 # Schrodinger’s Equation and Wave Functions

Erwin Schrodinger’s fundamental impact on the development of quantum theory.

Schrödinger's equation
An interference pattern
Differential equation
The discrete energy emission spectra
Quantum superposition
Schrödinger's Cat

## The Shortcomings of Classical Systems

Before quantum mechanics, things were a whole lot simpler. For example, Newton’s laws of motion were naively thought to be all we needed to – at least in theory – describe motion of any kind.

All physicists had to do was apply his second law of motion F = m × a linking the force on a body with its mass and acceleration, and they could answer pretty much any question one could think to ask about the world! Except, they couldn’t.

As soon as scientists zoomed in to view reality at the smallest possible scales, Newton’s laws fell apart and things became very weird indeed. In fact, things only started to make sense again when the theory and equations of quantum mechanics were employed. The core equation of this new theory, and the analogue of Newton’s second law, is known as ‘Schrödinger’s equation’.

## The Need for Uncertainty

It’s usual in classical mechanics to describe the state of a physical system using the quantities of position and momentum. If we know the initial conditions of such a system, we can in theory use Newton’s laws to work out the ‘dynamical evolution’ of said system at any later time.

In other words, it’s ‘deterministic’. It turns out that it’s nowhere near as easy to predict or understand the future evolution of quantum systems, and concepts of position and momentum are no longer the correct variables to use to describe them. Unfortunately, quantum objects don’t behave like tiny billiard balls, and sometimes it’s better to think of them as waves! To fully embrace the quantum way of thinking, we need to become comfortable with uncertainty and a fuzzy ‘probabilistic’ way of talking about the behavior of systems. How can such probabilities be calculated? All we need to know about quantum systems is contained in the solution to Schrödinger’s equation, known as a ’wave function’.

## The De Broglie Wavelength

In 1923, French physicist Louis de Broglie defined his ‘de Broglie wavelength’, a wavelength which all objects in quantum mechanics manifest. It’s related to Planck’s constant and the momentum ‘p’ of a particle through λ = h / p. Since in quantum theory, all matter exhibits wave-like behavior, such waves are also described as ‘matter waves’.

The de Broglie wavelength indicates the length scale at which wave-like properties are important. For example, if a particle is interacting with something significantly larger than its “wavelength”, then its wave-like properties won’t be noticeable!

His ideas were later proven correct, most notably by experiments demonstrating the diffraction – historically, a wave property! – of electrons and therefore their ‘wave-particle duality’. de Broglie arrived at his equation by combining Einstein’s famous energy equation E = mc^2 with the energy of a photon E = h × f, making the first theoretical hypothesis that matter too may demonstrate this incredibly strange wave-particle duality.

## The Double-slit Experiment Revisited & Wave-particle Duality

Evidence of de Broglie’s idea came with an updated version of the double-slit experiment, in which a beam of electrons was fired at the slits instead. If electrons only behaved like particles, you’d expect them to pile up in two straight lines behind both slits.

But what you actually see on the detector screen is an interference pattern! Diffraction and interference are two properties only possible with waves, so the presence of an interference pattern was direct proof that electrons must be behaving as waves!

Bizarrely, this experiment even worked when electrons were fired one by one! It was as if each electron passed through both slits simultaneously as a matter wave and interfered with itself as the constituent waves spread out again on the other side.

This experiment’s result is identical to what was seen all historically with a light source: a pattern of ‘bright’ – where electrons hit the detector screen most often – and ‘dark’ patches arranged horizontally.

## Introduction to Schrödinger's Equation

Inspired by de Broglie, Austrian physicist Erwin Schrödinger reasoned in 1925 that since classical wave systems like sound waves or waves on an oscillating string have well-established equations governing their motion over time and space, so too must matter waves have their own equation. The solution to such an equation is a ‘wave function’, which in theory would tell you everything important about your quantum system. For simple systems, like one particle moving in three dimensions, the equation can be written in its more complex time-dependent form where ‘V’ is the particle’s ‘potential energy’. If the potential energy of the system is constant, we can simplify things further and use the time-independent form. Schrödinger’s Equation is an example of a ‘differential equation’ in mathematics, and the methods used to solve it can be quite sophisticated. When a suitable solution is found (the wave function for a particular system, whether one particle or multiple), that’s where the real fun begins…

## How is Schrödinger's Equation Used?

The solution to Schrödinger’s Equation for a system is called its ‘wave function’, due to its similarity to classical equations describing waves. However, a quantum wave function doesn’t provide a precise location for your particle(s) at any time ‘t’ as in classical physics.

Instead, the wave function outputs an abstract value for every point in three-dimensional space. Schrödinger’s Equation contains four variables: time ‘t’, and three spatial coordinates ‘x’, ‘y’ and ‘z’.

Nobody knew how to interpret these values, until in 1926, Max Born conceived a profound ‘probabilistic interpretation’. He postulated that the squared absolute value of the wave function tells you the ‘probability density’ associated with finding a particle at a given position (x, y, z).

In quantum theory, we must surrender any notion of certainty and instead embrace a probability-based view of reality. We no longer know where a particle will be, but with Schrödinger’s Equation, we can be reasonably sure of where it’s most likely to be.

## Why Should We Trust Schrödinger's Equation?

Why is Schrödinger’s Equation worth trusting as the correct view of reality? After all, Richard Feynman once said that the equation – which wasn’t derived from anything previously known – “came out of the mind of Schrödinger only”.

Well, the equation has held its own in every single experiment carried out so far to test it. In fact, it’s one of the most successful equations in history, with predictions that have been verified countless times. Therefore, it has earned its place as the fundamental equation in quantum mechanics and the starting point for every quantum system we seek to describe.

One of the equation’s earliest successes was in helping to elucidate one of the phenomena that gave rise to quantum theory in the first place: the discrete energy emission spectra of the hydrogen atom. Amazingly, the solution to Schrödinger’s Equation when applied to a hydrogen atom exactly reproduced the same quantized energy levels theorized by Bohr in his quantum atomic model!

## Quantum Superposition

Imagine a particle in a box. Unlike a billiard ball, which exists at a scale large enough that it obeys the classical laws of physics very well, a quantum particle doesn’t have a clearly defined trajectory. Upon opening the box and looking inside, we will find the particle at a particular point, but as per Schrödinger’s Equation, we have no way of predicting in advance where this point will be. All we have are probabilities!

Another consideration: if we decide not to open the box and spot the particle in a particular location, then where is it? According to the maths and theory of Schrödinger’s Equation, the answer to that is that the particle exists in all the places we could have potentially seen it, simultaneously. This wacky idea that a particle can be said to be in several places at once is known as ‘quantum superposition’, and it was the inspiration for Schrödinger’s famous thought experiment involving a cat.

## The Schrödinger's Cat Thought Experiment

Beyond his equation, ‘Schrödinger’s Cat’ is the esteemed scientist’s second-best known claim to fame! He conceived this now household-name thought experiment as a critique of sorts, to demonstrate just how absurd and counterintuitive the idea of superposition as implied by quantum physics is. It goes as follows: imagine a cat in a steel chamber, next to a ‘Geiger counter’ containing a tiny amount of radioactive substance so small that perhaps it decays over the next second, and perhaps it doesn’t…

In this probabilistic arrangement, if the substance does decay, poison is released from the flask and the unfortunate cat is killed. So, according to quantum superposition, as long as we don’t look inside the chamber, the system evolves into a superposition state of radioactive atoms that have simultaneously decayed and not decayed. It follows from this logic that Schrödinger’s cat will be simultaneously dead and alive. Weird.

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