Here is a typical textbook question. Your car has run out of petrol. With how much force do you need to push it to accelerate it to a given speed?

The answer comes from Newton’s second law of motion:

where is acceleration, is force and is mass. This wonderfully straightforward, yet subtle law allows you to describe motion of all kinds and so it can, in theory at least, answer pretty much any question a physicist might want to ask about the world.

Schrödinger's equation is named after Erwin Schrödinger, 1887-1961.

Or can it? When people first started considering the world at the
smallest scales, for example electrons orbiting the nucleus of an
atom, they realised that things get very weird indeed and that Newton's laws no longer
apply. To describe this tiny world you need quantum mechanics,
a theory developed at the beginning of the twentieth century. The
core equation of this theory, the analogue of Newton's second law, is called
*Schrödinger's equation*.

### Waves and particles

"In classical mechanics we describe a state of a physical system using position and momentum," explains Nazim Bouatta, a theoretical physicist at the University of Cambridge. For example, if you’ve got a table full of moving billiard balls and you know the position and the momentum (that’s the mass times the velocity) of each ball at some time , then you know all there is to know about the system at that time : where everything is, where everything is going and how fast. "The kind of question we then ask is: if we know the initial conditions of a system, that is, we know the system at time what is the dynamical evolution of this system? And we use Newton’s second law for that. In quantum mechanics we ask the same question, but the answer is tricky because position and momentum are no longer the right variables to describe [the system]."

The problem is that the objects quantum mechanics tries to describe don't always behave like tiny little billiard balls. Sometimes it is better to think of them as waves. "Take the example of light. Newton, apart from his work on gravity, was also interested in optics," says Bouatta. "According to Newton, light was described by particles. But then, after the work of many scientists, including the theoretical understanding provided by James Clerk Maxwell, we discovered that light was described by waves."

But in 1905 Einstein realised that the
wave picture wasn't entirely correct either. To explain the
*photoelectric effect* (see the *Plus* article Light's identity crisis)
you need to think of a beam of light as a stream of particles, which
Einstein dubbed *photons*. The number of photons is proportional to the intensity of the light, and the energy *E* of each photon is proportional to its frequency *f*:

The double slit experiment: The top picture shows the interference pattern created by waves passing though the slits, the middle picture shows what you'd expect to see when particles are fired through the slits, and the bottom picture shows what actually happens when you fire particles such as electrons through the slits: you get the interference pattern you expect from waves, but the electrons are registered as arriving as particles.

Einstein's result linked in with the age-old endeavour, started in the 17th century by Christiaan Huygens and explored again in the 19th century by William Hamilton:
to unify the physics of optics (which was all about waves) and mechanics (which was all about particles). Inspired by the schizophrenic behaviour of light the
young French physicist
Louis de
Broglie took a dramatic step in this journey: he postulated that not only light, but also matter
suffered from the so-called *wave-particle duality*. The tiny
building blocks of matter, such as electrons, also behave like particles
in some situations and like waves in others.

De Broglie's idea, which he announced in the 1920s, wasn't based on experimental evidence, rather it
sprung from theoretical considerations inspired by Einstein's theory of relativity. But experimental evidence
was soon to follow. In the late 1920s experiments involving particles
scattering off a crystal confirmed the wave-like nature of electrons (see the *Plus* article Quantum uncertainty).

One of the most famous demonstrations of wave-particle duality
is the *double slit experiment*. In it electrons (or other particles like photons or neutrons) are fired one
at a time all over a screen containing two slits. Behind the screen
there's a second one which can detect where the electrons that made it
through the slits end up. If the electrons behaved like particles,
then you would expect them to pile up around two straight lines behind the
two slits. But what you actually see on the detector screen is an
*interference pattern*: the pattern you would get if the electrons were waves,
each wave passing through both slits at once and then interfering
with itself as it spreads out again on the other side. Yet on the detector screen, the electrons are registered as arriving just as you would expect: as particles. It's a very weird result indeed but one that has been replicated many
times — we simply have to accept that this is the way the world
works.

