The presentation opens up by introducing Schrodinger's time dependent and independent wave equation. Then it covers the derivation of time independent wave equation, followed by its applications.

1. Schrödinger’s Time
Independent Wave Equation

2. SCHRÖDINGER’S WAVE EQUATION
Schrödinger’s Time Dependent
Equation
 In most situations, potential
energy is a function of position as
well as time.
 (-
ℏ𝟐
𝟐𝒎
𝛁𝟐 + 𝑽)𝜳(𝒙, 𝒚, 𝒛, 𝒕) = 𝒊ℏ
𝜕𝛹
𝜕𝑡
Schrödinger’s Time Independent
Equation
 In several situations, potential energy
is a function of position alone and
doesn’t depend upon time.
Schrödinger’s time independent
equation is also known as the steady
state equation.
 (-
ℏ𝟐
𝟐𝒎
𝛁𝟐
+ 𝑽)𝜳(𝒙, 𝒚, 𝒛) = 𝑬𝜳(𝒙, 𝒚, 𝒛)
Schrödinger’s equation is a wave equation in terms of the variable 𝛹 and
represents the fundamental equation in quantum mechanics.

3. DERIVATION
Consider a system of stationary waves associated with a particle
moving with velocity, ‘v’ and 𝛹 = 𝛹 (x, y, z, t) be the wave function.
The differential equation of the 3-D wave motion is given as:
𝜕2
𝛹
𝜕𝑡2
= 𝑣2
𝜕2
𝛹
𝜕𝑥2
+
𝜕2
𝛹
𝜕𝑦
2 +
𝜕2
𝛹
𝜕𝑧2
= 𝑣2∇2𝛹
The solution of the above equation is of the form:
𝛹 = 𝛹0 sin 𝜔𝑡 = 𝛹0 sin 2𝜋𝜃𝑡
⇒
𝜕𝛹
𝜕𝑡
= 𝛹0 2𝜋𝜃 cos 2𝜋𝜃𝑡

4. 𝜕2𝛹
𝜕𝑡2 = −𝛹0 2𝜋𝜃 2
sin 2𝜋𝜃𝑡 = −4𝜋2
𝜃2
𝛹 = −4𝜋2 𝜈2
𝜆2 𝛹
On equating equations (1) and (3), we get,
𝑣2
∇2
𝛹 = −4𝜋2 𝜈2
𝜆2 𝛹 ⇒ 𝑣2
∇2
𝛹 + 4𝜋2 𝜈2
𝜆2 𝛹 = 0
∇2
𝛹 +
4𝜋2
𝜆2 𝛹 = 0 ⇒ ∇2
𝛹 + 4𝜋2 𝑚2𝜈2
ℎ2 𝛹 = 0 {From 𝜆 =
ℎ
𝑚𝑣
}
Now, E =
1
2
𝑚𝜈2
+ 𝑉 ⇒
1
2
𝑚𝜈2
= E-V ⇒ 𝑚2
𝜈2
= 2m(E-V)
Substituting for 𝑚2
𝜈2
from equation (5) into (4), we have,
∇2𝛹 +
4𝜋2
ℎ2 2𝑚(𝐸 − 𝑉)𝛹 = 0 ⇒ ∇2𝛹 +
8m𝜋2
ℎ2 (𝐸 − 𝑉)𝛹 = 0

5. ⇒ ∇2
𝛹 +
2m
ℏ2 (𝐸 − 𝑉)𝛹 = 0
⇒ -
ℏ2
2𝑚
∇2𝛹 + (𝑉 − 𝐸)𝛹 = 0
⇒ (-
ℏ𝟐
𝟐𝒎
𝛁𝟐
+ 𝑽)𝜳 = 𝑬𝜳
OR 𝑯𝜳 = 𝑬𝜳
Where 𝑯 is Hamiltonian operator.
For a free particle, V = 0 so the above Schrodinger equation
reduces to:
-
ℏ𝟐
𝟐𝒎
𝛁𝟐
𝜳 = 𝑬𝜳
⇒ ∇2𝛹 +
2mE
ℏ2 𝛹 = 0 {For a free particle}

6. APPLICATION OF
SCHRÖDINGER’S TIME
INDEPENDENT WAVE
EQUATION

7. 1. Particle in a box/ Infinite potential well:
Using Schrödinger’s Time Independent
Equation, we get the following wavefunction and eigen
values for this case,
𝛹n(x)=
2
𝑎
sin
𝑛𝜋𝑥
𝑎
where n = 1,2,3,…
𝐸𝑛 =
𝑛2𝜋2ℏ2
2𝑚𝑎2

8. 2. Quantum Harmonic Oscillator:
Using Schrödinger’s Time Independent
Equation, we get the following wavefunction and eigen values for
this case,
𝐸𝑛 = 𝑛 +
1
2
ℏ𝜔

9. 3. Quantum Mechanical Tunnelling Effect:
• Quantum mechanically, if E > 𝑉0 , then there is
always some probability of reflection at x = 0 and
at x = a.
• Also, if E < 𝑉0 , then there is always some
probability of penetration into the barrier.
• In this case we apply Schrödinger’s Time
Independent
• Equation to understand quantum mechanical
tunnelling effect.
V 𝒙 =
𝑽𝟎 𝟎 < 𝐱 < 𝐚
∞ 𝐞𝐥𝐬𝐞𝐰𝐡𝐞𝐫𝐞

10. MCQ 1: Which of these is the mathematical
representation of Hamiltonian operator?
1) -
ℏ𝟐
𝟐𝒎
∇ + 𝑽
2) -
ℏ𝟐
𝟐𝒎
𝛁𝟐
+ 𝑽
3) -
ℏ𝟐
𝟒𝒎
𝛁𝟐
+ 𝑽
4) -
ℏ𝟐
𝟒𝒎
∇ +𝑽

11. MCQ 2: Which of the following option is
true?
Statement 1: Schrödinger’s time independent equation is also known as the steady
state equation.
Statement 2: Schrödinger’s time dependent equation is also known as the steady
state equation.
1) Both the statements are true.
2) Only Statement 2 is true.
3) None of them is true.
4) Only Statement 1 is true.

12. THANK YOU