Contents:
Multiplying together the errors in the measurements of these values the errors are represented by the triangle symbol in front of each property, the Greek letter "delta" has to give a number greater than or equal to half of a constant called "h-bar". Planck's constant is an important number in quantum theory, a way to measure the granularity of the world at its smallest scales and it has the value 6. One way to think about the uncertainty principle is as an extension of how we see and measure things in the everyday world. You can read these words because particles of light, photons, have bounced off the screen or paper and reached your eyes.
Each photon on that path carries with it some information about the surface it has bounced from, at the speed of light. Seeing a subatomic particle, such as an electron, is not so simple.
You might similarly bounce a photon off it and then hope to detect that photon with an instrument. But chances are that the photon will impart some momentum to the electron as it hits it and change the path of the particle you are trying to measure. Or else, given that quantum particles often move so fast, the electron may no longer be in the place it was when the photon originally bounced off it.
Either way, your observation of either position or momentum will be inaccurate and, more important, the act of observation affects the particle being observed. The uncertainty principle is at the heart of many things that we observe but cannot explain using classical non-quantum physics. Take atoms, for example, where negatively-charged electrons orbit a positively-charged nucleus. By classical logic, we might expect the two opposite charges to attract each other, leading everything to collapse into a ball of particles.
The uncertainty principle explains why this doesn't happen: This means that the error in measuring its momentum and, by inference, its velocity would be enormous. In that case, the electron could be moving fast enough to fly out of the atom altogether. Heisenberg's idea can also explain a type of nuclear radiation called alpha decay. Alpha particles are two protons and two neutrons emitted by some heavy nuclei, such as uranium Usually these are bound inside the heavy nucleus and would need lots of energy to break the bonds keeping them in place.
But, because an alpha particle inside a nucleus has a very well-defined velocity, its position is not so well-defined.
That means there is a small, but non-zero, chance that the particle could, at some point, find itself outside the nucleus, even though it technically does not have enough energy to escape. When this happens — a process metaphorically known as "quantum tunneling" because the escaping particle has to somehow dig its way through an energy barrier that it cannot leap over — the alpha particle escapes and we see radioactivity.
A similar quantum tunnelling process happens, in reverse, at the centre of our sun, where protons fuse together and release the energy that allows our star to shine. The temperatures at the core of the sun are not high enough for the protons to have enough energy to overcome their mutual electric repulsion. But, thanks to the uncertainty principle, they can tunnel their way through the energy barrier. Perhaps the strangest result of the uncertainty principle is what it says about vacuums. If I decrease the uncertainty of the position, so I decrease it to one, so the uncertainty in the momentum must increase to four, because one times four is equal to four.
If I decrease the uncertainty in the position even more, so if I lower that to point five, I increase the uncertainty in the momentum, that must go up to eight. So point five times eight gives us four. And so, what I'm trying to show you here, is as you decrease the uncertainty in the position, you increase the uncertainty in the momentum. So another way of saying that is, the more accurately you know the position of a particle, the less accurately you know the momentum of that particle.
And that's the idea of the uncertainty principle. And so let's apply this uncertainty principle to the Bohr model of the hydrogen atom. So let's look at a picture of the Bohr model of the hydrogen atom. Alright, we know our negatively charged electron orbits the nucleus, like a planet around the sun. And, let's say the electron is going this direction, so there is a velocity associated with that electron, so there is velocity going in that direction. Alright, the reason why the Bohr model is useful, is because it allows us to understand things like quantized energy levels. And we talked about the radius for the electron, so if there's a circle here, there's a radius for an electron in the ground state, this would be the radius of the first energy level, is equal to 5.
So if we wanted to know the diameter of that circle, we could just multiply the radius by two. So two times that number would be equal to 1. And this is just a rough estimate of the size of the hydrogen atom using the Bohr model, with an electron in the ground state. Alright, we also did some calculations to figure out the velocity.
So the velocity of an electron in the ground state of a hydrogen atom using the Bohr model, we calculated that to be 2. And since we know the mass of an electron, we can actually calculate the linear momentum. So the linear momentum P is equal to the mass times the velocity. So we have point one here. If I want to know the uncertainty of the momentum of that electron, so the uncertainty in the momentum of that particle, momentum is equal to mass times velocity. So let's go ahead and do that. So we would have the mass of the electron is 9.
The velocity of the electron is 2. We're gonna multiply all those things together.
So we take the mass of an electron, 9. So the uncertainty in the momentum is 2.
And the units would be, this is the mass in kilograms, and the velocity was in meters over seconds, so kilograms times meters per second. Alright, so this is the uncertainty associated with the momentum of our electrons. Let's plug it in to our uncertainty principle here: So we can take that uncertainty in the momentum and we can plug it in here. So now we have the uncertainty in the position of the electron in the ground state of the hydrogen atom times 2.
This product must be greater than or equal to, Planck's Constant is 6. Alright, divide that by four pi. So we could solve for the uncertainty in the position.
So, Delta X must be greater than or equal to, let's go ahead and do that math. So we have Planck's Constant, 6. So we also need to divide by the uncertainty in momentum, that's 2. So the uncertainty in the position must be greater than or equal to 2.
Let's go back up here to the picture of the hydrogen atom. So the uncertainty in the position would be greater than the diameter of the hydrogen atom, using the Bohr model. So the Bohr model is wrong.