### Schrödinger's equation

The radical new picture proposed by de Broglie required new physics. What does a wave associated to a particle look like mathematically? Einstein had already related the energy of a photon to the frequency of light, which in turn is related to the wavelength by the formula Here is the speed of light. Using results from relativity theory it is also possible to relate the energy of a photon to its momentum. Putting all this together gives the relationship

between the photon’s wavelength and momentum ( again is Planck’s constant).

(See Light's identity crisis for details.)Following on from this, de Broglie postulated that the same relationship between wavelength and momentum should hold for any particle.

At this point it's best to suspend your intuition about what it really means to say that a particle behaves like a wave (we'll have a look at that in the third article) and just follow through with the mathematics.

In classical mechanics the evolution over time of a wave, for example a sound wave or a water wave, is described by a *wave equation*: a differential equation whose solution is a *wave function*, which gives you the shape of the wave at any time (subject to suitable boundary conditions).

For example, suppose you have waves travelling through a string that is stretched out along the -axis and vibrates in the -plane. In order to describe the wave completely, you need to find the displacement of the string in the -direction at every point and every time . Using Newton’s second law of motion it is possible to show that obeys the following wave equation:

where is the speed of the waves.

A snapshot in time of a string vibrating in the *xy*-plane. The wave shown here is described by the cosine function.

A general solution to this equation is quite complicated, reflecting the fact that the string can be wiggling around in all sorts of ways, and that you need more information (initial conditions and boundary conditions) to find out exactly what kind of motion it is. But as an example, the function

describes a wave travelling in the positive -direction with an angular frequency , so as you would expect, it is a possible solution to the wave equation.

By analogy, there should be a wave equation governing the evolution of the mysterious "matter waves", whatever they may be, over time. Its solution would be a wave function (but resist thinking of it as describing an actual wave) which tells you all there is to know about your quantum system — for example a single particle moving around in a box — at any time . It was the Austrian physicist Erwin Schrödinger who came up with this equation in 1926. For a single particle moving around in three dimensions the equation can be written as

Here is the potential energy of the particle (a function of , , and ), is the mass of the particle and is Planck’s constant. The solution to this equation is the wave function

In some situations the potential energy does not depend on time In this case we can often solve the problem by considering the simpler *time-independent* version of the Schrödinger equation for a function depending only on space, i.e.

where is the total energy of the particle. The solution to the full equation is then

These equations apply to one particle moving in three dimensions, but they have counterparts describing a system with any number of particles. And rather than formulating the wave function as a function of position and time, you can also formulate it as a function of momentum and time.

### Enter uncertainty

We'll see how to solve Schrödinger's equation for a simple example in the second article, and also that its solution is indeed similar to the mathematical equation that describes a wave.

But what does this solution actually mean? It doesn't give you a precise location for your particle at a given time , so it doesn't give you the trajectory of a particle over time. Rather it's a function which, at a given time gives you a value for all possible locations . What does this value mean? In 1926 the physicist Max Born came up with a *probabilistic* interpretation. He postulated that the square of the absolute value of the wave function,

*probability density*for finding the particle at position at time . In other words, the probability that the particle will be found in a region at time is given by the integral

Werner Heisenberg, 1901-1976.

This probabilistic picture links in with
a rather shocking consequence of de Broglie's formula for the wavelength and momentum of a particle, discovered by Werner Heisenberg in 1927.
Heisenberg found that there is a fundamental limit to the
precision to which you can measure the position and the momentum of a
moving particle. The more precise you want to be about the one, the
less you can say about the other. And this is not down to the quality
of your measuring instrument, it is a fundamental uncertainty of
nature. This result is now known as Heisenberg's *uncertainty
principle* and it's one of the results that's often quoted
to illustrate the weirdness of quantum mechanics. It means that in quantum mechanics we simply cannot talk about the location or the trajectory of a particle.

