Derivation of Schrödinger's Equation

Schrödinger's Equation in Quantum Physics

• Schrödinger's equation is one of the most important ideas in quantum physics. It shows how a quantum system changes over time.

I. Introduction

• Newton's rules, which show how an object's position, speed, and acceleration are related, describe how a particle moves in classical mechanics.
• When we get to the quantum world, though, the rules of traditional physics no longer hold true. Wave-particle duality is introduced by quantum physics.
• Particles can behave in both wave-like and particle-like ways. Schrödinger's equation gives us a new way to think about maths that lets us explain this behaviour.

II. Two Types of Particles and Waves

• Albert Einstein came up with the special theory of relativity in 1905. It went against the long-held belief that time and space are fixed.
• Later, in 1924, Louis de Broglie came up with the idea of wave-particle duality, which says that particles like electrons can behave like waves.
• Experiments, like the famous double-slit experiment, helped to prove this idea even more.

III. Classical Wave Equation

• We need to start with the classical wave equation in order to understand how Schrödinger's equation came to be.
• The classical wave equation shows how waves, like sound waves or light waves, move through a substance.
• It is written as:

∇²ψ = (1 / v²) × (∂²ψ / ∂t²)

• Where:
• ψ is the wave function
• v is the wave's speed
• ∇² is the Laplacian operator

IV. Quantization of Energy

• Quantum physics says that energy is quantized, which means that it comes in discrete packets (quanta) instead of being a constant stream.
• In classical physics, energy is thought to be continuous, so this is a big change.
• An equation that shows how energy is quantized is given below:

E = h × f

• Where:
• E is energy
• h is Planck's constant
• f is the frequency of the wave

V. Derivation of Schrödinger's Equation

• We need to put together the classical wave equation and the idea of quantized energy to get Schrödinger's equation.
• First, let's say that the wave function (ψ) depends on both time (t) and position (x).
• After that, we put the energy formula (E = h × f) into the classical wave equation:

(1 / v²) × ∇²ψ = ∂ψ / ∂t

• A new form of the energy equation that uses the de Broglie relation (p = h / λ) is:

E = p² / 2m

• This formula fits into the classical wave equation, giving us:

∇²ψ = (1 / v²) × (∂²ψ / ∂t²) = (2m / h²) × (E – V)ψ

• • Where:
    • V is the potential energy

VI. Time-Dependent Schrödinger Equation

• Introducing the idea of time dependence is the last step.
• There is a chance that the wave function (ψ) changes with both time (t) and place (x).
• The time-dependent Schrödinger equation is given by:

Hψ = iℏ × (∂ψ / ∂t)

• Where:
• i is the imaginary unit
• ℏ is the reduced Planck's constant (h / 2π)
• ψ is the wave function

VII. Time-Independent Schrödinger's Equation

• The potential energy (V) doesn't always change with time, and the wave function (ψ) can be split into:
• A spatial part: ψ(x)
• A time-dependent part: e^(-iEt/ℏ)
• The time-independent Schrödinger equation is given by:

Hψ(x) = Eψ(x)

• The energy eigenstates of the system are shown by this equation.

Explore Subject Categories

• StudySpot360 Home: https://studyspot360.com/

• Geology Study Materials: https://studyspot360.com/geology

• Physics Notes and Learning Resources: https://studyspot360.com/physics

• English Grammar and Study Materials: https://studyspot360.com/english

About StudySpot360

StudySpot360 is a trusted educational platform providing high-quality study materials, detailed notes, MCQs, quizzes, question papers, and academic resources across multiple subjects. Our mission is to make reliable educational content accessible to students, educators, researchers, and lifelong learners while supporting academic excellence, examination preparation, and continuous learning.

Explore more educational articles, tutorials, practice questions, quizzes, and subject-wise learning resources to strengthen your knowledge and continue your learning journey with StudySpot360.