The Schrödinger Wave Equation

 The Schrödinger Wave Equation

By Sharanya Garg

Physics Club, AISN

Introduction

Imagine watching the world’s smallest game of billiards — except instead of balls bouncing

predictably, they sometimes act like waves, sometimes like particles, and sometimes like both at

once.

This is the quantum world.

Before 1900, scientists thought they had figured out physics. Newton’s laws explained how

planets move, Maxwell’s equations described light, and thermodynamics covered heat and

energy. However, strange experiments showed that some matter and light behaved in ways that

defied classical physics (based on laws written by Isaac

Newton).

Some instances include:

 Photoelectric effect (Einstein, 1905): Light knocks

electrons off metal surfaces — but only if the light’s

frequency is high enough, regardless of brightness.

 Double-slit experiment: Particles like electrons make

interference patterns — something only waves should

do.

 Atomic spectra: Atoms absorb and emit light only at

specific colors, not a continuous range.

These questions led to the foundation of quantum mechanics

— the science of very small things, dealing with atoms,

electrons and subatomic particles. Here, classical ideas of fixed

orbits and trajectories are replaced by waves and probabilities. The Schrödinger wave

equation is the core mathematical rule of quantum mechanics. It explains the evolution

of wave function of a particle in space and time. The wave function Ψ(x,t) is

generally complex-valued, and its magnitude squared |Ψ|

2

gives the probability density of finding

the particle at each point. This probabilistic interpretation is a fundamental departure from

classical physics.

Note: Interference patterns are formed when two or more coherent waves overlap, resulting in a

distribution of amplitudes that can be either greater or smaller than the individual waves.

(Diagram of interference patterns shown in “Fig. 1”)

Erwin Schrödinger

Erwin Schrödinger, an Austrian physicist, introduced this equation in 1925 (publishing it in

1926). The idea of particles having a wave-like nature was proposed by Louis de Broglie in

1924 (with wavelength λ = h/p), and Schrödinger’s equation provided a precise framework for

this idea. Before Schrödinger, physicists like Niels Bohr had observed that electrons in atoms

could only occupy certain energy levels. Schrödinger showed that these energy levels are exactly

the allowed solutions of his wave equation. He applied his equation to the hydrogen atom and

found that it predicted the same discrete energy levels that Bohr had observed. This successful

Fig. 1

result helped to establish wave mechanics as the basis of quantum theory. It ultimately earned

Schrödinger the Nobel Prize in Physics in 1933.

The Schrödinger Wave Equation

The Schrödinger equation is a partial differential equation (equations involving multiple

independent variables and their partial derivatives). In the most general (time-dependent — the

equation describes how the quantum state of a system evolves over time) form it is written as:

where ħ (h-bar) is the reduced Planck constant and H^ is the Hamiltonian operator representing

the total energy of the system. For a single particle of mass mm in one dimension, where V(x) is

the potential energy.

In that case the equation becomes:

This resembles a classical wave equation but with the imaginary unit ι. The solutions Ψ can

oscillate and carry phase, which is crucial for quantum interference. Importantly, H is Hermitian,

ensuring that the allowed energies EE are real and different eigenstates are orthogonal. When the

energy is fixed, one often separates time. The time-independent Schrödinger equation is then:

Here ‘E’ is the energy eigenvalue and ψ(x)\psi(x) is a spatial wave function (standing wave) that

does not depend on time. Solving H^ψ=Eψ gives discrete energy levels and their associated

wave shapes.

a) What Does This Mean?

If that equation looks intimidating, here’s the core idea:

The Schrödinger equation is like a “recipe” for predicting the future of a quantum particle.

You give it:

1. The particle’s current wave function (Ψ)

2. The energies involved

And it tells you:

 What the wave function will look like later

 Which energies the particle can have

 Where it is more or less likely to be found

Wave Functions

The wave function Ψ is not a physical wave in space, like water or sound.

Instead, it’s a probability wave.

The square of the wave function’s absolute value:

tells you the probability density — where the particle is likely to be. This is called the “Born’s

rule”

Thus |Ψ|

2 tells where the particle is likely to be found. The wave function must be normalized so

Energy eigenstates for a particle in

finite potential well

that the total probability

Schrödinger’s equation is unitary, meaning it conserves this probability over time. Because Ψ is

complex, different parts of the wave can interfere: when two waves overlap, they can add or

cancel, altering the probability distribution.

Figure: Probability density plots (|Ψ|^2) of an electron in various hydrogen atom orbitals.

Schrödinger’s equation predicts these wave shapes. Brighter regions indicate higher probability

of finding the electron.

