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.