1 00:00:00,414 –> 00:00:03,711 Hello and welcome back to the next mathematical symbol.

2 00:00:03,786 –> 00:00:06,516 Which is given by these pointy brackets.

3 00:00:07,086 –> 00:00:12,637 Often there is a comma in between, but sometimes you also see a straight line

4 00:00:12,837 –> 00:00:16,896 and this is in mathematics how we denote an inner product.

5 00:00:17,371 –> 00:00:22,077 This means it’s a special multiplication we find in a vector space.

6 00:00:22,586 –> 00:00:28,898 So you see one vector you would put here on the left-hand side and another one here on the right-hand side.

7 00:00:29,343 –> 00:00:36,652 Hence if we call the vector space V, we can see it as a map defined on the Cartesian product, V with itself

8 00:00:37,043 –> 00:00:42,388 and then the outcome of this inner product calculation should be a real or complex number.

9 00:00:42,871 –> 00:00:49,321 Hence we immediately recognize it’s a very abstract concept, but it has a lot of concrete applications.

10 00:00:49,914 –> 00:00:56,555 For example it’s used a lot in physics and especially this notation you will see in quantum mechanics.

11 00:00:56,986 –> 00:01:04,283 As an example we can say in quantum mechanics, they would calculate the inner product of 2 states. Psi and Psi tilde

12 00:01:04,743 –> 00:01:08,461 and then they might say: this is the bracket notation of quantum mechanics,

13 00:01:08,661 –> 00:01:12,203 but mathematically it’s just the inner product in a vector space.

14 00:01:12,714 –> 00:01:17,845 However, then in physics you also see a funny thing. They separate this bracket.

15 00:01:18,045 –> 00:01:26,969 On the one hand they say we have the called bra vector Psi and on the other hand the so called ket vector Psi tilde

16 00:01:27,514 –> 00:01:33,266 and then you can argue, putting both things together gives us a so called bra-ket.

17 00:01:33,466 –> 00:01:38,121 In short you can just say, this is another way to explain the inner product.

18 00:01:38,686 –> 00:01:42,590 In physics this is usually called the so called Dirac notation

19 00:01:42,943 –> 00:01:49,201 and if you want to learn more about this and inner products, I have a whole functional analysis course you can watch.

20 00:01:49,543 –> 00:01:54,403 So I really hope that I see you there or in the next video about mathematical symbols.