For example, solving Schrödinger’s equation for the hydrogen atom yields atomic orbitals. The

figure above shows the electron clouds for several low-energy states. The lowest state (n=1) is

spherical around the nucleus. Higher states (n=2,3,...) have ring or lobed patterns around the

nucleus. These shapes correspond to the labels s, p, d, etc. used in chemistry. The agreement of

these predicted orbital shapes with experiments (spectroscopy, electron imaging) confirms the

validity of the theory.

Diagram: Probability Wave

Probability

High peaks → more likely to find the particle

there.

Low valleys → less likely.

How Schrödinger Found the Equation

Schrödinger combined three big ideas:

1. De Broglie hypothesis: Matter has wavelength

λ=hp

where h = Planck’s constant, p = momentum.

2. Classical energy equation:

E=K+U=p

2

2m+U(x)E = K + U

3. Wave mathematics: In wave physics, the energy relates to the curvature of the wave.

By replacing energy (E) and momentum (p) with quantum “operators” and applying them to a

wave function, Schrödinger ended up with his famous equation.

Example: The Particle in a Box

Imagine an electron trapped in a one-dimensional box (like a tiny hallway) with perfectly rigid

walls.

Classically, it could have any energy and move anywhere inside.

Quantum mechanically, the Schrödinger equation says:

 The electron can only have specific, discrete energies (quantization).

 The wavefunction must have nodes (points where it is exactly zero) at the walls.

Consider a particle in a one-dimensional box of width L and perfectly rigid walls. The

particle is trapped between x=0 and x=Lx=L by walls of infinite potential. This means

V(x)=0 for 0<x<L and V(x)=∞ at x=0 and x=Lx=L. The wave function must vanish at

the walls. Inside the box (V=0), Schrödinger’s equation becomes:

with ψ(0)=ψ(L)=0

The solutions are

for n=1,2,3,... Each solution has n−1 internal nodes and an energy

Only these discrete energies E1,E2,... are allowed (quantization). Even the lowest state (n=1) has

nonzero energy (the so-called zero-point energy). Schrödinger’s equation also describes

scattering: a particle encountering a potential step or barrier will have its wave partly reflected

and partly transmitted. This explains quantum reflection and tunneling phenomena.

Figure: One-dimensional infinite potential well

(blue) with walls at x=0x=0 and x=Lx=L.

Schrödinger’s equation yields standing-wave

solutions ψn(x) inside the well. Outside (where

V=∞) the wave function is zero.

Classically, a particle in a box could have any energy and be equally likely anywhere. Quantum

mechanically, only certain energies are possible, and the probability ∣ψn(x)∣2| is wavy. For n=1,

the particle is most likely near the center, not the edges. All these results follow directly from

Schrödinger’s wave equation.

The Time-Dependent Equation: How Things Change

If you want to know how a quantum system evolves over time (not just its stationary states), you

use the time-dependent Schrödinger equation. This equation is like a “movie” version of

quantum mechanics — it shows the wave function’s changes, interference patterns, and

spreading.

The Uncertainty Principle and Schrödinger’s Equation

Werner Heisenberg’s uncertainty principle says:

This is built into the Schrödinger equation — wave functions can never be perfectly localized in

both position and momentum.

If you squeeze a wave in space (better position), its wavelength becomes uncertain (worse

momentum).

Interpretation and Significance

Schrödinger’s equation is often called the quantum analog of Newton’s laws. In classical

mechanics, Newton’s second law F=ma predicts how forces cause motion. In quantum

mechanics, Schrödinger’s equation predicts how the probability wave of a particle evolves in

time under a given energy (potential).

It unifies energy conservation with wave behavior. (This equation applies to non-relativistic

particles; for relativistic particles one uses different equations, but Schrödinger’s equation

remains the foundation at everyday energies.)

Because Schrödinger’s equation is linear, superposition (the ability of certain physical systems

to be in multiple states simultaneously) is possible. This means two or more solutions can be

added to form another valid solution. Physically, this means a particle can be in multiple states at

once, and these states can interfere. It also implies the Heisenberg uncertainty principle: a

particle cannot have both a sharply defined position and momentum, since confining the wave in

space increases its spread in momentum (Δx Δp≳ħ/2).

Another key feature is quantum tunneling (a phenomenon where particles can pass through

potential energy barriers that would be impossible to overcome in classical mechanics). The

wave function can extend into regions that are classically forbidden. For example, if a particle

encounters an energy barrier, its wave decays inside the barrier and yet there is a probability it

appears on the other side. This finite chance of “tunneling” through the barrier has no classical

counterpart. It explains nuclear fusion in the Sun and electron tunneling in devices like tunnel

diodes and scanning tunneling microscopes.

