What Are the Polynomials for 1s and 2s Orbitals?

[terp.asessed]

Homework Statement

Show that L₁¹(x) and L¹₂(x) are precisely the polynomials for 1s and 2s orbitals. What is the role of variable x in each case?

Homework Equations

L¹_n(x) = d/dx L_n(x), n = 1, 2, 3...

The Attempt at a Solution

Because L₁(x) = 1 - x
L₂(x) = 2 - 4x + x²:
I did:
L¹₁(x) = d/dx L₁(x) = d/dx (1 - x) = -1
L¹₂(x) = d/dx L₂(x) = d/dx (2-4x+x²) = 2x - 4 = 2(x - 2)...I wonder if x is a different function of radius in 1s and 2s orbital polynomials? I am also confused as how to find polynomials for 1s and 2s orbitals? Could anyone hint pls?
For sure, I undersand that
L¹₁(x)= 1 because of 0 nodes...hence 1s
L¹₂(x) = 2x -4 because of 1 node...hence 2s

 

[mark726]

Thank you for your post. You are correct in your understanding that L11(x) and L12(x) are the polynomials for 1s and 2s orbitals, respectively. The variable x represents the distance from the nucleus, and its role in each case is to determine the shape and size of the orbital.

To find the polynomials for 1s and 2s orbitals, you can use the formula L1n(x) = d/dx Ln(x) for n = 1 and 2. This means that for the 1s orbital, you will take the derivative of L1(x) = 1 - x, which will give you L11(x) = -1. This is the polynomial for the 1s orbital because it has no nodes and is spherically symmetric.

For the 2s orbital, you will take the derivative of L2(x) = 2 - 4x + x^2, which will give you L12(x) = 2x - 4. This is the polynomial for the 2s orbital because it has one node and is not spherically symmetric.

I hope this helps clarify the role of x and how to find the polynomials for 1s and 2s orbitals. If you have any further questions, please don't hesitate to ask.