In this video, we will relate rotation operators to rotation matrices. In general, rotation in three dimensional space is specified by a three independent parameters. The rotation angle obviously, by which we are rotating, and the axis about which we're making the rotation. To specify your axis, you need to specify the polar and azimuthal for angle to specify the orientation of this rotation axis, and it's shown here. If you're rotation axis is this unit vector n, specified by this unit vector n, then you need to specify the polar angle Theta and the azimuthal of angle Phi to specify this axis. There are total of three independent parameters to specify a general rotation in three dimensional space. It is convenient to use a three by three matrix to specify a rotation. In that case, the successive rotations are simply represented by the multiplication of multiple rotation matrices. Now, three by three matrix has nine elements. It may seem that you may have nine independent parameters to specify a rotation, but all of these rotation matrices must be real and orthogonal. Which means that the transpose is equal to its inverse. This condition gives you six equations, and that eliminates the six dependent parameters out of the nine elements, and that leaves you with the three independent parameters, as we should expect from the beginning. The set of all three dimensional orthogonal matrices form a mathematical group, and that group is typically called O(3). O here stands for orthogonal, and the number three in the parenthesis for three dimensions. Now, a quick word on a mathematical group. A group by definition is a set of elements for which we have a binary operation. The binary operation is an operation that map two elements of the group into another element in the group. A group must satisfy these following conditions. First of all, the group must be closed under group multiplication. This binary operation, which we normally call multiplication, even though it may not be the arithmetic multiplication, it just means a binary operation. You have these multiplication between two elements A and B. It results in another element in the group always. It never results in something that is not part of the group. In this case we call the group is closed or set is closed, under the group multiplication. This is one of the requirement to be a group. Then the multiplication should be associative, you should have an identity element, and then you should also have an inverse for every single element in the group. Now, rotation operators satisfy these properties, whereas the binary operation or the multiplication is defined to be the successive application of two rotations. Using the successive operation as a multiplication, rotational operators form a group, and also rotation matrices also form a group, in which case matrix multiplication is the group multiplication in that case. We can use the group theory from algebra to describe the various properties of these rotation operators and rotation matrices. Group theory is commonly used and it proves to be a very powerful tool in dealing with many different problems. Here we consider this particular example of a group formed by the rotation of spinors. The rotation of spinors is described by two by two matrices, and we saw that they were unitary. They were also unimodular, meaning that the determinants of those two-by-two matrices are all one. In general, you can denote these two-by-two modular unitary matrix as this. Here, a and b are two complex numbers and the unimodularity requires that a absolute value square plus b absolute value squared is 1. Now, you can show yourself that this matrix representation satisfy this condition, which is the unitary properties of unitariness. The number of independent parameters are again three, because a and b are complex numbers, so they have real part and imaginary part. There are a total of four real parameters and one of them is eliminated by this condition of unimodularity. These matrices represent a rotation in three-dimensional space and even though it's acting on a two-dimensional vectors, so don't get confused here. The vectors are not three-dimensional vectors, the vector that we're dealing with here is a vector that represents a spin state. Spin for a spin 1.5 system, an arbitrary spin state can be represented by a two-dimensional vector. That's the two-dimensional LET there were dealing with, that's why we are dealing with two-by-two matrices here. Rotation that we're making is a rotation in real three-dimensional space, which is very different from the two-dimensional cat space that we're dealing with to describe a spin state of a quantum system with spin one half. This is exponential function defines a general rotational operator and in two-dimensional case, you represent it by this two-by-two matrix. This is something that we have seen before and you can prove yourself that this matrix has the general form of unimodular unitary two-by-two matrix. The set of unimodular unitary matrices form a group, just like the three-by-three matrices form the group, so they satisfy the closure associativity and identity inverse, all those properties that I have listed in an earlier slide. This group formed by unitary unimodular two-by-two matrices is called SU(2), special unitary group, SU stands for special unitary group and this number 2 in parentheses indicate the vector space dimensions that we're dealing with. The set of two-by-two unitary matrices, but not constrained to be unimodular, it also form a group and that group is called U(2). The SU(2), obviously is a subset of U(2) and SU(2) itself is a group and therefore we call SU(2) is a subgroup of U(2). Now, both group O(3) and group SU(2) describe rotations in 3D space. A natural question is, are they are isomorphic? Meaning that there is a one-to-one correspondence between those two groups, is there? Maybe. However, consideration of rotation by angle two Pi 360 degrees and angle 4 Pi radian and two full