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Okay, so today will be really interesting because we are going to deal with quantum mechanical waves propagating along a transmission line.

And we already started describing a transmission line last time.

So let me remind you of what we did. In the end we will be interested in having a line, say, of two metallic wires next to each other.

And we know that waves can propagate along this line. For example, there will be a charge distribution on one of the wires and an opposing charge distribution on the other wire.

And you can have a pattern which then in the end will propagate as a wave as we will learn.

So in order to describe this situation we made up a discrete model. So in our mind we think of introducing a grid and describing each element separately.

And we agreed that the following kind of model description would be appropriate. We have a network that contains both capacitors and inductances.

And the idea would be that the capacitances stand for the fact that if there is some extra charge here then there will be an electric field and there will be extra energy stored in the electric field.

So energetically this is unfavorable to have extra charge there and charges try to repel each other and to disperse.

So that's the purpose of introducing these capacitances. And then we know that if the charges start to move they will create a magnetic field.

If they create a magnetic field this leads to an electric field that tries to oppose the beginning motion of the charges. So that is encoded by having these inductances.

And now the idea is to attach a charge to any of these nodes. Say qn is the charge sitting at this particular node.

And that would be qn plus one and qn minus one.

And then of course there will be currents flowing, for example in this direction. And in my notation I chose to call this current in. And that was in plus one.

So then we derive two equations. The first one is simply charge conservation.

So we know that the charge qn can change only because there is current flowing in from the left, this is im.

And then there is current flowing out towards the right, so that would be im plus one.

That is charge conservation. And then if there is a voltage difference between these two nodes then this voltage difference will try to drive a current through the inductance.

And that is the correct equation, that the voltage drop equals L times the time derivative of the current.

And so what is the voltage drop? Well we just take the difference between the two voltages.

And these voltages can be obtained by dividing the charge by the capacitance.

So that would then be here qn minus one minus qn divided by c.

So this is a system of discrete equations.

And we have already learned that if we take say the time derivative of the second equation and then insert the first equation, we get a second order and time equation for the current.

And furthermore we learned how to go from the discrete description to a continuum description.

And then we found that the current actually obeys a wave equation.

So the second time derivative of the current we found equals one over little n times little c, which is the inductance per unit length times the capacitance per length times the second space derivative.

So this is a wave equation and it tells us what is the speed of the waves. The square of the speed of the waves is one over lc.

And so indeed we find waves. For example you could build a wave packet that propagates with a velocity v to the right.

You could have another wave packet that comes from the right and propagates to the left. When they feed each other you will see an interference pattern and so on.

Now very briefly I already made a remark last time about how would the continuum version of these equations look like and let me finish the discussion there.

So q dot, that's the first equation, would be equal to, now whenever we have a difference here it would turn into a first order derivative in the sense of a Taylor series.

And since the spacing between different nodes is supposed to be given by a, then this would be a times the derivative.

And since n, that is the point to the left, stands in front, this is actually minus the derivative. So this is minus a times the space derivative of the current.

Now this looks a little bit strange maybe still because in the end we want to let a go to zero.

But now we know that q in this limit will also go to zero because the charge contained in one element will also shrink if we let the element go to zero.

And so it's more convenient to define a charge density which would be q over a.

And then you see this equation just means that the time derivative of the charge density plus the space derivative of the current equals zero.

And now this is nothing but the usual continuity equation. I should point out that the current is the same as the current density in one dimension.

And the reason is the current density as you know is the current per area in three dimensions, the current flowing through an area.

If you transfer this to two dimensions it would be the current flowing through a line section.

And in one dimension it's the current flowing through a point but a point has no dimension so here the current density is the same as the current.

Okay, so that would be the equation of continuity.

And then we can do the same to the second equation.

So again this would turn into a derivative, again minus a times q divided by c equals error times i dot.

And now you can introduce q over c as the voltage.

Again this is convenient because in the limit of a tending to zero both the charge and the capacitance will go to zero in the same manner.

And the voltage will be a finite value.

So this is one thing and then of course if you divide the right hand side by the a that stands here then you'll get the inductance per unit length.