Main content

## AP®︎/College Physics 2

### Unit 8: Lesson 2

Atoms and electrons- De Broglie wavelength
- Quantum Wavefunction
- Atomic Energy Levels
- Bohr model radii (derivation using physics)
- Bohr model radii
- Bohr model energy levels (derivation using physics)
- Bohr model energy levels
- Absorption and emission
- Emission spectrum of hydrogen
- Bohr's model of hydrogen
- The quantum mechanical model of the atom

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# De Broglie wavelength

Is it a particle or a wave? This is the question that physicists of the 1920s were asking about light. In 1924, Louis de Broglie took this question to another level as he explored how electrons - which are matter, and were thought to be simply particles - can behave like waves.

## Want to join the conversation?

- Does the De Brogiie wavelength have any practical applications today? Like do we use it with technology?(23 votes)
- Yes! A great example is the electron microscope.

A big problem with microscopes is that you cannot see any object smaller than the wavelength of the waves you are using to illuminate it, even with an otherwise ideally perfect instrument. Since visible light has a wavelength of about 500 nanometers, this means that visible light microscopes can not see a lot of interesting small things, like viruses or the atomic structure of crystals or other solids. We could use light with a shorter wavelength, like x-rays or gamma rays, but as we shorten the wavelength, light pretty quickly becomes both destructive and penetrating, so that it just passes through the sample and destroys it.

Instead what we do is to use electrons. Since electrons have a rest mass, unlike photons, they have a de Broglie wavelength which is really short, around 0.01 nanometers for easily achievable speeds. This means that a microscope using electron "matter waves" instead of photon light waves can see much smaller things.(98 votes)

- What does the wavelength of an electron even mean? How do I visualise it? What am I supposed to imagine when I think about electrons now?(15 votes)
- Its a great question and I think you need to come up with your own way of visualizing it and making your own sense of it. Just as Einstein and others did. I think the way to do it is this: Take in as much of the experimental data as you can (such as the info in Andrews answer) and then allow your own 'model' to emerge.

Personally, my way of seeing it is like this: when the electron (or any fundamental particle) is 'stationary' or in a situation such as the photoelectric effect, then I see it as a small particle... like a tiny, steel ball. (even though it has zero volume... this idea is helpful)

When it is in the atom changing energy levels for example, then I see it as a wave-thing. A cloud of 'something'. maybe analogous to a cloud of water vapour. And, under certain conditions, this wave-like cloud will rapidly condense into the tiny ball-like particle. For example in the double slit experiment.... electron waves diffract and then 'condense' onto the 'screen' as a particle.

I know that my expression of the model is incomplete and inadequate but a) I think that is true for us all since it is impossible to express the 'truth' using our limited, 3 dimensional ability to conceptualise and b) our models continue to develop as our understanding grows.

IM(25 votes)

- This is great, but I think I got a gap here in my understanding of the whole thing, I mean why couldn't they explain the photoelectric effect by saying that the light is a wave that carries energy, and this energy has been transmitted to the electrons of the atoms of the metal, causing its ionization and thus freeing the electrons?(8 votes)
- Light is a electromagnetic wave with continous distribution of energy, the absorbtion of energy of electrons takes place continuosly over the wavefront. As large number of electrons absorb energy, energy absorbed per electron per unit time becomes small. Hence if u illuminate light for more time say 1 hour or a day the energy acquired by electrons will overcome workfunction and gets detached from metal. But in photoelectric effect there must be no time lag between illumination of light and detachment of electrons(8 votes)

- how does light decide when to act like particle and when to act like wave ?(8 votes)
- Light actually doesn't have to act like a particle or a wave for any specific experiment. Because light always is a wave and a particle at the same time. I know this sound weird but quantum mechanics is weird. David said that way just to make things simple.(3 votes)

- If we throw sand particles with the speed of light, then do these particles behave like waves as electrons behave?(2 votes)
- Actually, it's more the opposite. As particles with mass move closer and closer to the speed of light, more and more energy must be input, adding more and more mass. This would increase its momentum toward infinity, decreasing any potential wavelength further towards zero.(9 votes)

- I was seeing the proof of the de broglie wavelength. He equated E=mc^2 and E=hf. Then,

hf=mc^2

If I'm not crossing a line, if we separate the constants to one side and the variables to the other, we get:

f/m=c^2/h which is a constant.

What does that even mean? Is f/m always a constant?

#Sorry to have posted this doubt again. :-)(4 votes)- I think your thinking is right, but remember that it applies only for particles that travel at near-light speed because in the relation you used 'c' represents the velocity of light. So being an electron or a neutrino, you can say that the relationship between the frequency and the mass is constant.

In fact we really have total energy E = Mc^2 = mc^2/sqrt(1 - (v/c)^2) = hF = hc/L; so that F/m = c^2/(h sqrt(1 - (v/c)^2)), which varies with v the relative speed of the particle.

But a more interesting relationship is that the frequency varies F = mc^2/(h sqrt(1 - (v/c)^2). [See source.] And hF increases as v increases, which comes as no surprise as we're putting energy into the particle and that results in increasing kinetic energy.

Source(s): https://arxiv.org/pdf/1208.0119.pdf is an excellent source that reviews how DB derived the relationship in his PhD dissertation no less.

So finally, f/m is always a constant for photons and near light speed particles.

Hope this helped!

