Schrödinger Wave Equation and Derivation: Class 12 Physics Notes

Before delving into the Schrödinger equation and its definition, let’s dive into a contextual understanding of experimental evidence before it. That’s covered in Chapter 11 Physics, Class 12. 

Experimental Evidence from Hertz to Einstein 

• In 1887, Heinrich Hertz observed that ultraviolet light influenced the production of electric sparks and currents. 
• Following that, Philipp Lenard and Wilhelm Hallwachs showed that certain metals emitted electrons when illuminated by ultraviolet light. That’s what we know as the photoelectric effect. One critical observation here is that it depended critically on the light frequency having a threshold. Below this, no electrons could emerge, no matter how intense the light would be. 
• J.J. Thomson's experiments established the electron as a universal negatively charged particle. This was supported by Millikan’s precise charge measurements with the oil-drop experiment. 
• Albert Einstein’s 1905 explanation of the photoelectric effect introduced the idea of discrete photons, packets of energy hν (h is Planck's constant; ν (nu) is the frequency of the electromagnetic radiation in hertz (Hz)). That proved why electron emission depended on frequency, not intensity.

de Broglie’s Hypothesis

Louis de Broglie proposed in 1924 that if light can behave both as a wave and a particle, then electrons and other matter particles should also have wave-like properties. 

λ = h/p = h/mv

Here,

• λ (lambda) is the de Broglie wavelength in meters
• h is Planck's constant (6.626 × 10⁻³⁴ J·s)
• p is the momentum of the particle (kg·m/s)
• m is the mass of the particle in kilograms
• v is the velocity of the particle in m/s

Electron diffraction experiments later confirmed these matter waves, introducing the concept of wave-particle duality for matter. That could signify a fundamental symmetry in nature.
You would recall this from the Class 11 chemistry topic, Quantum Model of Atom. 

The Schrödinger wave equation tells us how particles behave at the atomic scale. Instead of telling us exactly where a particle is, it gives us probabilities where it could be. 

The name of this equation comes from the Austrian physicist, Erwin Schrödinger (1887–1961). He built upon de Broglie’s matter waves, formulated the wave equation that mathematically governs the behaviour and interactions of quantum particles such as electrons. 

Because of this equation, we can mark the departure from classical determinism by describing the evolution of the particle’s wave function over space and time.

The Schrödinger equation provides a probabilistic description of micro-particles. That is helpful in both physics and chemistry, when calculating atomic structures, chemical bonding, and the behaviour of particles at microscopic scales.

The Schrödinger equation arises from fusing three key physical pieces. 

• The properties of classical plane waves (oscillating functions representing waves)
• The conservation of energy principle applied within quantum contexts
• The de Broglie relation linking momentum and wavelength

Let’s see how. 

The concept of Schrödinger's wave equation will be more relatable when you recall the mechanical wave equation (y = A sin(kx - ωt)) and concepts like Simple Harmonic Motion. 

The Schrödinger wave equation tells us probabilities instead of physical displacement, which you learn in the displacement relation of a progressive wave. 

Earlier, you had also learnt that particles move in periodic and predictable movements when learning about Oscillations. The Schrödinger equation describes oscillatory behaviour. Instead of physical displacement, it tracks the oscillation of probability amplitudes. 

In quantum mechanics, energy conservation still holds true. The total energy of an isolated quantum system remains constant over time, even when particles exist in probability states. 

Additionally, you remember Newton’s Second Law, F=ma. It can provide the acceleration at any instant. Now, through integration, we can calculate the complete trajectory of classical objects. First, we integrate to find velocity, then the position over time.

Similarly, the Schrödinger equation provides the rate of change of the quantum wave function. With this, we can predict how the probability distributions of particle locations change over time. You can say that this equation serves as the quantum equivalent for describing motion.

Let us first consider a complex plane wave theory,

Ψ (x, t) = Aei (kx – wt)

We know that the Hamiltonian is

H = T + V

In this, V stands for potential energy,

T = Kinetic energy, and

H = total energy

This equation can be rewritten as:

E = p2 / 2m + V (x)

Let us take the derivatives,

∂Ψ / ∂t = iw Aei (kx – wt) = -iw Ψ (x, t)

∂2Ψ / ∂x2 = -k2 Aei (kx – wt) = -k2 Ψ (x, t)

We also know that,

P = 2 Πh / λ and k = 2 Π / λ

Here, λ = wavelength and k is the wavenumber.

Now we have,

K = p / h

So,

∂2Ψ / ∂x2 = -p2 / h2 Ψ (x, t)

After multiplying it with Ψ (x, t), we will get

EΨ (x, t) = p2 / 2m Ψ (x, t) + V (x) Ψ (x, t)

It can be written as,

EΨ (x, t) = -h2 / 2m ∂2Ψ / ∂x2 + V (x) Ψ (x, t)

We know, energy wave of the matter is,

E = hw,

Now we can say that,

EΨ (x, t) = hw / -iw Ψ (x, t)

After combining the right hand side, we will get,

ih ∂Ψ / ∂t = -h2 / 2m ∂2Ψ / ∂x2 + V (x) Ψ (x, t)

This is the Schrödinger Wave equation

Here are the notes and solutions aligned with the latest NCERT curriculum in 2025-26. 

1	Class 12 Maths NCERT Solutions
2	Class 12 Physics NCERT Solutions
3	Class 12 Chemistry NCERT Solutions
4	Class 12 Physics CBSE Notes
5	Class 12 Chemistry CBSE Notes
6	Class 12 Maths CBSE Notes
7	Class 12 Physics Question Papers
8	Class 12 Chemistry Question Papers
9	Class 12 Mathematics Question Papers