"If we believe in this uncertainty picture, then we have to accept a probabilistic account [of what is happening] because we don’t have exact answers to questions like ’where is the electron at time ?’," says Bouatta. In other words, all you can expect from the mathematical representation of a quantum state, from the wave function, is that it gives you a probability.

Whether or not the wave function has any physical interpretation was and still is a touchy question. "The question was, we have this wave function, but are we really thinking that there are waves propagating in space and time?" says Bouatta. "De Broglie, Schrödinger and Einstein were trying to provide a realistic account, that it's like a light wave, for example, propagating in a vacuum. But [the physicists], Wolfgang Pauli, Werner Heisenberg and Niels Bohr were against this realistic picture. For them the wave function was only a tool for computing probabilities." We'll have a closer look at the interpretation of the wave function in the third article of this series.

### Does it work?

Louis de Broglie, 1892-1987.

Why should we believe this rather fantastical set-up? In this article we have presented Schrödinger's equation as if it were plucked out of thin air, but where does it actually come from? How did Schrödinger derive it? The famous physicist Richard Feynman considered this a futile question: "Where did we get that [equation] from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger."

Yet, the equation has held its own in every experiment so far. "It's the most fundamental equation in quantum mechanics," says Bouatta. "It's the starting point for every quantum mechanical system we want to describe: electrons, protons, neutrons, whatever." The equation's earliest success, which was also one of Schrödinger's motivations, was to describe a phenomenon that had helped to give birth to quantum mechanics in the first place: the discrete energy spectrum of the hydrogen atom. According to Ernest Rutherford's atomic model, the frequency of radiation emitted by atoms such as hydrogen should vary continuously. Experiments showed, however, that it doesn't: the hydrogen atom only emits radiation at certain frequencies, there is a jump when the frequency changes. This discovery flew in the face of conventional wisdom, which endorsed a maxim set out by the 17th century philosopher and mathematician Gottfried Leibniz: "nature does not make jumps".

In 1913 Niels Bohr came up with a new atomic model in which electrons are restricted to certain energy levels. Schrödinger applied his equation to the hydrogen atom and found that his solutions exactly reproduced the energy levels stipulated by Bohr. "This was an amazing result — and one of the first major achievement of Schrödinger's equation." says Bouatta.

With countless experimental successes under its belt, Schrödinger's equation has become the established analogue of Newton's second law of motion for quantum mechanics. Now let's see Schrödinger's equation in action, using the simple example of a particle moving around in a box. We will also explore another weird consequence of the equation called *quantum tunneling*.

*Read the next article: Schrödinger's equation — in action
But if you don't feel like doing the maths you can skip straight to the third article which explores the interpretation of the wave function.*

### About this article

Nazim Bouatta is a Postdoctoral Fellow in Foundations of Physics at the University of Cambridge.

Marianne Freiberger is Editor of *Plus*. She interviewed Bouatta in Cambridge in May 2012. She would also like to thank Jeremy Butterfield, a philosopher of physics at the University of Cambridge, and Tony Short, a Royal Society Research Fellow in Foundations of Quantum Physics at the University of Cambridge, for their help in writing these articles.

## Comments

## Thanks for the interesting, and informative article.

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## Great Article

Was just browsing the web looking for a quick blurb about schrodinger and found this. Very nice setup and nice flow. Reminded me of my college days.

## Thank you very much for this

Thank you very much for this very nice illustration, developed step by step chronically. Reading this nice article, one can see how quantum mechanics evolves and understand it better. Thank you very much.

## This article has been

This article has been extremely helpful with the readings of Einstein's essays. Thank you

## Thank U

The article is really helpful !!!!

## Excellent article

A very concise article giving a big picture description of the basic tenets of QM. Thanks. Would have been more impactful if the article was written straight instead of writing quotes from the interview.

## Although I am a PhD student

Although I am a PhD student in physics and have been studying quantum physics for several years, this article gives me a better view and summary of quantum concepts. With special thanks

## Excellent Article

Thanks Marianne. I was looking for a straight forward explanation of Schrodinger's equation, and here is your thoughtful article.

