The time independent schrodinger wave equation

1. Time independent Schrodinger
wave equation
Dr. Mithil Fal Desai
Shree Mallikarjun and Shri Chetan Manju Desai
College Canacona Goa
Ĥ𝚿 = 𝑬𝚿

2. 𝛙 = wave function
𝐦 = mass
h = plank constant
E = total energy
V = potential energy
Schrodinger time independent wave equation
𝐝𝟐
𝛙
𝐝𝐱𝟐
+
𝐝𝟐
𝛙
𝐝𝐲𝟐
+
𝐝𝟐
𝛙
𝐝𝐳𝟐
+
𝟖𝛑𝟐
𝐦
𝐡𝟐
(𝐄 − 𝐕)𝛙 = 𝟎

3. Sin (0) = 0
Sin (90) = 1
Sin (x) = y
f(x) = y
Remember f(x)
-1.5
-1
-0.5
0
0.5
1
1.5
0 90 180 270 360 450 540 630 720 810 900
Sine wave

4. -1.5
-1
-0.5
0
0.5
1
1.5
0 90 180 270 360 450 540 630 720 810 900
sinx (dsinx/dx)=cosx d(cosx)/dx =-sinx
Understanding first and second derivative
Difficult?

5. Understanding first and second derivative
(easy way)
0
50
100
150
0 2 4 6
distance speed accelaration
Time
Distanc
e
speed acceleration
t x
dx/dt
=speed
d speed/dt=
acceleration
0 0 2 1
1 2 3 2
2 5 5 10
3 10 15 15
4 25 30 20
5 55 50 30
6 105 80 35
7 185 115
8 300

6. Schrodinger wave equation
𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒘𝒂𝒗𝒆 𝒏𝒖𝒎𝒃𝒆𝒓
𝒌 =
𝟐𝝅
λ
𝐝𝟐𝒇(𝒙)
𝐝𝐱𝟐 = −
𝟒𝝅𝟐
λ𝟐 𝒇 𝒙 …….1
A standing wave having wavelength (λ) that has an amplitude at any point
along x direction is mathematically described as a function f(x)

7. 𝐝𝟐ψ
𝐝𝐱𝟐 = −
𝟒𝝅𝟐
λ𝟐 ψ
--2
If a wave function f(x) is represented as 𝝍 (psi) the
equation can be written as
Schrodinger wave equation

8. 𝐝𝟐𝛙
𝐝𝐱𝟐 +
𝐝𝟐𝛙
𝐝𝐲𝟐 +
𝐝𝟐𝛙
𝐝𝐳𝟐 = −
𝟒𝝅𝟐
λ𝟐 ψ
--3
When this standing wave is considered in 3 dimensions
having x, y and z coordinates
Schrodinger wave equation

9. 𝛁𝟐
𝛙 = −
𝟒𝝅𝟐
λ𝟐
ψ
--4
Schrodinger wave equation
If,
𝐝𝟐𝛙
𝐝𝐱𝟐
+
𝐝𝟐𝛙
𝐝𝐲𝟐
+
𝐝𝟐𝛙
𝐝𝐳𝟐
= 𝛁𝟐
𝛙
𝛁 is called Del Operator

10. λ =
𝒉
𝒎𝒗
𝜆2
=
ℎ2
𝑚2𝑣2
--5
Schrodinger wave equation
𝒗 𝒊𝒔 𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚
de Broglie states that

11. 𝛁𝟐
𝝍 = −
𝟒𝝅𝟐
𝒉𝟐
𝒎𝟐𝒗𝟐
𝝍
𝜵𝝍 +
𝟒𝝅𝟐
𝒎𝟐
𝒗𝟐
𝒉𝟐
𝝍 = 𝟎
--6
Schrodinger wave equation
From equation 4 and 5

12. 𝑬 = 𝑲. 𝑬. + 𝑽
K. E.= 𝑬 − 𝑽
𝟏
𝟐
𝒎𝒗𝟐
= 𝑬 − 𝑽
𝒗𝟐
=
𝟐
𝒎
(𝑬 − 𝑽)
--7
Schrodinger wave equation
The total energy (E) of system is given as

13. 𝛁𝟐
𝝍 +
𝟒𝝅𝟐𝒎𝟐
𝒉𝟐
{
𝟐
𝒎
𝑬 − 𝑽 }𝝍 = 𝟎
𝛁𝟐
𝝍 +
𝟖𝝅𝟐
𝒎
𝒉𝟐
𝑬 − 𝑽 𝝍 = 𝟎
---8
Schrodinger wave equation
From equation 6 and 7
Time independent
Schrodinger wave equation

14. Schrodinger wave equation
general form
Ĥ= Hamiltonian operator,
E= Eigen value, specific values of energy
Ĥ𝚿 = 𝑬𝚿
Can you state Ĥ from equation 8?