Ψ represents the wave function in the Schrodinger wave equation and describes the state of an electron. Its square, Ψ2, gives the probability of finding the electron in a particular region of space. Acceptable solutions to the Schrodinger equation must have properties such as continuity, finiteness, and being single valued. The wave function and its square are represented graphically. Several wave functions can satisfy the Schrodinger equation, each with a corresponding energy level. These wave functions are called orbitals.

1. Significance of ψ and ψ2
Dr. Mithil Fal Desai
Shree Mallikarjun and Shri Chetan
Manju Desai College Canacona Goa
ψ ψ2

2. Ψ and Ψ2
Ψ (Psi) is a wave function represented in the
Schrodinger wave equation, which represents the
state of the electron. Its corresponding
probability density (Ψ2) gives the probability of
finding the electron in a particular region in
space.

3. Schrodinger equation
𝐝𝟐
𝛙
𝐝𝐱𝟐
+
𝐝𝟐
𝛙
𝐝𝐲𝟐
+
𝐝𝟐
𝛙
𝐝𝐳𝟐
+
𝟖𝛑𝟐
𝐦
𝐡𝟐
(𝐄 − 𝐕)𝛙 = 𝟎

4. Acceptable solutions to the
Schrodinger wave equation
must have certain properties
 Ψ must be continuous
 Ψ must be finite
 Ψ must be single valued
 0
∞
Ψ𝟐
= 1

5. Continuous and noncontinuous function
-1
-0.5
0
0.5
1
0 100 200 300 400
sin(θ)
-6
-4
-2
0
2
4
6
0 100 200 300 400
tan(θ)
Continuous
function
noncontinuous
function
θ θ

6. Finite and infinite function
A = {0, 1, 2, 3,4, 5} finite set
A = {0, 1, 2, 3,4, 5,……} infinite set
 𝑓(𝑥) = 𝑥 + −1𝐴 = 𝜋𝑟2
 𝑓(𝑥) =
𝑥
0
 𝑓(𝑥) = 𝑥2
×(∞)

7. -10
-5
0
5
10
0 20 40 60 80 100
More than one value for the function
y
x
At same value of x, we observe two different value of Y
𝑦2
= 4𝑝𝑥
27

8. Integrating function from a to b
Integrating any function gives area under the curve
formed by that function
𝑎
𝑏
𝑓𝑥 𝑑𝑥 = 𝑎𝑟𝑒𝑎
3
4
π
1
2
π
1
4
π π
0π
0.314 0.314 0.314 0.314
0.314 0.314 0.314 0.314
0.314 0.314
0
0.4
0.8
0
π
𝑠𝑖𝑛𝑥 𝑑𝑥 = 2
𝑓(𝑥) = 𝑠𝑖𝑛𝑥

9. Function and function square
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
0 100 200 300
sin(θ)
or
[sin(θ)]
2
θ

10. End note
Several wave function satisfy these properties and each of
these have corresponding energies. These wave functions
are called orbitals. In hydrogen atom the wave function
which has lowest energy level(E1) is 𝝍1.
There are number of acceptable solutions of the wave
function and each orbital is described by set of three
quantum numbers n, l and m.