 Consider any one-dimensional wave function psi of x describing a quantum mechanical particle. The value of the wave function, say at point x1, is psi of x1. At point x2 is the function value psi of x2. At point x3 is the function value psi of x3 and so on. You can assign a function value to each x value in this way. We can then represent all function values as a list. We can take this list of values to be a column vector psi that lives in an abstract space. The vector then has the components psi of x1, psi of x2, psi of x3 and so on. We can even visualize this vector as in linear algebra. The first component psi of x1 forms the first coordinate axis. The second component psi of x2, the second axis. And the third component psi of x3, the third axis. We'll stick with just three components because I can't draw a four-dimensional coordinate system. Each component is assigned a coordinate axis. In this way, the three components span a three-dimensional space. As soon as we consider an additional function value psi of x4, the space becomes four-dimensional and so on. We call the vector psi, representing a wave function psi of x, a state vector. Theoretically, of course, there are infinitely many x values. Therefore, there are also infinitely many associated function values psi of x. If there are infinitely many function values, then the space in which the state vector psi lives is infinite-dimensional. This abstract space in which various quantum mechanical state vectors psi live is called Hilbert space. In general, this is an infinite-dimensional vector space. However, it can also be finite-dimensional. For example, the spin-up and spin-down states, which describe a single particle, live in a two-dimensional Hilbert space. That means the state vectors like the spin-up state have only two components. So we can represent a quantum mechanical particle in two ways, as a wave function and as a state vector. In order to better distinguish the description of the particle as a state vector, from the description as a wave function, we write the state vector psi inside an arrow-like bracket. Wave function psi of x, represented as a column vector, is called cat vector, and the arrow-like bracket points to the right. So when you see the cat notation, then you know that it means the representation of the particle state as a state vector. On the other hand, if you see psi of x, then you know that it means the representation of the particle state as a wave function. The vector adjoined to the cat vector is called bra vector. This symbol is pronounced as dagger, so we say psi dagger. For a compact notation, we write the bra vector with an inverted arrow. To get the bra vector adjoined to the cat vector, you need to do two operations. Transpose the cat vector, this makes it a row vector, and then complex conjugate the transposed cat vector. This operation adds asterisks to the components. So let's summarize. The wave function psi in the vector representation corresponds to the cat vector, and the row vector adjoined to the cat vector is the bra vector. Since we have interpreted the wave function psi as a cat vector, we can work with it practically in the same way as with usual vectors you know from linear algebra. For example, we can form a scalar product or a tensor product between the bra or cat vectors. The thing that is probably new to you is that the components of the vector can be complex, and the number of components can be infinite. You can form the scalar product between a bra vector phi and a cat vector psi. Here, we do not need to include the scalar product point and can omit one vertical line. If the state vectors between which you form the scalar product live in an infinite dimensional Hilbert space, then we do not call this operation scalar product but inner product. However, the bra cat notation of the inner product remains the same as in the case of the scalar product. In a finite n dimensional Hilbert space, the scalar product between an arbitrary bra vector phi and a cat vector psi looks like this. The indices 1, 2, 3 up to n and the components are just a short notation for the function values. For example, the component psi 1 stands for the function value psi of x1. You can multiply out the vectors just as you do with the usual matrix multiplication. You can write this equation shorter with a sum sign. Here n is the dimension of the Hilbert space, that is the number of components of a state vector living in this Hilbert space. The scalar product with a sum sign is not exact for states for the infinite dimensional Hilbert space because we would just omit many function values between x1 and x2 points. With infinite dimensional states, we must switch to an integral. Therefore, we replace the sum sign by an integral sign. Of course, we now consider the function values phi i and psi i, not at discrete points xi, but at all points x. So to calculate the inner product of two states phi and psi, we need to calculate this integral. What does this inner product or scalar product actually mean? The inner product like a scalar product is a number that measures how much two states overlap. Another important operation between a bra and a cat vector is the tensor product, or more precisely, outer product. We can omit the tensor symbol because it is immediately clear from the bra-cat notation that it is not a scalar or inner product. The bra and cat vectors are swapped here. The result of the tensor product is a matrix. Just as you know from matrix multiplication, here we multiply a cat vector phi, which is a column vector, by a bra vector psi, which is a row vector. You will encounter such matrices very often in quantum mechanics, for example when learning about quantum entanglement. If we take a normalized state psi, that is, the magnitude of this vector is one, and form a tensor product of this state with itself, we get a projection matrix. When we apply it to any cat vector phi, we multiply a matrix by a column vector. The special feature of a projection matrix is it projects the state phi onto the state psi. In other words, it yields the part of the wave function phi that overlaps with the wave function psi. The result of the projection is thus a cat vector describing the overlap of the wave functions phi and psi. Projection matrices are thus an important tool in theoretical physics to study the overlap of quantum states. Probably the most important use of the projection matrices is the very simple change of bases. If we have some quantum state phi, and we want to look at it from a different perspective, or mathematically speaking represented in a different basis, then of course the first thing we do is choose the desired basis. This is as you hopefully know from linear algebra a set of orthonormal vectors, psi 1, psi 2, psi 3 and so on. Their number is equal to the dimension of the Hilbert space in which these vectors live. For the sake of demonstration, let us assume that our desired basis consists of only 3 basis vectors, psi 1, psi 2, psi 3. With each of these basis vectors, we can construct projection matrices. To represent the quantum state phi in these bases, we form the sum of the projection matrices. As we know from mathematics, the sum of the projection matrices forming a basis is a unit matrix i. The fact that the sum results in a unit matrix is very important when changing the basis, because we do not want to change the quantum state phi. A unit matrix multiplied by a column vector phi does not change this vector. Now we substitute the sum of the basis projection matrices into the unit matrix. The resulting state phi, while notated identically to the state before the basis change, is now represented in the basis psi 1, psi 2, psi 3. We can also, for example, if we want to emphasize the new basis, give it an index psi. I hope you now understand how useful the concept of projection matrices is. In general, we can write the basis change into a basis with n basis vectors by simply replacing the number 3 at the summation sign with n. That's it. Such a basis change with a finite number of basis vectors is of course exact only for states phi and psi living in finite dimensional Hilbert spaces. But how does the basis change work for states with infinitely many components, with an integral? For this, replace the discrete summation with a sum sign by a continuous summation with an integral. Now you should have a solid basic knowledge of bra-cat notation, what bra and cat vectors are, how you use it to form scalar product and inner product, how you construct projection matrices with it, how to perform a basis change with projection matrices in bra-cat notation. In the next video, you will learn about operators used in quantum mechanics, namely, Hermitian operators in bra-cat notation, of course.