In 1923, the physicist Louis de Broglie suggested that not only did waves exhibit particle-like aspects, but that the reverse was also true — all material particles should display wave-like properties.
How does this work? For a photon, momentum
![image0.png](https://www.dummies.com/wp-content/uploads/395162.image0.png)
And the wave vector, k, is equal to
![image1.png](https://www.dummies.com/wp-content/uploads/395163.image1.png)
![image2.png](https://www.dummies.com/wp-content/uploads/395164.image2.png)
De Broglie presented these apparently surprising suggestions in his Ph.D. thesis. Researchers put these suggestions to the test by sending a beam through a dual-slit apparatus to see whether the electron beam would act like it was made up of particles or waves. In the figure, you can see the setup and the results.
![An electron beam going through two slits.](https://www.dummies.com/wp-content/uploads/395165.image3.jpg)
In Figure a, you can see a beam of electrons passing through a single slit and the resulting pattern on a screen. In Figure b, the electrons are passing through a second slit. Classically, you’d expect the intensities of Figures a and b simply to add when both slits are open:
I = I1 + I2
But that’s not what happened. What actually appeared was an interference pattern when both slits were open (Figure c), not just a sum of the two slits’ electron intensities.
The result was a validation of de Broglie’s invention of matter waves. Experiment bore out the relation that
![image4.png](https://www.dummies.com/wp-content/uploads/395166.image4.png)
and de Broglie was a success.
The existence of matter waves says that you add the waves’ amplitude,
![image5.png](https://www.dummies.com/wp-content/uploads/395167.image5.png)
not their intensities, to sum them:
![image6.png](https://www.dummies.com/wp-content/uploads/395168.image6.png)
You square the amplitude to get the intensity, and the phase difference between
![image7.png](https://www.dummies.com/wp-content/uploads/395169.image7.png)
is what actually creates the interference pattern that’s observed.