In this video, we will revisit orbital angular momentum and construct the matrix elements for the rotation operator generated by orbital angular momentum operators. When spin angular momentum is zero or negligible, then obviously the entire angular momentum is the orbital angular momentum and orbital angular momentum is given by this classical definition cross product between the position and linear momentum vector. We first check the computational relation between various components X, Y, Z, components of the orbital angular momentum, use the defination and use the commutational relations between the position and momentum vector. We can derive that they do not commute the X, Y, Z components of the orbital angular momentum operators do not commute. Instead, they satisfy this general angular momentum commutation relations which we showed for the general angular momentum operator J. Orbital angular momentum is a special case of a general angular momentum vector, vector operator, and we anticipate that they satisfy the same commutation relations and we just show that. Now, next we consider the infinitesimal rotation operator defined with the Z component of the orbital angular momentum. Now, let's apply this infinitesimal rotation to a position eigenket, so x prime y prime, Z prime represents the position of this position eigenket. Once again, we express Else of Z as the cross product and now because the position operator and the momentum operator for different components, they commute so we can simply switch the order and rearrange like this. Then we rewrite this whole expression as this and notice that this part I divide it by hi and then quantity piece of y times this, plus piece of X times this. That is just a dot product of a momentum vector with a certain position vector. That is just a infinitesimal translation operator. This operator simply translate this position eigenket by the translation vector defined by this position vector dot product with the momentum vector here. The X component translation is by this amount to multiply to this piece of x operator and that's this. Then the y translation is by this amount multiplied to the piece of y vector that's here. The x component is translated, shifted by this amount and then y component is shifted by this amount. You immediately notice that this represents a rotation of a position vector by an angle Delta V about the axis. Now, recall the general expression of a wave function of a quantum state, Alpha, quantum state specified by a state can Alpha the spatial wave function. Wave function in position representation is generally expressed by this inner product of the car with the position eigenket. This should give you the position of representation, which is the familiar wave function as a function of spatial coordinate. If we do this, so if we multiply that inner product with this position eigenbra to the rotated ket. Ket rotated by this infinitesimal rotation operator and then we take on the inner product with this position eigenbra. This will give us the wave function of this rotated ket and this operator can operate on to the left and to this bra and then we already found what it does. It does this. Now, to see things more clearly, we change the position coordinates from the Cartesian coordinates to the spherical polar coordinate system. This is just the same ket, same position eigenket, this time specified by the spherical polar coordinates r theta. In this case, the infinitesimal rotation about the axis is simply represented by this small change in V , angle V. Because delta phi is supposed to be small, we can take a Taylor expansion and represent like this. Now, because these r, theta, phi, bra are any arbitrary position, it could be any arbitrary position. This equation should be valid for any arbitrary position. We can identify this one here, identify the operator with this part and the second term containing the L_z operator with the second term, the differential operator, and identify that the L_z acting on a Ket alpha. Then if you represent this L_z alpha in a position representation by taking our inner product with this position eigen bra, it will give you this differentiation. From this we identify this L_z operator can be represented in position or representation by this differential operator with respect to phi, which is something that we derived earlier using the classical definition of L_z and deriving on the basis of the wave mechanics. We obtain the same equation simply by considering an infinitesimal rotation generated by these orbital angular momentum operator L_z. We can do the same thing for L_x. This time the L_x will rotate your y and z coordinates like this and converting that x, y, z coordinates into spherical polar coordinate system. Once again, we obtain this familiar differential operator form for L_x. Likewise, we derive the same differential operator expression for L_y. Now we proceed to the ladder operator and the ladder operator if you recall, is defined by this equation, L_x plus minus i times L_y is the definition. Using this differential operator definition, we immediately find the differential operator expression of these ladder operator, L_x plus and L minus. Finally, we write down L square operator as this. L square is obviously L_x square plus L_y square plus L_z square. L_z square is here, L_x square plus L_y square is expressed in terms of the ladder operator here. We just found the differential operator form of L_z, L_+, L_-, plugging in all to that, simplifying it, you will see that this is the differential operator form for L square operator and this is the angular part of the laplacian operator L square, which again is something that we derived earlier from the white mechanics formalism. Earlier, we obtained the same differential operator forms for all of these angular momentum operators using the differential operator form of the momentum operator. We plugged in the differential operator form of the linear momentum operator. We convert that into spherical polar coordinate system and we obtain this operator form for L square and all the other differential operator forms that we showed in the previous slide. This time we obtain the same exact operator form simply by using the computation relation and the infinitesimal rotation generated by these angular momentum operators. Either way, the L square and L_z, the differential operator forms were found to be identical and therefore the simultaneous eigenkets or eigenfunctions of these operators are given by the spherical harmonics functions. Now we can solve the eigenvalue equation to obtain spherical harmonics as before, because we have the same differential equations and the solution obviously will be the same spherical harmonics, we already know that. Now, alternatively, we can use a ladder operator and we derive or obtain