Derivation of the Schrödinger Wave Equation

by Erwin Schrödinger

4

Schrödinger derived his wave equation (1926) by combining classical energy relations with wave–particle duality proposed by Louis de Broglie.

We’ll derive the time-dependent Schrödinger equation first, then obtain the time-independent form.

???? Step 1: Start from Classical Energy Relation

For a particle of mass ????m moving in potential ????V:

????=????22????+????E=2mp2?+V

Where:

• ????E = total energy

• ????p = momentum

• ????V = potential energy

???? Step 2: Use de Broglie Relations

From wave–particle duality:

????=?????p=?k????=?????E=?ω

Where:

• ?=?2?????=2πh?

• ????k = wave number

• ????ω = angular frequency

???? Step 3: Assume a Wave Function

Assume the particle behaves like a plane wave:

Ψ(????,????)=????????????(????????−????????)Ψ(x,t)=Aei(kx−ωt)

Now compute derivatives.

???? Step 4: Take Partial Derivatives

Time derivative:

∂Ψ∂????=−????????Ψ∂t∂Ψ?=−iωΨ

Multiply both sides by ?????i?:

?????∂Ψ∂????=?????Ψi?∂t∂Ψ?=?ωΨ

Since ????=?????E=?ω:

?????∂Ψ∂????=????Ψi?∂t∂Ψ?=EΨ

Second spatial derivative:

∂2Ψ∂????2=−????2Ψ∂x2∂2Ψ?=−k2Ψ

Multiply both sides by −?22????−2m?2?:

−?22????∂2Ψ∂????2=?2????22????Ψ−2m?2?∂x2∂2Ψ?=2m?2k2?Ψ

Since ????=?????p=?k:

?2????22????=????22????2m?2k2?=2mp2?

???? Step 5: Substitute into Energy Equation

From classical relation:

????=????22????+????E=2mp2?+V

Multiply by ΨΨ:

????Ψ=????22????Ψ+????ΨEΨ=2mp2?Ψ+VΨ

Substitute results from derivatives:

?????∂Ψ∂????=−?22????∂2Ψ∂????2+????Ψi?∂t∂Ψ?=−2m?2?∂x2∂2Ψ?+VΨ

? Final Form: Time-Dependent Schrödinger Equation

?????∂Ψ∂????=−?22????∇2Ψ+????Ψi?∂t∂Ψ?=−2m?2?∇2Ψ+VΨ?

(Here ∇2∇2 is the Laplacian operator in 3D.)

???? Time-Independent Schrödinger Equation

If potential ????V does not depend on time:

Assume:

Ψ(????,????)=????(????)????−????????????/?Ψ(x,t)=ψ(x)e−iEt/?

Substitute into time-dependent equation:

−?22????∇2????+????????=????????−2m?2?∇2ψ+Vψ=Eψ?

???? Physical Meaning

• ΨΨ = wave function

• ?Ψ?2?Ψ?2 = probability density

• Equation governs motion of microscopic particles

• It replaces Newton’s laws at atomic scale

???? Why This Equation Is Important

It forms the foundation of:

• Quantum mechanics

• Atomic structure

• Semiconductor physics

• Quantum computing