###### By Don Lincoln**, **Fermilab

## Over the years, researchers have come to embrace the idea that in the quantum realm, the distinction between waves and particles is a tenuous one, with photons and electrons and other denizens of the microcosm being a little bit of both. In the quantum world, the connection between cause and effect is different than our intuition.

### Probability in Subatomic World

Let’s start with the idea that the quantum realm is not governed by the laws of causality that we’re so familiar with. In the world that we experience, if one drops a glass, it will fall, and perhaps break. Effect follows cause.

However, in the subatomic world, events are not connected so linearly. The subatomic world is governed, in part, by probability. Again, this is different than the probability that one might experience in daily life. This goes deeper than not being able to control the roll of a dice, or simply just not knowing when the roulette wheel will hit your number.

### Louis de Broglie’s Discovery of Waves

The concept of probability starts in quantum mechanics in history and conjecture, but quickly it is tested. It begins with French physicist Louis de Broglie, who proposed a conjecture in his 1924 PhD thesis that matter, specifically electrons, were not only a particle, but also a wave.

We were later introduced to some of the data that proved the same conjecture.

### Erwin Schrӧdinger’s Equation

Another quantum luminary, Austrian physicist Erwin Schrӧdinger devised what is now called the Schrӧdinger equation. This equation is now the basis for modern atomic theory.

Schrӧdinger’s equation was quite general. It described quantum particles in all situations. If, for example, an electron existed more or less in free space, the solution to Schrӧdinger’s equation was a classical and mathematical sine wave, with a wavelength, frequency, and amplitude.

### Introduction to Wave Function

The Schrӧdinger equation predicts an infinite number of sine wave solutions, with an infinite number of different wavelengths. Because of this, the solution to the Schrӧdinger equation is called the wave function.

On the other hand, if one uses the Schrӧdinger equation to describe an electron inside an atom, the solution is no longer a sine wave. There are lots of solutions, with all sorts of curious and complicated shapes, but we still call the solution a wave function.

### Max Born’s Theory of Wave Function and Probability

The problem staring at the research community was that they had no real understanding of the physical significance of the wave function.

It was in 1926, not long after Schrӧdinger’s paper describing his equation was published, that German physicist Max Born proposed that the wave function was related to the probability that the electron would be detected in a particular configuration. That statement could be a bit confusing, so let’s explore an analogy that will help.

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### Understanding Born’s Theory

Suppose you’re in your house at night. The house is dark. You’re at the top of a set of carpeted stairs holding your phone, whose battery is dead by the way. You drop the phone and it tumbles down the stairs, landing on some unknown stair. It’s dark, so you can’t see where the phone landed.

Now in classical physics, the phone ended up where it ended up. You don’t know where it is, but it’s somewhere. You’ll find out when you walk down the stairs and step on it.

But in quantum mechanics, the situation is a bit different. In quantum mechanics, the wave function would say that there is one probability that the phone landed on the first stair, a different probability that it landed on the second stair, a third probability it landed on the third stair, and so on.

Fudging things a bit, we would say that the wave function was the probability of landing on stair one times the state of landing on stair one, plus the probability of landing on stair two, times the state of landing on stair two and so on. We’d add up all of the possible outcomes, which is to say possible stairs and associated probabilities, and that’s the wave function.

Born’s contribution was to say that the numbers in front of the configurations in the equation were related to the probability of finding each configuration.

### Atomic Probability

The situation is very similar at the atomic level. If you have an electron orbiting an atomic nucleus, the electron could be orbiting in a number of different ways, basically like different stairs.

Each manner of orbiting, what scientists call orbitals, has a different energy and an associated probability of finding the electron in that configuration.

### Copenhagen Interpretation

The situation got more complicated when Danish physicist Niels Bohr and German physicist Werner Heisenberg got involved. They collaborated over the period 1925 to 1927, trying to unravel the true meaning of the work of de Broglie, Schrӧdinger, and Born.

Their proposed solution, which eventually became known as the Copenhagen interpretation, said that the wave function didn’t describe probabilities in the way we ordinarily think about them. In this common case, there is a probability that the object will end up in that outcome a certain percentage of the time. But, if we were to watch the object during the shaking and movement and settling, we’d see it move in a complex but fixed path and end up in some configuration. Taking up the stair example, the phone would bounce a bit and end up on a particular stair. It’s never on more than one stair at the same time.

In the Copenhagen interpretation, however, what happens is that the object doesn’t bounce around into a single outcome. Instead, what happens is the object simultaneously bounces into all configurations, or at least that’s what the wave function says. It’s as if the phone bounces down the stairs and is simultaneously and literally on every possible stair, some percentage of the time.

### Common Questions about Probability in Quantum Mechanics

**Q: What led to the beginning of probability in quantum mechanics?**

Probability in quantum mechanics begins with French physicist Louis de Broglie, who proposed a conjecture in his 1924 PhD thesis that matter, specifically electrons, were not only a particle, but also a wave.

**Q: What was Max Born’s theory of probability and wave function?**

Max Born proposed that the wave function was related to the probability that the electron would be detected in a particular configuration.

**Q: What was Schrӧdinger’s equation?**

Schrӧdinger’s equation described quantum particles in all situations. If, for example, an electron existed more or less in free space, the solution to Schrӧdinger’s equation was a classical and mathematical sine wave, with a wavelength, frequency, and amplitude.