Schrödinger's Cormorant

4.6 Getting to know the General

Considers the general features of any one dimensional wave. Introduces the calculus of functions with more than one independent variable: partial differentiation.

So far, every wave we have contemplated has been some kind of sine wave. But not every wave is as regular as that and it would be hopelessly over-optimistic to expect that our matter waves will always be so simple, so we need a more general definition of a wave and – most importantly - a mathematical approach to deriving the equation for a wave.

Consider, for example, a function that looks like this:

This could become a wave if it shifts along the x-axis as a function of time. The key thing that would make it a wave, rather than any old time-evolving function, is if it retains its form as it goes. Like this:

We know how to express this retention of form as the wave moves in time, because we have already used the idea to write our equation for a sine wave: in the example above, we can choose an instant and call it time 0. We can then define the waveform at this instant as a function, y = f(x). The wave is travelling to the right with a constant velocity v, so after time t it will have moved by a distance vt (since distance = velocity x time). The value of y at any point x in the waveform as it is at time t will therefore be equal to the value of y at the point x-vt in the waveform as it was at time zero. In other words, we can write a general equation for the wave moving to the right:

y = f(x-vt)

What we need to do is to find a way to deduce the general nature of a function f which will form a wave. The key to this can be seen in the picture below:

Here, the wave is moving to the right, so the dashed line shows a snapshot of it at a slightly later time than the full line. If we look at position x1, we can see that the gradient of the waveform is positive at this point (ie, as x increases, so does y). Because of this, as the wave shifts to the right it brings a portion of the waveform with smaller y to position x1: in other words, as time t increases, y decreases.

On the other hand, if we look at position x2, we can see the opposite pattern of behaviour: the gradient is negative, so y decreases as x increases. Consequently, as t increases it brings a portion of the waveform with higher y values to position x2.

What this shows is that these two properties – the change in y with x and with time are related and – in the case of a wave moving to the right - they are opposite in sign. If we could express this relationship mathematically, it could be the key to a general definition of a wave. “Let’s differentiate!” shouts Cormorant excitedly, appreciating that this is the mathematical way to quantify the sensitivity of a dependent variable (y) to a change in another variable – time or distance. But Nefertiti has a cautionary tale:

It is October in England and Nefertiti tells you and Cormorant that she wants to know how the midday temperature varies with time in this part of the year. You both enthusiastically agree to help with the great project but while you are happy to sit in your deck chair and monitor the temperature as each day goes by, Cormorant is eager to get on with his migration to his winter home in Casablanca, so he decides that he will interrupt his flight south each day, wherever he may be, to measure the midday temperature and report back to Nefertiti. The results are in:

You can see the issue here – if we measured the average gradient of these two lines, we would conclude that (if temperature is T and time is t), dT/dt is positive as measured by Cormorant but negative as measured by you. Neither of you is wrong – the point is that the temperature is a function of more than one variable (time and latitude, at the very least) and we can’t differentiate meaningfully with respect to one of these variables without considering what is happening with the other one. No doubt you can see Nefertiti’s point – the value of y in a wave is a function of both displacement (x) and time. So if we’re going to make progress with the mathematics of waves, we have to find a way to deal with this problem – how does differentiation work when you’ve got more than one variable?

We deal with this by introducing the idea of a partial derivative. Suppose we have a function z which depends on two or more variables: z = f(x, y).

We define the partial derivative, ∂z/∂x, to be equal to dz/dx for the case where y is held constant. This seems quite restrictive but is actually extremely useful and it makes the differentiation process easy because constants are generally straightforward to deal with. Because you’ll no doubt be chatting about this with your friends and family, it’s useful to know that the curly ∂ is called “del”. Here’s how it works:

First suppose we have a function of a single variable, such as z = 2sin x + 3

We can differentiate, knowing what to do with the constants 2 and 3, to get: dz/dx = 2cos x

Now compare a function of two variables, z = 2y2sin x + 3y

We can differentiate with respect to x under the condition that y is constant (and hence 2y2 and 3y are constants), to get:

∂z/∂x = 2y2cos x

We can also differentiate with respect to y, with x held constant (hence sin x is constant):

∂z/∂y = 4ysin x + 3

This is all we need to know about partial derivatives, for our purposes, so if you're eager to move on and see how they fit in to to our story, go ahead . If, however, you're intrigued and would like to think a bit more about their meaning and significance, have a look at this little appendix, first:

Extension: more about partial derivatives

Back

Chapter Guide

Page updated

Google Sites

Report abuse