordinates?

The Laplacian in spherical coordinates is important in many fields of science, including physics, engineering, and mathematics. It is used to solve differential equations and describe physical phenomena such as heat flow, electrostatics, and fluid dynamics.

## 4. Are there any alternative coordinate systems where the Laplacian can be expressed?

Yes, the Laplacian can also be expressed in cylindrical coordinates and Cartesian coordinates. Each coordinate system has its own unique form of the Laplacian operator.

## 5. How is the Laplacian in spherical coordinates related to other mathematical concepts?

The Laplacian in spherical coordinates is related to other mathematical concepts such as gradient, divergence, and curl. These concepts are used to describe the behavior of vector fields in three-dimensional space.

• Introductory Physics Homework Help
Replies
4
Views
362
• Introductory Physics Homework Help
Replies
16
Views
447
Replies
9
Views
2K
• Introductory Physics Homework Help
Replies
30
Views
764
• Introductory Physics Homework Help
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
3
Views
941
• Introductory Physics Homework Help
Replies
17
Views
658
• Calculus
Replies
3
Views
1K
• Introductory Physics Homework Help
Replies
9
Views
2K