 The angular momentum or more precisely orbital angular momentum L of a classical particle is given by the cross product between the distance r of the particle from the axis of rotation and the particle linear momentum P. If we write out this cross product, we get the individual components of the angular momentum vector Lx, Ly and Lz, which indicate the magnitude of the angular momentum in x, y and z direction. This is the magnitude of angular momentum in x direction. This is the magnitude of angular momentum in y direction. And this is the magnitude of angular momentum in z direction. In quantum mechanics, such physical quantities as angular momentum, position and linear momentum are understood as operators. The momentum components Px, Py and Pz in classical mechanics are scalars that is ordinary numbers. In quantum mechanics, they are replaced by operators and marked with a hat to distinguish them from scalars. The classical momentum component Px becomes the operator Px hat and is equal to minus i h bar del x. Py becomes the operator Py hat with a derivative with respect to y. Pz becomes the operator Pz hat with a derivative with respect to z. Here i is the imaginary unit, h bar is the reduced Planck's constant and del x is the derivative operator which when applied to a function yields the derivative of that function with respect to x. Therefore, such an operator takes effect only when applied to a position-dependent function, for example on Psi of x, y, z. The result is the derivative of this function multiplied by minus i h bar. What about the positions x, y and z? The positions are of course now also operators with hats. Applied to a position-dependent wave function Psi, they return the wave function scaled by the position. So the operator here is equal to the position itself. To transform the classical angular momentum components Lx, Ly and Lz into quantum mechanical angular momentum components, we need to convert them into operators. For this purpose, we equip them with hats and insert the momentum operators and position operators. So we get this result for the Lx component. The classical angular momentum component Ly becomes this operator and the classical angular momentum component Lz becomes this operator. All right, we have constructed angular momentum operators. The question now is, are Lx, Ly and Lz Hermitian operators? This is important because only if they are Hermitian operators, they represent physical quantities and can be measured in the experiment. A Hermitian operator h hat is equal to its complex conjugate h dagger. So let us calculate Lx dagger. We know that the position operator and momentum operators are Hermitian operators we can therefore use the property of anti-linearity and pull the dagger into the parentheses. In the next step, we use anti-distributivity. This swaps the two operators. Since the momentum and position operators are Hermitian, we can omit the dagger. The two operators Pz and y may be interchanged because the momentum operator Pz differentiates with respect to the z-coordinate and y does not depend on z and therefore acts like a constant which can be placed behind Pz. The obtained expression on the right hand side corresponds exactly to the Lx operator. Analogously, we can show that Ly and Lz operators are also Hermitian operators. So now we have shown that it is possible to measure the angular momentum components of quantum mechanical systems in an experiment. Perfect. In the classical physics of our macroscopic world, the values of all three angular momentum components, for example of a circling particle, exist and can therefore be determined exactly and simultaneously. In quantum mechanics, on the other hand, we have the Heisenberg uncertainty principle which makes it impossible to determine certain physical quantities exactly all at once because one of the quantities does not have an exact value by nature if the other quantities measured exactly. Momentum Px and position x of a particle are an example of such measured quantities that underlie Heisenberg uncertainty principle. Mathematically formulated, if we first apply the position operator to the wave function psi and then the momentum operator Px, then we get something different if we first apply the momentum operator and then the position operator. Thus it matters whether we first measure the position or the momentum of a quantum mechanical particle. As soon as we reverse the order of measurement, we get something quite different for the momentum and position. We say the momentum and position are affected by the Heisenberg uncertainty principle. The difference of the two measurements is provided by a commutator. For this, we form the difference of the two measurements, bracket out the wave function and the difference of the operators is then the commutator of x and Px. If the commutator is 0, then it is in principle possible to know both observables at the same time exactly. If the commutator is not 0, then it is impossible to know both observables exactly. Only one of the observables can be determined exactly. In the case of the position and momentum operator, the commutator is equal to ih bar. So now we can ask, can we know exactly all three angular momentum components of a quantum mechanical particle? Short answer, no. To prove this, we need to calculate the commutators of the angular momentum components. We will find that they are all non-zero. Therefore, it is impossible to know two angular momentum components at the same time. Let's look at the commutator of Lx and Lz to demonstrate that it is not 0. First, we use the definition of a commutator and insert the determined expressions for the angular momentum operators. Let's multiply out the brackets. Then we swap the operators so that some terms cancel out. In the first term, we