Feel free to ask more questions!(3 votes)

- did de Broglie done any experiments to say that wavelength =h/mv

but I read in one book that ,,,de Broglie only said that this formulae is only applicable for EMW or EMR ,,,,,,and when he went near to Einstein for saying about this then Einstein has said that this formulae is not only applicable for EMW. but this is applicable for all the things in UNIVERSE ....

this was read by me in 1 of my books

SO, please give me clarity about this(4 votes)- As we have no object that is moving with the speed of light or photon,due to which we say that photon have no rest mass.(1 vote)

- Ok. Then let's say ψ squared means the probability density of finding the electron in a certain place. That's why it's always positive. But is there any way to interpret a negative ψ (when not squared), or it's also something that physicists are not sure about?(2 votes)
- The wave function itself has no physical interpretation, it is not measurable.(3 votes)

- Can object moving in straight line may have angular momentum with respect to a specific point?(2 votes)
- How can we find final momentum of an electron using its potential difference value?(1 vote)
- accelerate an electron through a potential difference V

the kinetic energy gained = work done by the potential difference = eV

so KE=eV

1/2 m v>2 = eV

Can you find the momentum from this?(3 votes)

## Video transcript

- [Narrator] So in the early 20th century, physicists were bamboozled because light, which we thought was a wave, started to behave in certain experiments as if it were a particle. So, for instance, there
was an experiment done called the photoelectric effect, where, if you shine light at a metal, it'll knock electrons out of the metal if that light has sufficient energy, but if you tried to explain
this using wave mechanics, you get the wrong result. And it was only by resorting
to a description of light as if it could only deliver
energy in discrete packets that Einstein was able to describe how this photoelectric effect worked, and predict the results that they actually measure in the lab. In other words, light
was only giving energy in certain bunches, equal to something called planks constant, multiplied by the frequency of the light. It either gave all of this
energy to the electron, or it gave none of the
energy to the electron. It was never half and half. It never gave half of this energy, it was sort of all or nothing. But this was confusing to people. 'Cause we thought we had established that light was a wave, and we thought that
because if you shine light through a double slit, if it were a particle, if light were just a bunch of particles, you would expect particles to just either go through the top hole and give you a bright spot right here, or go through the bottom hole, give you a bright spot right here, but what we actually measure when you do this experiment with light, is that the light seemingly
diffracts from both holes, overlaps, and it gives you a diffraction pattern on the screen. So instead of just two bright spots, it gives you this constructive
and destructive pattern that would only emerge if the light beam were passing through both slits, and then overlapping, the way waves would, out of two holes on this
other side of the double slit. So this experiment showed that
light behaved like a wave, but the photoelectric effect showed that light behaved more like a particle, and this kept happening. You kept discovering different experiments that showed particle-like behavior, or different experiments that showed wave-like behavior for light. Finally, physicists resigned to the fact that light can seemingly have
particle-like properties, and wave-length properties, depending on the
experiment being conducted. So that's where things sat when in 1924, a young French physicist, a
brilliant young physicist, named Louis de Broglie, now, it looks like you pronounce
this "Louis de Broe-glee", and that's what I always said. I always read this and I said
"de Broe-glee" in my mind, and I knew that wasn't right. If you look it up, it's
more like "Louis de Broy", so get rid of all that, replace
it with a "y" in your mind, Louis de Broglie, in 1924, wrote a paper, and he did something
no one else was doing. Everyone else was worried about light, and light behaving like
a particle or a wave, depending on the experiment; Louis de Broglie said this,
"What about the electron? "You got this electron flying off here," he said, "if light, which
we thought was a wave, "can act like a particle, "maybe electrons, which
we thought were particles, "can act like a wave." In other words, maybe
they have a wavelength associated with them. He was trying to synthesize these ideas into one over arching framework in which you could describe
both quanta of light, i.e. particles of light, and particles, which we thought were just particles, but maybe they can behave
like waves as well. So maybe, he was saying,
everything in the universe can behave like a particle or a wave, depending on the experiment
that's being conducted. 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 6.626 times 10, to the negative 34th joule seconds. It's really small. 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. 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? "They don't have any mass. "I know momentum is just m times v, "if the mass of light is zero, "doesn't that mean the
momentum always has to be zero? "Wouldn't that make this
wavelength infinite?" And if we were dealing
with classical mechanics, that would be right, but it turns out, this is not true when you travel near the speed of light. 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?" That's hard to even comprehend. 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.e., just take these electrons, shoot them through the double slit. 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. "These are just flat out
particles, de Broglie was wrong." But that's not what they discovered. 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. 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. 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. Now, sometimes, de Broglie
is given sort of a bum rap. People say, "Wait a minute, all he did "was take this equation that
people already knew about, "and just restate it
for matter particles?" And no, that's not all he did. If you go back and look at
his paper, I suggest you do, he did a lot more than that. The paper's impressive,
it's an impressive paper, and it's written beautifully. He did much more than this, but this is sort of the thing people most readily recognize him for. And to emphasize the importance of this, before this point,
people had lots of ideas and formulas in quantum mechanics that they didn't completely understand. After this point, after this pivot, where we started to view matter
particles as being waves, previous formulas that worked, for reasons we didn't
understand, could now be proven. In other words, you
could take this formula and idea from de Broglie, and show why Bohr's atomic
model actually works. And shortly after de Broglie's paper, Schrodinger came around and basically set the stage for the entire
rest of quantum physics. And his work was heavily influenced by the ideas of Louis de Broglie. So recapping, light can have particle-like or wave-like properties, depending on the experiment, and so can electrons. The wavelength associated
with these electrons, or any matter particle, can be found by taking Planck's constant, divided by the momentum
of that matter particle. And this wavelength can
be tested in experiments, where electrons exhibit
wave-like behavior, and this formula accurately
represents the wavelength that would be associated
with the diffraction pattern that emerges from that wave-like behavior.