I have heard that humans have a wavelength. Is this true?
The wavelength λ = h/p is called the de Broglie wavelength, and the relations λ The wave function also has a well defined deBroglie frequency f = E/h, which. Until the eighteenth century, light was thought of purely as a wave, like sound. energy to the frequency of waves, and by Einstein's famous equation E=mc2, Thus de Broglie supposed that matter might also be related to the frequency of. The individual frequency of one atom interferes with the frequencies of all the others, resulting in an almost flat dispersion relation. The group velocity is then . That's how de Broglie got the idea of the wave in the first place.
Well, people already knew for light, that the wavelength of a light ray is gonna also equal Planck's constant, divided by the momentum of the photons in that light ray. So the name for these particles of light are called photons.
I'm drawing them localized in space here, but don't necessarily think about it that way. Think about it just in terms of, they only deposit their energy in bunches.
They don't necessarily have to be at a particular point at a particular time. This is a little misleading, this picture here, I'm just not sure how else to represent this idea in a picture that they only deposit their energies in bunches.
De Broglie wavelength
So this is a very loose drawing, don't take this too seriously here. But people had already discovered this relationship for photons. And that might bother you, you might be like, "Wait a minute, how in the world can photons have momentum? Because parallel to all these discoveries in quantum physics, Einstein realized that this was actually not true when things traveled near the speed of light. The actual relationship, I'll just show you, it looks like this.
The actual relationship is that the energy squared, is gonna equal the rest mass squared, times the speed of light to the fourth, plus the momentum of the particles squared, times the speed of light squared. This is the better relationship that shows you how to relate momentum and energy. This is true in special relativity, and using this, you can get this formula for the wavelength of light in terms of its momentum. It's not even that hard. In fact, I'll show you here, it only takes a second.
Light has no rest mass, we know that, light has no rest mass, so this term is zero. We've got a formula for the energy of light, it's just h times f.
So e squared is just gonna be h squared times f squared, the frequency of the light squared, so that equals the momentum of the light squared, times the speed of light squared, I could take the square root of both sides now and get rid of all these squares, and I get hf equals momentum times c, if I rearrange this, and get h over p on the left hand side, if I divide both sides by momentum, and then divide both sides by frequency, I get h over the momentum is equal to the speed of light over the frequency, but the speed of light over the frequency is just the wavelength.
And we know that, because the speed of a wave is wavelength times frequency, so if you solve for the wavelength, you get the speed of the wave over the frequency, and for light, the speed of the wave is the speed of light. So c over frequency is just wavelength. That is just this relationship right here. So people knew about this. And de Broglie suggested, hypothesized, that maybe the same relationship works for these matter particles like electrons, or protons, or neutrons, or things that we thought were particles, maybe they also can have a wavelength.
And you still might not be satisfied, you might be like, "What, what does that even mean, "that a particle can have a wavelength? How would you even test that? Well, you'd test it the same way you test whether photons and light can have a wavelength.
You subject them to an experiment that would expose the wave-like properties, i. So, if light can exhibit wave-like behavior when we shoot it through a double slit, then the electrons, if they also have a wavelength and wave-like behavior, they should also demonstrate wave-like behavior when we shoot them through the double slit.
And that's what people did. There was an experiment by Davisson and Germer, they took electrons, they shot them through a double slit. If the electrons just created two bright electron splotches right behind the holes, you would've known that, "Okay, that's not wave-like. Davisson and Germer did this experiment, and it's a little harder, the wavelength of these electrons are really small.
So you've gotta use atomic structure to create this double slit. It's difficult, you should look it up, it's interesting. People still use this, it's called electron diffraction. But long story short, they did the experiment.
- Deriving the de Broglie Wavelength
- I have heard that humans have a wavelength. Is this true?
They shot electrons through here, guess what they got? They got wave-like behavior. They got this diffraction pattern on the other side. And when they discovered that, de Broglie won his Nobel Prize, 'cause it showed that he was right.Frequency, Wavelength, and the Speed of Light - a video course made easy by Crash Chemistry Academy
Matter particles can have wavelength, and they can exhibit wave-like behavior, just like light can, which was a beautiful synthesis between two separate realms of physics, matter and light.
Turns out they weren't so different after all. Photons are the particles of light. Matter is made of atoms, and atoms are made protons, neutrons and electrons. These are not macroscopic particles.
The deBroglie Equation: Relating a Particle's Energy to its Wavelength
Typical atomic dimensions are on the order of m, nuclear dimensions are on the order of m, and the electron seems to be a point particle with no size at all. How do these particles behave? If a wave equation describes the behavior of photons, maybe a wave equation also describes the behavior of other microscopic particles. InLuis deBroglie Nobel Prize in Physics in proposed that a wave function is associated with all particles.
Where this wave function has nonzero amplitude, we are likely to find the particle.
The standard interpretation is that the intensity of the wave function of a particle at any point is proportional to the probability of finding the particle at that point. The wavelengths of the harmonic waves used to build the wave function let us calculate the most likely momentum of the particle and the uncertainty in the momentum.
The wave function for a material particle is often called a matter wave. A spread in wavelengths means an uncertainty in the momentum. The uncertainty principle also holds for material particles.
For any particle, we cannot predict its position and momentum with absolute certainty. We will learn about complex functions in the next section. This wavelength is approximately times smaller than the typical size of an atom. If you double the kinetic energy of a particle, how does the deBroglie wavelength change?
If you double the speed of a particle, how does it deBroglie wavelength change? An electron and a proton have the same speed.
Which has the greater deBroglie wavelength? The wave function for a particle for which we have some position information is a wave packet, and the particle's momentum now has some uncertainty. Wave packets build from harmonic waves with smaller wavelength can be smaller in size. We can know the position of more massive particles moving with approximate speed v more precisely than the position of less massive particle moving with the same speed since the more massive particles have shorter deBroglie wavelengths.
Electrons are confined in atoms. The uncertainty principle tells us that they cannot be sitting still, but must be moving around with high speeds. It must be jittering around a bit, but there are few instruments capable of detecting this jitter. As we increase the mass of the object further, the uncertainty in the speed of the object becomes completely negligible. So while the uncertainty principle also holds for massive, macroscopic particles, it has no practical consequences. Dispersion Photons always move with the speed of light c.