can put x at the beginning because x commutates with y as well as with Pz. We can also put the operator Py in front of Pz, but not in front of y because the commutator of y and Py is not 0 but is equal to ih bar. Therefore, we must replace yPy with ih bar plus Pyy. Next, we multiply out the bracket. This term cancels out. In this term, we can interchange all operators without problems and eliminate it with the other term. And also in this term, operators can be interchanged so that it cancels out with this term. Now, we arrive again at a term where the interchange cannot be done just like that. First, we can interchange y and Px and then z with Py. Now, to swap y with Py, we have to replace it with ih bar plus Py times y because of the non-vanishing commutator. Multiply out the bracket. In this term, we swap z with Px and can thus cancel it out with the other term. We can factor out ih bar. The expression in the bracket corresponds to the Ly operator. As you can see, the commutator between Lx and Lz is not 0, so it is impossible to know Lx and Lz simultaneously with arbitrary precision. Analogously, you can derive the other two commutators. We can easily illustrate this uncertainty of angular momentum components using a classical particle. So, let us consider a particle moving on a circular path. So, it has an angular momentum L. Let's assume that we have measured its angular momentum component Lz. Because of the non-vanishing commutators, the components Lx and Ly have no definite value. The direction of the total angular momentum vector L is no longer uniquely given, but lies somewhere on this cone mantle. From the illustration, we can already guess that although the direction of L is not unique, the length of the L vectors is. We can determine the length of the L vector with the sum of the squares of the angular momentum operators, Lx squared plus Ly squared plus Lz squared. This sum is briefly notated as the L squared operator. This operator is Hermitian, so it represents a physically measurable quantity, namely the length of the angular momentum vector squared. And the great thing is, this operator computes with each angular momentum component Lx, Ly and Lz. This is very very good, because it allows us to know exactly not only one angular momentum component for a quantum mechanical particle, but also the magnitude of the total angular momentum. This would be very bad for physics if it were not the case. Because without a fixed total angular momentum, the law of conservation of angular momentum in quantum mechanics would not work at all. A commutator tells us not only whether two observables are accurately measurable simultaneously, but whether the associated operators have common eigenfunctions psi. Now you have to recall your knowledge from linear algebra. You can imagine all operators as matrices and the wave functions as vectors. Now if an operator such as L squared is applied to a state that is an eigenstate of that operator, then the result is a scaled eigenstate. That is, the eigenstate psi is not changed at all, but only multiplied by the number lambda. And this number is physically quite special, because it corresponds to a possible measured value of this operator. In the case of L squared operator, this number yields the magnitude of the total angular momentum squared. And if Lz is applied to the state psi, which is also an eigenstate of Lz, then we again get the scaled eigenstate out with a different eigenvalue Mu. This eigenvalue represents the magnitude of the angular momentum component, Lz. If the commutator of L squared and Lz vanishes, and it does, then we know that there is a state psi that is simultaneously both an eigenstate of L squared and an eigenstate of Lz. With the help of the latter operators, we can determine these eigenvalues L squared and Lz a bit more precisely. How this is done, you'll learn in a separate video or on my website. I show here the famous result, which probably every chemistry student knows. The eigenvalues L squared are a multiple of h bar squared. L times L plus 1 times h bar squared. The factor L times L plus 1 is determined by an integer or half integer number L, which we call the angular momentum quantum number. It can take the values 0, 1 half, 1, 3 and a half, 2 and so on. So the total angular momentum squared can only have these values. Thus also the magnitude L of the total angular momentum is quantized and can have only these values. The eigenvalues Lz of the Lz operator are a multiple of h bar. Lz is equal to mh bar. The integer m is called the magnetic quantum number and it can only take values between minus L and L in plus one steps. For example if L is equal to 2 then m can take the values minus 2, minus 1, 0, 1 and 2. The Lz angular momentum component is also quantized. It has a further constraint by the total angular momentum represented by the quantum number L. If the total angular momentum is at L equal to 2 then the Lz angular momentum component can have only five possible values minus 2h bar minus 1h bar, 0, 1h bar or 2h bar. So let's summarize what you've learned. You now know how to turn classical angular momentum into a quantum mechanical angular momentum operator. You know how to show that angular momentum components are physically measurable quantities. You have learned that all three angular momentum components are affected by the uncertainty principle and that only one of the components can be determined exactly. You have also learned how to work with angular momentum commutators. You have learned about the L squared operator which can be used to calculate total angular momentum and you have learned the allowed eigenvalues of L squared and Lz operators. In another video we will look at the quite revolutionary meaning of the half integer values of the quantum number L.