# Wave-Particle Duality

**Claimed by Arghya Roy (Spring 2022)**

Wave-particle duality is the concept that states every elementary particle has both wave and particle-like properties.

## The Main Idea

In the 1920s, French physicist Louis de Broglie suggested that all matter has wave-like properties. This conclusion was largely the result of two landmark experiments whose outcomes classically contradicted each other. The first experiment was Thomas Young's double slit experiment, which demonstrated that light behaved like a wave. The second experiment was conducted by Albert Einstein, who showed, through his research on the photoelectric effect, that light was made up of discrete packets of energy called photons: indicating light also shared the properties of a particle. Due to these experiments, physicists came to recognize that classical descriptions of particles and waves don't strictly apply to quantum mechanical entities.

### Double slit experiment

The double slit experiment is an experiment that was originally conducted by Thomas Young in the 17th century. In the experiment, Young simply sent a beam of light through two slits and observed the pattern on the surface behind the slits. The experiment revealed an interference pattern that only could have been present if a wave was passed through the two slits. The bright spots occur where the amplitudes of the two waves match (both waves are at their peaks) and the dark spots occur when one wave is at its maximum amplitude, while the other is at its minimum.

### Photoelectric effect

It was known that when light struck a metal, electrons were liberated from the surface. The intuition was that increasing the intensity of light (shining more light) would liberate more electrons. Albert Einstein found something interesting, though. Varying intensity of light had no effect on how many electrons were liberated. Rather, the *frequency* of the light determined how many electrons, if any, would be freed. Furthermore, the original theory was that the electrons that would be freed was continuous -- even the smallest amount of light would free some electrons. In fact, this was not the case. Einstein found that there was a minimum threshold frequency that must have been present in order to release electrons at all. This implied there was a *minimum amount of energy*, or **quantum** involved in the interaction. This pointed to the fact that light in fact behaved as particles (called photons) which were packets of these quantum energies. This directly conflicted with the double slit experiment.

PhET Simulation for Photoelectric effect

### A Mathematical Model

Now that we can treat these particles at the quantum level as waves, we can use many different equations from wave mechanics to describe their behavior. One of the most important equations in dealing with wave like properties of these quantum systems and particles is the Schrödinger equation. The Schrödinger equation is the analog of Newton's second law (**F** = *m***a**) in quantum mechanics, and describes the wave function over time of a system such as a particle moving in a magnetic field. But rather than a simple linear equation, the Schrödinger equation is a linear partial differential equation:

[math]\displaystyle{ i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat H \Psi(\mathbf{r},t) }[/math]

is the general, relativistic (works for particles moving up to close to the speed of light) equation, where [math]\displaystyle{ i }[/math] is the square root of negative 1, [math]\displaystyle{ ħ }[/math] is the Planck constant divided by [math]\displaystyle{ 2pi }[/math], the symbol ∂/∂t indicates a partial derivative with respect to time, Ψ is the wave function of the quantum system, and [math]\displaystyle{ Ĥ }[/math] is the Hamiltonian operator, which represents the total energy of the wave function at different times.

Using the Schrödinger equation involves using the proper form of the Hamiltonian operator that accounts for the kinetic and potential energy of the particles, and using that operator to then solve the partial differential equation. The output wave function contains information about the system at different times.

### A Computational Model

The author of this page has written up some code to simulate the double slit experiment. In the model, shot particles are depicted, and as the model runs, you can see how there are two distinct clusters where the particles land.

The Glowscript Code: https://trinket.io/glowscript/9dc7cdc15a

## Examples

Wave-Particle Duality is a principle that can encompass a number of problems. Past editors of this page have included the following three problems, but for the final 'hard' problem, the current author would like to include a traditional problem involving the two-slits.

Two very straightforward formulas involving Planck's constant *h*, which has a value of [math]\displaystyle{ 6.62607004*10^-34 m^2 }[/math] m^2 kg / s, can be used to relate fundamental properties such as energy *E*, frequency [math]\displaystyle{ \nu }[/math], and wavelength [math]\displaystyle{ \lambda }[/math].

- [math]\displaystyle{ E = h \nu }[/math] (1)

- [math]\displaystyle{ \lambda = \frac{h}{p} . }[/math] (2)

Another very useful equation is that the frequency and the wavelength of a particle are inversely proportional, and multiply to the speed of light, *c*.

- [math]\displaystyle{ c = \lambda\nu }[/math] (3)

### Ex. 1

Microwave ovens emit microwave energy with a wavelength of 12.9 cm. What is the energy of exactly one photon of this microwave radiation?

Here we need to use equations 1 and 3.

Next we define our constants.

