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HAZEL SIVE: The last thing I want to tell you about in this class is probably the coolest. And that is that macromolecules carry information. And they carry information because they have got different ends and because they have got direction. And if you think about it, all languages have got some kind of start and stop signals to them. And there is some direction associated with them. And the language of life is exactly the same. So let us write down that macromolecules may have order and polarity. The polarity would give some kind of direction. And these two attributes together comprise information that can be used by the cell. The two classes of macromolecules that do this par excellence are the nucleic acids and the proteins. And so we're going to talk about both of those-- nucleic acids and proteins. The nucleic acids as I said carry the information of hereditary. The proteins carry the information of just about everything else. So these two classes are what we...

PROFESSOR: So one more thing let's do with this operator. So we're getting accustomed to these operators, and these permutation operators can also act on operators themselves. So that's important. So consider the action on operators. So for example, an operator B that acts belonging to the linear operator some V, when acting on V tensor V, we define two operators, B1 and B2. And you define them in an obvious way, like B1 acting on Ui 1 tensor Uj 2. OK, B as an operator knows how to act on every vector on the vector space capital V. So when you say 1, you're meaning that this operator acts on the first Hilbert space. So this is equal to B times Ui 1 tensor Uj 2. So it just acts on the first state. How does it act? Via B, that is an operator, in the vector space V. Similarly, if you have B2 of Ui tensor Uj 2 you have Ui 1 tensor BUj 2 to OK. So these are operators that act either on the first state or in the second state. So the permutation operators can do ...

PROFESSOR: Important thing to do is to just try to understand one more thing. The creation and annihilation operators-- what do they do to those states? You see, a creation operator will I add one more a dagger, so somehow must change phi n into phi n plus 1. A destruction operator with an a will kill one of these factors, and therefore it will give you a state with lower number of phi n minus 1. And we would like to know the precise relations. So look at this. Let's do with an A on phi n. And we know it should be roughly phi n minus 1. This is one destruction operator, but we can do it. Look-- this is 1 over square root of n. A times a dagger to the n phi 0. A with a dagger to the n phi 0, we can replace by a commutator again. Commutator of a, a dagger to the n phi 0. This is 1 over square root of n factorial, and here we get a factor of n times a dagger to the n minus 1 phi 0. You know, it's all a matter of those commutators we on the left blackboard. But this s...

PROFESSOR: Now, the standard thing you do with a recursive data type in programming is you define recursive procedures on them, so let's look at how that works. I'm going to define a function f on a recursively defined data type R, and the way I'm going to do it is I'm going to define f of b explicitly in terms of b and operations that are already understood for all of the base cases of b in R. And then I'm going to define f of a constructor applied to x in terms of x and f of x. And if I keep to that structure, that gives me a recursive definition of the function f on the recursively defined data set R. Let's look at an example to make this recipe explicit and clearer. Let's think about a recursive function on a set of matched brackets. This is a somewhat interesting one. Let's define the depth of a string as follows, and the idea is it's how deeply nested are the successive pairs of left and right brackets. Well, the depth of the empt...