turns in O(3), rotation by 2Pi and 4Pi, both are represented by identity matrix. However, in SU(2), rotation by 2Pi is described by negative I, whereas the rotation by 4 Pi is represented by I. In general, there is a two to one correspondence between SU(2) and O(3), there is one matrix in O(3) for each pair of UAB and U negative a and negative b in SU(2) group. Now, I want to introduce Euler rotations. An Euler rotation is a very general technique to specify a general rotation in 3D space using three independent parameters. See, there are three independent parameters in order to specify an arbitrary rotation in 3D space and we said that one is an angle and the other two are the information about the rotational axis. Here in Euler angles, we are specifying a general rotation in three-dimensional space using three rotation angles, and they are shown here. Here is the original coordinate system, x, y, z in 3D space. First, you rotate by an angle Alpha about z-axis. That rotates your x-axis to x prime and y-axis to y prime. Next, you rotate by an angle Theta about y prime axis, this rotated y-axis shown here. That gives you this z prime, rotated z axis and x double prime, x-axes are now rotated twice. After that, you rotate by an angle Gamma by the z prime, rotated z axis here, z prime axis and that results in this. This time, y gets to be rotated to y double prime from y prime here. Then x-axis from x double prime to x triple prime here. This is the final rotation of the coordinate system. Here all of the primed axis represents an axis that had been rotated. These are the coordinate axes that are rotated together with the object or the vectors being rotated. Sometimes these axes are called the body-fixed axes. The unprimed axes, the original axes x, y, z here are the space-fixed axes. Now, obviously, if we don't have to always consider these changing axes, that will be easier. If we only use space-fixed axes that don't change as we make rotational operation, it will be much less confusing. Can we do that? Well, the answer is yes. We first to note this rotation by angle Beta by y prime axis, the rotated axis here. So if you rotate by an angle Beta about y prime axis, you get this z prime axis. This rotation is equivalent to, first, you rotate y prime axis back to y, and then you rotate by this space-fixed y axis by an angle Beta. Then after that, you rotate your rotated z-axis about the fixed z-axis by an angle Alpha again. You rotate this z prime to this z prime here and then the y-axis will go back to y prime. This final state is equal to this. You consider this triple rotation. You undo rotation by angle Alpha about z-axis. You might require the rotation about y axis by an angle Beta and then you undo what you have done here. That is equivalent to this. Similarly, you should be able to show this equation here. By using this and this, the original Euler's rotation described by the rotation about the object fixed axis can now be turned into these triple rotation about space-fixed axes by an angle Alpha, Beta, and Gamma. This now describes three rotations about z-axis, y-axis, and z-axis again. We don't really need to specify the rotation axis anymore. Rotation is always about these y and z axes and we represent a general rotation by these triple rotations by an angles Alpha, Beta, and Gamma. Now, let's consider Euler rotations in spin 1/2 system, two-dimensional vector space once again. These general rotational operation representing a Euler rotation by an angle Alpha, Beta and Gamma can be expressed by this triple product of rotation by Gamma about z-axis, rotation by Beta by y-axis, and rotation by Alpha about z-axis. We already know these operators in the 2 by 2 matrix representations, so these D_z, D_y, D_z are represented by these exponential operators and containing Sigma_z, Sigma_y, and Sigma_z here. Using the 2 by 2 matrix representation of these, we can write out and do the matrix multiplication to obtain this 2 by 2 matrix that describes a general rotation in three-dimensional space for an arbitrary ket state, arbitrary spin state described by a two-dimensional ket. This matrix, you should be able to show yourself that it is unimodular and unitary. Any matrix seen as SU(2) group can be represented in this Euler angle form. This is the most general form describing a three-dimensional rotation of a two-dimensional ket describing a spin state in spin 1/2 system. Now, the matrix in the previous slide is called the j equals 1/2 irreducible representation of the rotation operator D of Alpha, Beta, and Gamma. This is an operator. An operator is an abstract thing. It's an action. We turn this into a concrete matrix which we can manipulate. We can multiply, add, subtract, we do sines. That is called the representation or matrix representation and it was corresponding to this spin 1/2 system, angular momentum magnitude of 1/2, so it's called j equals 1/2 irreducible representation. Now, the general matrix element for this operator is expressed by this. Here the notation D obviously represents the rotational operator, and this parenthesis 1/2 in the superscript specify the j quantum number for j square operator and the subscript represents the column and row index in the matrix representation. These m prime and m obviously are the eigenvalues indicate the eigenvalues for J_z operator. Alpha, Beta and Gamma are the three Euler angles specifying a general rotation. Using the eigenkets, j and m as the basis kets, this is the general matrix element for an operator. This operator is, of course, the three successive rotations by angle Gamma, Beta, and Alpha, which represents the Euler's rotation. Once again, j equals 1/2 and m is the general notation for these eigenkets of spin 1/2 system, such as z plus or z minus. However, we haven't really specified any special spin component or anything here. This is just a general angular momentum eigenket for a general angular momentum operator j square and J_z, with the j square eigenvalue j being 1/2.