## Best article

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## Thank You

Thank You

## Schrodinger Eqn.

Thanks for the nice article. I just notice that the solution of the Schrodinger equation in the special case is a function of time whereas it should be a function of space only.

## Schrodinger Eqn.

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## Best explanation of schrodinger equation

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## Schröndinger Equation

No we don't know everything there is to know! We know a lot about the obvious movement but the transfer of intelligence or information between the elements, (not confined to the same time), is mentioned about this.

## Beautiful explanation!

Been searching a long time for an adequate explanation of Schrodinger's equation. Thanks a million for the clarity and context.

## The Schrodinger equation

In need of some detailed explanations of the Schrodinger equation I luckily found these articles. Very nice, clear and detailed, and with much more that I could image. Thank you very much indeed.

## Mathematics

Very good introduction. Thank you.

## Schrödinger

At the beginning of the twentieth century, experimental evidence suggested that atomic particles were also wave-like in nature. For example, electrons were found to give diffraction patterns when passed through a double slit in a similar way to light waves. Therefore, it was reasonable to assume that a wave equation could explain the behaviour of atomic particles.

Schrodinger was the first person to write down such a wave equation. Much discussion then centred on what the equation meant. The eigenvalues of the wave equation were shown to be equal to the energy levels of the quantum mechanical system, and the best test of the equation was when it was used to solve for the energy levels of the Hydrogen atom, and the energy levels were found to be in accord with Rydberg's Law.

It was initially much less obvious what the wavefunction of the equation was. After much debate, the wavefunction is now accepted to be a probability distribution. The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position.

Answered by: Ian Taylor, Ph.D., Theoretical Physics (Cambridge), PhD (Durham), UK

The Shrodinger equation is:

The solution to this equation is a wave that describes the quantum aspects of a system. However, physically interpreting the wave is one of the main philosophical problems of quantum mechanics.

The solution to the equation is based on the method of Eigen Values devised by Fourier. This is where any mathematical function is expressed as the sum of an infinite series of other periodic functions. The trick is to find the correct functions that have the right amplitudes so that when added together by superposition they give the desired solution.

So, the solution to Schrondinger's equation, the wave function for the system, was replaced by the wave functions of the individual series, natural harmonics of each other, an infinite series. Shrodinger has discovered that the replacement waves described the individual states of the quantum system and their amplitudes gave the relative importance of that state to the whole system.

Schrodinger's equation shows all of the wave like properties of matter and was one of greatest achievements of 20th century science.

It is used in physics and most of chemistry to deal with problems about the atomic structure of matter. It is an extremely powerful mathematical tool and the whole basis of wave mechanics.

Answered by: Simon Hooks, Physics A-Level Student, Gosport, UK

The Schrodinger equation is the name of the basic non-relativistic wave equation used in one version of quantum mechanics to describe the behaviour of a particle in a field of force. There is the time dependant equation used for describing progressive waves, applicable to the motion of free particles. And the time independent form of this equation used for describing standing waves.

Schrodingerï¿½s time-independent equation can be solved analytically for a number of simple systems. The time-dependant equation is of the first order in time but of the second order with respect to the co-ordinates, hence it is not consistent with relativity. The solutions for bound systems give three quantum numbers, corresponding to three co-ordinates, and an approximate relativistic correction is possible by including fourth spin quantum number.

Answered by: Dan Summons, Physics Undergrad Student, UOS, Souhampton

## Great article - much

Great article - much appreciated

## Metallurgical and Material Engineering

Very nicely expressed in a concise form.

## Problem

In the article, as a solution to the Schödinger equation, the function is given (in Maple notation):

Psi (t): = psi*exp(-(2*Pi*I*E/h)*t);

If one then takes (|Psi(t)|)**2, one would expect the integral for t from zero to infinity to be equal to 1. The integral (in Maple notation):

Int ((abs(psi*exp(-(2*Pi*I*E/h)*t)))**2, t = 0 .. infinite);

however, is equal to infinity. What error did I make?