So with that in mind, I'm going to give you an opportunity to assess yourself
how, how well you've followed that last concept.
You'll get a little quiz to come up in a moment, and you can look at the
explanation for why the correct answer is the correct answer.
And then we'll move on again. Alright, I want to wrap up now by
explaining how do we get these quantized energy levels.
How can we use quantum mechanics to predict allowed energies for systems.
And the answer to that is the Schrodinger Equation.
So here's Erwin Schrodinger, and he was a German physicist who first explored how
to take wave mechanics and use it to describe the properties of energy and
matter. And so, here is the one dimensional
Schrodinger equation. And it's not terribly important that you
memorize this equation. I'll just call out some key features of
it It says that there is this thing, it's called a wave function, psi, operating on
the wave function by taking its second derivative.
And it's one dimensional, so it depends on x, and I'm taking the second
derivative with respect to x. Multiplying by h bar squared.
So here's Planck's constant. But h bar is Planck's constant divided by
2 pi, it's just a short-hand way to keep track of a factor of 2 pi.
A mass associated with the system, whatever it be.
A potential energy term, so there's a kinetic energy term and a potential
energy term. Those are the two kinds of energy that
you might find in a physical system. And this wave function, when operated on
in this fashion, gives rise to a loud energy levels, that satisfy this
equation. So, what we do is solve the Schrodinger
equation for a given potential, and so depending on the nature of the system,
there are different potentials: particles confined in boxes, free particles,
particles rotating in a circle. That defines what the potential might be.
It also defines a set of boundary conditions.
So that's typically quite important in quantum mechanics.
Where do we know the system is not allowed to be.
So particle in a box for instance, can't be outside the box.
Given those potentials and boundary conditions we solve for the wave
functions, and each of those wave functions they're indexed by n, just some
integer, we count them, one, two, three, four, five.
And every one of the wave functions has an associated energy.
And those are the allowed energy levels, we say they are allowed.
If you're wondering what, what this wave function really contains within it, it
turns out that the wave function when you take its square modulus over a certain
volume, or this is one dimensional, so over a certain length.
You determine the probability that you'll find the system within whatever small
volume or length you're surveying. And ,it's an interesting concept and one
that troubles people in the sense that it says there's no objective reality until
you actually do an experiment. There's only a probability that something
may be somewhere, and when you do the experiment you either find it or you
don't. But it's not that it really was or wasn't
there ahead of time. It's just probabilistically distributed.
It exists in what's called a superposition of states.
So, some of you may have heard of Schrodinger's cat.
And this is a thought experiment that Schrodinger proposed.
That if you put a cat in a box with a radioactive nucleus that could decay
randomly at any moment, and when it decays it breaks a vial of poison and
that kills the cat. But until you open the box, you don't
know if the decay has happened or not. And so is the cat alive or dead?
It's neither. It's in a superposition of states.
And until you open it, you won't know. Now, I don't like cruelty to animals so
I'm going to just imagine that cat is alive.
But it's a fundamentally different way of thinking about the universe obviously.
And that's one of the fascinating things about quantum mechanics.
So, actually, defining the relevant Schrodinger equations and doing the
solutions to get the energy levels, that's what a course in elementary
quantum mechanics is about. We are not going to do that.
All we do need to do is be familiar with the allowed energy levels and for the
systems that we'll encounter. And so, we will be going through some of
those in the very near future. But before that, we're going to do a
demonstration that will help to illustrate the concept of quantized
energy levels. And in particular, we're going to look at
the hydrogen chloride cannon. So this is a reasonably exciting
demonstration I think. And I hope you'll enjoy it as much as I
enjoy doing it. In this demonstration, I want to show you
a chemical reaction that we will look at in more detail, to see how thermodynamics
helps us explain, in quantitative detail, what we observe qualitatively.
In this apparatus is a tube that has been filled with a mixture of hydrogen gas,
H2, and chlorine gas, Cl2. The tube is further encased in a clear
plastic sleeve, for reasons that will become clear shortly.
These two gases can react with one another to make a new gas hydrogen
chloride, dissolving hydrogen chloride in water is how hydrochloric acid is made.
Just as was true for the thermite reaction, demonstrated in this course's
introductory video, reaction of H2 and Cl2 to make HCl is favorable, but it
requires and initial kick to get the reaction started, and that kick is the
energy required to break the bond between two chlorine in Cl2, which amount of
energy is 242 kilojoules per mole. I'm holding here three different laser
pointers. One lazes red light.
One green light and one blue light. As we'll see in some exercises in an
early lecture, the energy of a photon, which is the fundamental unit of light,
is determined by its wavelength. The red, green, and blue laser pointers
emit wavelengths of 650, 532, and 405 nanometers respectively.
Let's see which, if any, of these wave lengths seem sufficient to break the
chlorine-chlorine bond. Here's red shining into the gas mixture.