the same spherical harmonics solutions, all in a completely different way. Start with this, when M quantum number M is at its maximum value, which is l, then operating a lateral operator L_plus raising operator L_plus will give you a null ket. Using the differential operator form for L plus, and writing down these ll eigenket as these y sub ll spherical harmonics, and then equating it to zero. This is the position representation of this simple equation. Now, we already know that the Phi-dependence of these eigen function e to the i, l Phi from the L sub z eigenvalue equation. Therefore, we can plug that in, e to the i, l Phi here, and this differentiation with respect to Phi therefore will simply yield i times l falling out in front and, so then you get a very simple first order differential equations with respect to Phi, and by doing a simple substitution and change a variable, you will see that this solution is simply given by this, so the Phi part is once again, e to the i, l Phi, and then the Theta part is sine Theta, lth power. Then there is of course the constant to be determined. The constant is determined by normalization condition. You will find the constant is given by this, proper after proper normalization. Now, we obtained a functional form for y sub l, l, and now we apply the lowering operator L minus to that y sub l, l to generate all other spherical harmonics, so complete set of spherical harmonics functions. Now we call the equation for j sub m that we obtained in the previous lecture. We write in terms of spherical harmonics, so replace these kets with a spherical harmonics functions, and replace this J minus with L minus as shown here. Then that's this. Therefore, the Y sub l, m function is obtained by taking a derivative of sine Theta times 2l, and you take a derivative by l minus m times in order to achieve these value m here. This was the constant that we derive from the normalization condition. This was the Phi dependence that we obtained from the l of z eigenvalue equation. This is a general expression for the spherical harmonics function y sub l, m. Once again, this expression is valid for integer values of l and m. Half integer values are not allowed because we're only dealing with orbital angular momentum and If you have a half-integer value, then those include spin states, and those cannot be represented by these spatial functions of spherical harmonics function. This is valid for integer values of l and m and If you should be able to show yourself that this general of differential form does indeed reproduce the complete set of y, l, m spherical harmonics function. Now, let us consider a rotation operator that rotate the z-axis to an arbitrary axis m. This is the general equation. This is the rotation operator acting on z-axis and turn it into some arbitrarily oriented axis n. These are the direction eigenkets corresponding to a unit vector oriented in some direction. In terms of euler angles, notice that a general rotation operator that rotates z-axis is specified by this Alpha equals Phi, Beta equals Theta, and Gamma equals zero, so recall the general euler rotation of coordinate system, which we discussed earlier, you want to describe an arbitrary rotation by this arbitrarily oriented axis z prime. How do you obtain this z? You first rotate your y-prime axis back to the y-axis, and then you rotate your z-axis about this y-axis to obtain z-prime. This is the polar angle beta, and that's this. Then you rotate this z-prime axis about z-axis by an angle alpha. This is this angle alpha here and that's the azimuthal angle. That gives you this z-prime axis. The gamma is the final rotation about this z-prime axis, but here, we're only interested in the orientation of the z-axis rotated to a certain axis and prime, which is z-prime in this diagram here. Gamma, we don't consider so we don't rotate about. We ignore this final rotation by Gamma and then only consider this first two angles, Alpha and Beta. Alpha being the azimuthal angle phi, beta being the polar angle theta to specify this n-prime in hat axis, which is equal to z-prime axis here in this diagram. Now we re-express this equation by inserting the identity operator, this l, m cut and l, m bra outer product, summed over all values of l and m. This from the completeness, is equal to identity operator. We can insert identity operator between the operator and z cut. We can do that and multiply to this equation, both sides of this equation, with a l, m prime. This is supposed to be a prime here, so l, m prime. Left hand side is this. On the right hand side, you get this. This here is simply the matrix element of this rotation operator. We denote this as this. For all different values of l's, these matrix element is zero. The matrix is specified for a fixed value of l as we have shown before for the general angular momentum state. Then the row and columns are specified by this m-prime and m. Then you're left with this, l, m bra inner product with your original z-axis direction I can keep. Now this l, m bar inner product with the identity cut, that is simply the spherical harmonics evaluate at Theta equals zero. Z-axis corresponds to polar angle zero, theta equals zero. This here is simply your angular way function for theta equals zero. What is your angular way function? It's your spherical harmonics function. This here is the bra and that's why we have a complex conjugate here and l, m is the subscript l, m for your spherical harmonics function. Evaluate this at Theta equal to zero, which represents this z direction. Likewise, l,m prime inner a product with this arbitrary axis n is likewise y stared l, m evaluated at a particular theta and phi value corresponding to this new axis and hat. Now spherical harmonic function is zero at theta equal zero whenever m is non-zero. In other words, spherical harmonics is non-zero for theta equals zero only when m is equal to zero and therefore we can write like this. The non-zero value is given by this square root of 2l plus 1 over 4Pi and then Delta m zero. Whenever m is zero you get this, whenever m is not zero then this product is zero. Now, substitute this into the previous equation, so l, m prime inner producted with this arbitrary axis here is represented by this. So you are rotating the z-axis by this rotation matrix and the rotation matrix is represented by this notation here. Plugging in this equation here, we've finally obtained the matrix element that rotates on axis. Gamma is zero because once again, we're only interested in rotating the z-axis, so Alpha Beta are the the polar and azimuthal angle of the new rotated axis n. The rotation matrix that represents this particular rotation is simply given by this. Those are the elements. The matrix element is non-zero only when m is equal to zero. For all other matrix elements, they are zero.