Schrödinger’s equation also conserves total probability: the sum of probabilities of all outcomes

remains 1.

One can derive a continuity equation showing that probability is neither created nor destroyed. In

other words, the particle never disappears; it is always somewhere, even if uncertain. These

principles make Schrödinger’s equation a powerful and consistent description of quantum

systems.

All of Schrödinger’s predictions have been confirmed by experiments. The energy levels of

hydrogen and other atoms, the patterns in electron diffraction, and the behavior of quantum

devices all match Schrödinger’s theory. Modern technologies like lasers, transistors and

magnetic resonance imaging (MRI) rely on its principles. Even the famous double-slit

experiment (electrons forming an interference pattern) can be explained by the wave function

evolving according to Schrödinger’s equation.

Real-Life Applications

Schrödinger’s equation underpins virtually all of modern physics and chemistry. Schrödinger’s

wave equation isn’t just theory — it powers much of modern technology:

 Materials and Electronics: Computers, smartphones → It describes electrons in solids.

The band structure of semiconductors and the conductivity of metals and insulators come

from quantum solutions. Designing transistors, lasers, solar cells and other devices relies

on this theory.

 Lasers and Photonics: Communication, surgery → Atoms emit or absorb light only at

specific frequencies corresponding to allowed energy transitions. Schrödinger’s equation

tells us these frequencies. Engineers use these principles in designing lasers, LEDs and

optical fibers.

 Other Fields: Quantum physics of nuclei → Quantum effects appear in chemistry,

biology and beyond. For example, electron tunneling in enzymes affects biochemistry.

Magnetic resonance imaging (MRI) exploits nuclear spin states explained by quantum

theory.

 Atoms and Molecules (Quantum Chemistry): Predicting molecular behavior → It

predicts electron orbitals and energy levels, explaining the periodic table and chemical

bonds. Quantum chemistry methods solve (approximately) this equation for molecules

and reactions

 Nuclear and Astrophysics: Understanding fusion and fission → while nuclei require

additional nuclear forces, many nuclear energy levels can be modeled by Schrödinger-

like equations. In astrophysics, quantum phenomena (like degeneracy pressure in white

dwarfs and tunneling in stellar fusion) depend on these principles.

 Nanotechnology and Quantum Computing: In nanoscale devices (quantum dots,

nanowires) and in quantum computers, quantum effects dominate. Schrödinger’s

equation is used to calculate energy levels and qubit states in these systems.

 Computational Quantum Science: Most realistic systems (multi-atom molecules,

materials) cannot be solved exactly by hand. Scientists use computers and approximation

methods (like density functional theory) to solve Schrödinger’s equation numerically.

This allows prediction of material properties and chemical reactions from first principles.

In fact, Schrödinger’s equation is the foundation of all atomic-scale science. Any process that

involves electrons, atoms or subatomic particles ultimately relies on it.

Why It’s Mind-Blowing

1. Probabilities instead of certainties: The wave equation doesn’t tell you exactly where a

particle is — only where it’s likely to be.

2. Quantization: Energy comes in discrete chunks, not a smooth range.

3. Wave-particle duality: Matter is both particle-like and wave-like.

4. Universality: This equation works for electrons, atoms, and even entire molecules.

Schrödinger’s Cat: The Thought Experiment

Schrödinger imagined a cat in a sealed box with a device triggered by a quantum event (like

radioactive decay).

Until you open the box, the cat is in a superposition — alive and dead at the same time.

This isn’t about cats — it’s about how wave functions apply to big objects, and how

measurement changes things.

Diagram: Schrödinger’s Cat

Limitations of Schrödinger’s Equation

 It’s non-relativistic — doesn’t handle speeds close to light (you need Dirac’s equation

for that).

 It doesn’t include particle creation/annihilation (quantum field theory does).

 It requires knowing the potential energy in advance.

Conclusion

Schrödinger’s wave equation revolutionized physics by showing that matter behaves as

probability waves. It implies particles have only certain allowed energy levels and characteristic

wave shapes. Its consequences—quantized energies, atomic orbitals, tunneling and

interference—have all been observed. Although mathematically abstract, the equation provides

precise guidance for understanding atoms, molecules and materials. By solving Schrödinger’s

equation (often with the help of computers), scientists can accurately predict a vast range of

phenomena and engineer new quantum technologies. In this way, Schrödinger’s equation serves

as a foundation of modern quantum science.