[math]\displaystyle{ c= 2.998*10^8 m/s }[/math] (this problem wants us to use this number for speed of light), [math]\displaystyle{ h=6.626*10^34J-s }[/math]

Now we simply plug in, making sure that our units match (convert 12.9cm to meters = 0.129m)

[math]\displaystyle{ 2.998*10^8 m/s = .129 * v }[/math]

[math]\displaystyle{ v = 2,324,031,008 Hz }[/math]

Now that we found v, we can solve for E.

[math]\displaystyle{ E = 2,324,031,008 Hz * 6.626*10^-34 }[/math]

[math]\displaystyle{ E= 1.53990294*10^-24 }[/math]

[math]\displaystyle{ E= 1.54*10^-24 }[/math]

### Ex. 2

A radio station broadcasts at a frequency of 590 KHz. What is the wavelength of the radio waves?

We need to use equation 3.

First we convert KHz to Hz.

[math]\displaystyle{ 590 }[/math] KHz = [math]\displaystyle{ 590*10^3 }[/math] Hz

[math]\displaystyle{ (3*10^8)/(590*10^3) }[/math] = [math]\displaystyle{ 500 }[/math]m = [math]\displaystyle{ \lambda }[/math]

[math]\displaystyle{ \lambda }[/math] = 500m.

### Ex. 3: Two-slit experiment problem

Problem:

In a two-slit experiment, set the distance between the slits to 500 micrometers, and the screen is 2 meters away. If red light is used, what is the distance between the first and the second fringe on the screen?

Solution:

For this problem, you'll need to utilize Young's Double Slit Formula, which states [math]\displaystyle{ m*\lambda=d*sin(\theta) }[/math]

[math]\displaystyle{ d=5.0e-4 m, \lambda=650e-9 m }[/math]

From the formula, [math]\displaystyle{ sin(\theta)=m*\lambda/d }[/math] -> [math]\displaystyle{ \theta=sin^{-1}(m*\lambda/d) }[/math]

To isolate the two degree values for the first and second fringes, input the above values to find

[math]\displaystyle{ m=1, \theta_1=0.0744845^{\circ}, m=2, \theta_2=0.1489691^{\circ} }[/math]

From trigonometry, we know [math]\displaystyle{ tan(\theta)=h/l, y_1=h, tan(\theta)=y_1/l }[/math], which yields [math]\displaystyle{ y_1=l*tan(\theta) }[/math]

Substituting [math]\displaystyle{ l=2 }[/math] meters and the respective [math]\displaystyle{ \theta }[/math] values, we get

[math]\displaystyle{ y_1=0.0026 m, y_2=0.0052 m }[/math], yielding a difference between the fringes of 0.0026 meters.

## Connectedness

1. How is this topic connected to something that you are interested in?

Wave-Particle duality as a concept was the way that I first got introduced to quantum weirdness, and higher level physics. I'm personally interested in quantum computing, and a major issue in the current development of multi-qubit systems is decoherence. The Wave-Particle duality is central to measurement in quantum mechanics, because when the particle is located, the particle is forced to be in a more localized state due to Heisenberg Uncertainty Principle.

2. How is it connected to your major?

I'm a computer science major, so the adjacent subject would be Quantum Computing, where Wave-Particle Duality and other quantum mechanical properties are used to solve problems Classical computers can't address as easily. Beyond also developing simulators like the one I coded in the Computational Model section, there isn't too much of an overlap.

3. Is there an interesting industrial application?

Right now, since quantum computing is not effective or cheap enough for companies to use, industry use is limited. But common lab use is in electron microscopy - it is possible by exploiting the high frequencies of electrons, meaning that one can see objects much smaller than those that can only be seen with visible light.

## History

Throughout the 1800s, scientists, such as John Dalton and Ernest Rutherford theorized and discovered elementary particles. As discussed in the first section, the photoelectric effect and the double-slit experiment discovered the contradiction in definitely describing physically objects as particles or waves. Newton's classical mechanics had no way of explaining phenomenon like this, and quantum mechanics was developed to study physics on the minute stage. The early 20th century included scientists like Richard Feynman and Erwin_Schr%C3%B6dinger, who advanced the field. Werner Heisenberg later developed his uncertainty principle, which established many of the relationships we explored in the problem section of this page. More recently, scientists have worked on applying quantum effects to computing.

## See also

### Further Reading

The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta, by Albert Einstein and Leopold Infeld

Chapter 4, Introduction to Modern Physics Textbook, by Kenneth S. Krane

R. Nave. "Wave–Particle Duality" (Web page). HyperPhysics. Georgia State University, Department of Physics and Astronomy: http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html

### External Links

This topic is a big idea in the field of quantum mechanics, but there are many other interesting concepts to further explore:

## References

All pictures were from Wikimedia Commons, and references are already hyperlinked to key words in the text.