 This is how the mean value of an operator h in the state psi is calculated in quantum mechanics. This is, as you know, a bra-cat notation which hides an integral in the case of an infinite-dimensional Hilbert space. In this example, the mean value of the operator h is calculated using the one-dimensional spatial wave function psi of x and its complex conjugate. A mean value, as we know it from everyday life, is a real number. In quantum mechanics, however, we can also get a complex mean value with a real and imaginary part. For example, if we want to calculate the mean value of position or mean value of energy, then we expect quantum mechanics to give us a real value. A complex position or energy do not make any sense. In such cases, we require that the mean must be equal to the complex conjugate mean. This is a requirement that is only satisfied by a real number. As you know from mathematics, when we complex conjugate a real number, it remains unchanged. But if we complex conjugate a complex number, then the sign changes. So by this requirement, we exclude complex mean values. Our demand for real numbers has far-reaching consequences for the operator h in its properties. Let's find out what we will get if we manipulate the complex conjugate integral. First, we can swap the two scalar functions h psi and psi star. Then we apply the complex conjugation, that is, the star, to the individual integrants. Psi, twice complex conjugated, gives psi again. Let's also write the result in bra cat notation. The rewritten complex conjugate mean in bra cat notation must be equal to the original mean according to our requirement. As you can see, in order for our requirement for a real mean to be satisfied, it must be possible to interchange the operator h in the scalar product. So it must not matter whether we apply h to the cat or to the bra vector. The mean value remains the same. Such an operator, which can be shifted back and forth in the scalar product without changing the mean value, is called a Hermitian operator. Here you see the definition of an adjoint operator hdega. So this is an operator which is applied to the bra vector and gives the same scalar product as the operator h, which acts on the cat vector. Compare the definition of the adjoint operator with the definition of the Hermitian operator. And you will see that h must be equal to hdega. An operator that is equal to its adjoint is called a self-adjoint operator. Since it doesn't matter whether the Hermitian operator is applied to the bra or to the cat vector first, h can be placed between the bra and the cat. The operator then acts on either the left or right successor. Similarly, the adjoint operator can be written between the bra and cat vectors. A Hermitian operator has three other incredibly important properties. Property number one, Hermitian operators have real eigenvalues. Let's prove it quickly. So we take the eigenvalue equation for the Hermitian operator h. Phi is an eigenvector of h and lambda is the associated eigenvalue. Then we apply the bra vector phi to the eigenvalue equation. We may pull out the eigenvalue from the scalar product because it is just a number. Now we apply the property that defines the Hermitian operator. We can apply h to the bra vector instead. The scalar product remains the same. h applied to the bra vector yields the complex conjugate eigenvalue, lambda star. This is also a number that we can pull out of the scalar product. If we now equate both versions, bring everything to one side of the equation and factor out the scalar product phi with phi, then we find that lambda must be equal to lambda star so that this equation results in zero. The eigenvalue lambda is therefore equal to its complex conjugate and thus real. Property number two, eigenvectors of a Hermitian operator are orthogonal if they have different eigenvalues. Let's prove this quickly. Let's take two eigenvalue equations for the Hermitian operator h. Phi one and phi two are two different eigenvectors of h and lambda and mu are the corresponding eigenvalues. Let's apply the bra vector phi two from the left to the first eigenvalue equation. The eigenvalue lambda is a real number which we can place in front of the scalar product. We express the second eigenvalue equation with the bra vector. This is done by simply flipping the two kets, but in general we also need to take the adjoint operator of h and complex conjugate the eigenvalue. But we are dealing with a Hermitian operator here, so h dagger is equal to h and mu star is real, so we can just write mu. Let's set our two results equal. Put everything on one side and factor out the scalar product. If we assume that the eigenvectors are non-degenerate, that is they have different eigenvalues lambda and mu, then the scalar product between the two eigenvectors of h must be zero. The non-degenerate different eigenvectors of a Hermitian operator are therefore always orthogonal to each other. property number three. The set of eigenvectors of a Hermitian operator can be used as a basis. This property is so important that it has a name, the spectral theorem. You can find the proof in the article for this video. A linear operator h is, as you know, a linear map, for example, from a two-dimensional Hilbert space to the same Hilbert space. The basis for this Hilbert space in this case consists of two vectors. According to the spectral theorem, we can take two different eigenvectors of h as basis vectors and thus represent every possible cat from this Hilbert space. As you know from linear algebra, you can represent a linear map as a matrix. Thus, we can write our Hermitian operator, which maps the Hilbert space unto itself as a matrix. If the Hilbert space is two-dimensional, then the operator h corresponds to a two-by-two matrix. A Hermitian operator represented as a matrix is called a Hermitian matrix. Remember the property h equals h dagger? In the matrix representation, dagger stands for a transposed and complex conjugated matrix. A Hermitian matrix is equal to its transposed and complex conjugated matrix. Let's make a few examples. Here we see, for example, a Pauli-Sigma y matrix. This is clearly a Hermitian matrix, because if we transpose it and complex conjugate all the elements, we get back the same matrix. This is a Pauli-Sigma x matrix. This is a real matrix because it does not contain complex components. This is also a Hermitian matrix, because if we transpose it and complex conjugate all the elements, we get back the same matrix. You can also see from this example that real Hermitian matrices are equivalent to symmetric matrices. Here you can see an example of a non-Hermitian matrix. Because if you transpose it and complex conjugate it, you get a completely different matrix, which is not equal to the original one. So, that's it. If you want to decide which topic I should cover next, feel free to participate in the polls I do on the channel. With this in mind, bye and see you next time.