Note that 2πr is the circumference of the orbit of electron around the nucleus and nλ is Example 1: Calculate the de Broglie wavelength for an electron that is. DeBroglie's Formula for Calculating a Particle's Wavelength; Bohr's Special Orbits That is, the first orbit has a circumference of one deBroglie wavelength. de Broglie came up with an explanation for why the angular momentum might be it was. de Broglie realized that if you use the wavelength associated with the that an integral number of wavelengths must fit in the circumference of an orbit.
And he set out to figure out what would this wavelength be, he figured it out, it's called the "de Broe-lee" wavelength, oh, I reverted, sorry, "de Broy" wavelength, not the "de Broe-glee" wavelength. The de Broglie wavelength, he figured it out, and he realized it was this. So, he actually postulated it. He didn't really prove this. He motivated the idea, and then it was up to experimentalists to try this out.
So he said that the wavelength associated with things that we thought were matter, so sometimes these were called matter waves, but the wavelength of, say, an electron, is gonna be equal to Planck's constant, divided by the momentum of that electron.
And so, why did he say this? Why did he pick Planck's constant, which, by the way, if you're not familiar with Planck's constant, it is like the name suggests, just a constant, and it's always the same value, it's 6.
Matter wave - Wikipedia
This was a constant discovered in other experiments, like this photoelectric effect, and the original blackbody experiments that Planck was dealing with. It's called Planck's constant, it shows up all around modern physics and quantum mechanics.
So how did Louis de Broglie even come up with this? Why Planck's constant over the momentum? 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.
De Broglie wavelength
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. 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.
De Broglie wavelength (video) | Khan Academy
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.
We see that Bohr's model of the atom does a great job explaining both the emission and absorption spectra of hydrogen and hydrogen-like atoms. Also, it inspired the shell structure of atoms and gave a qualitative understanding of the periodic table of elements. However, it fell short in some important respects. The Bohr model cannot explain more complex features of the hydrogen atom nor can it explain atoms involving more than one "outer shell" electron. Also, as we shall learn, the Bohr Model says nothing about how to calculate the likelihood that an atom will undergo a transition involving the emission or absorption of photons.
This is left to the subject of Quantum Mechanics, which is the next major topic. Louis deBroglie made the complementary observation that particles, like electrons, might also have a wavelike behavior. He postulated that a wavelength could be associated with a particle once you know its mass and its velocity. The deBroglie wavelength is defined as follows: One could also express mv as the particle's momentum.
It works as follows. The 1st orbit consists of one complete wave wrapped around in a circle. That is, the first orbit has a circumference of one deBroglie wavelength.
It is analogous to the wave motion that occurs when a string in a musical instrument is plucked. Not all wavelengths are possible, only the ones that are related to the length of the string l. The vibration of a string with one of these wavelength is seen as a "standing wave.
A standing waves for a closed circular loop doesn't follow exactly the same formula that applies to a string, but a very rule applies.
A smaller fraction will accumulate fairly evenly at side regions as shown. Eventually after so many electrons go under diffraction at the slit, the familiar diffraction pattern will develop.
Homework Help: De Broglie wavelength question.
This means that the probability that the diffracted electrons fall on the centerline is higher than the probability of them falling on the side lines; however, the sum of the probabilities must equal one. As the number of electrons passing through the two slits increases the separation of the bright and dark fringes becomes more apparent.
Classical physics is based on the principle of determinism that means: Given the initial position and velocity of a particle as well as the forces acting on it, its future course can accurately be predicted, at least theoretically. It is not possible to exactly determine the position of a single particle at a given time.
Quantum mechanics can correctly predict the average values of physical quantities, not the results of individual measurements. The wavelength of a wave that extends over many cycles is well-defined but its position is not.
For example, if you see continuous ripples on a calm swimming pool, you will be able to measure the wavelength fairly accurately; however, you can not exactly measure the position of the moving wave. This is shown in Figure 1. Now, let's look at another scenario. When waves of different wavelengths interfere and add up, they can form a localized wave packet, but the wavelength is not well-defined.
An example for this case is when two waves of different wavelengths interfere waves generated by two moving boats on a calm lakethey generate waveforms that appear stationary or at least appearing at the same location periodically. In this example, the position of the wave packet can be determined, but not the wavelengths.