 After Albert Einstein explained the photoelectric effect by assuming that light consists of particles, photons, Louis de Broglie came up with a similar idea. He thought if it was possible to assign a particle character to the wave-like light, then it must also be possible to assign a wave character to a particle-like matter. Just like light, electrons are also able to generate an interference pattern in the double slit experiment. This observation that light and matter can behave in a wave-like and particle-like manner, depending on the situation, is called wave-particle duality. A photon of wavelength lambda has the momentum h over lambda. Here h is the Planck's constant, a physical constant with the value 6.6 times 10 to the minus 34 joule seconds, and lambda is the wavelength of light. Analogously, a wavelength can be assigned to a particle with mass, which has a momentum p. For this purpose, we rearrange the momentum with respect to the wavelength lambda. lambda is equal to h over p. The momentum p of a classical particle is defined as the product of its mass m in its velocity v. We insert m times v for momentum and interpret the wavelength lambda as the matter wavelength of a particle. It is also called de Broglie wavelength. lambda is equal to h divided by m times v. From this equation, you can see that fast and heavy particles, which therefore have a large momentum, have a shorter de Broglie wavelength than slow and light particles, which have a small momentum. With the help of the de Broglie wavelength, you can estimate whether an object will behave more wave-like or particle-like. If the velocity of the particle and thus the momentum of the particle approaches zero, then the de Broglie wavelength becomes very large. The particle starts to show its wave character and behaves like a quantum particle. For a large velocity and large mass of the particle, the de Broglie wavelength is negligible. The particle behaves like a real classical particle and can be described with classical mechanics. Quantum effects such as particle interference do not matter here. Let's make an example. As you know, free electrons exist in a metal. Together, they are also called electron gas. The electrons have a thermal velocity of 10 to the power of 6 meters per second at room temperature. By thermal is meant that the electrons in the metal are moving in a random manner. After a collision with a metal atom, it moves in one direction. After the other collision, it moves in the other direction. An electron has a rest mass of 9.1 times 10 to the power of minus 31 kilograms. Thus, the de Broglie wavelength of a free electron is 7.2 times 10 to the power of minus 10 meters. This is a matter wavelength of 0.72 nanometers. For comparison, the diameter of the DNA double helix is about 2 nanometers. Usually, you don't know the velocity of a particle directly, but you can eliminate it and bring voltage into play instead. Consider an electrically charged particle with charge q in an electric field of a plate capacitor. We apply a voltage u between the plates. If this charge passes through the voltage, moving from one plate to the other, then it gains a kinetic energy, given by q times u. We equate this energy with a classical formula for kinetic energy one-half times mv squared. Let's rearrange the equation for v and use it to replace the unknown velocity in the de Broglie formula. The result is a formula for de Broglie wavelength that depends on voltage. All the constants, h, m and q of an electron are known. You can adjust the voltage at the voltage source and thus increase or decrease the de Broglie wavelength of the particle. If we use an electron with its corresponding mass and charge and set a voltage of 1000 volts, the de Broglie wavelength becomes 38.7 picometers. With a voltage of 1 volt, the de Broglie wavelength would be only 1.2 nanometers.