This document discusses the radial and angular parts of the hydrogenic wave functions for the 1s, 2s, 2p, 3s, 3p and 3d orbitals. It explains that the radial component, R(r), gives the distribution of the electron as a function of radius r from the nucleus. The angular component, Y(θ,φ), gives the distribution as a function of the angles θ and φ. It describes how R(r) depends on the principal and azimuthal quantum numbers, while Y(θ,φ) depends on the azimuthal and magnetic quantum numbers. Graphs are presented showing the radial variation of the wave functions for different orbitals.

1. Radial and angular parts of the hydrogenic wave
functions, variations for 1s, 2s, 2p, 3s, 3p and 3d
orbitals ?
Mr. Mithil Fal Desai, Ph. D.
Assistant Professor in Chemistry
Shree Mallikarjun and Shri Chetan Manju Desai College
Canacona, Goa

2. 𝐝 𝟐
𝛙
𝐝𝐱 𝟐
+
𝐝 𝟐
𝛙
𝐝𝐲 𝟐
+
𝐝 𝟐
𝛙
𝐝𝐳 𝟐
+
𝟖𝛑 𝟐
𝐦
𝐡 𝟐
(𝐄 − 𝐕) = 𝟎
-----------1
Schrodinger equation with Cartesian coordinates

3. Cartesian coordinates and Polar coordinates
θ
φ r
(x, y, z) or (r, θ, φ)
x axis
zaxis
x = r sin θ cos ϕ
y = r sin θ sin ϕ
z = r cos θ
m = μ

4. 𝟏
𝐫 𝟐
𝐝
𝐝𝐫
(𝐫 𝟐
𝐝𝛙
𝐝𝐫
) +
𝟏
𝐫 𝟐 𝐬𝐢𝐧 𝛉
𝐝
𝐝𝛉
(𝐬𝐢𝐧 𝛉
𝐝𝛙
𝐝𝛉
) +
𝟏
𝐫 𝟐 𝐬𝐢𝐧 𝟐 𝛉
𝐝 𝟐 𝛙 𝟐
𝐝𝛟 𝟐
+
𝟖𝛑 𝟐 𝛍
𝐡 𝟐
(𝐄 − 𝐕) = 𝟎
-----------2
𝛙(r, 𝛉, 𝛟) = R(r), Θ(𝛉), Φ(𝛟)
𝛙(r, 𝛉, 𝛟) = R(r), Y(𝛉, 𝛟)
Schrodinger equation with polar coordinates
We have,

5. 𝛙(r, 𝛉, 𝛟) = R(r), Y(𝛉, 𝛟)
𝛙n𝐥𝐦𝐥 = Rnl(r), Ylm(𝛉, 𝛟)
Schrodinger equation with polar coordinates
We have,

6. Radial component ‘R(r)’ of wave function ′𝛙’ gives the distribution of
electron as a function of radius ‘r’(distance from the nucleus)
Radial wave function = R(r)
Radial component of wave function
Radial wave function depends on principle quantum number ‘n’ and
azimuthal quantum number ‘l’ and have a common function.

7. Angular component ‘Y(𝛉, 𝛟)’ of wave function ′𝛙’ gives the distribution of
electron as a function of angle (𝛉, 𝛟).
Angular component of wave function = Y(𝛉, 𝛟)
Angular component of wave function
Angular component depends on azimuthal quantum number ‘l’ and
magnetic quantum number.

8. 8
http://web.pdx.edu/~pmo
eck/lectures/modern/TRM
-7.ppt
Wave function
of Hydrogen
atom

9. Radial variation of wave function
Plot the graph of R(r) v/s r for
1s, 2s, 3s, 3p and 3d orbital

10. Angular variation of atomic orbital