In this video we'll discuss spinor representation, so let Us construct matrix representation for a spin one and half system. And using the eigenkets of s of z as basis, so the in the basis of sz I can catch the plus icon Cat is simply represented by +10 z minus as 01. And we're going to specify this 10 column vector, two dimensional column vector, 10 Azcuy plus and 01 column vector as chi minus the bra cats, bra vectors that are dual to the Z plus and the minus cats are of course then represented by two row vectors one, zero and 01. Now represent an arbitrary tap alpha as a superposition of Z plus and the minus co options of course are once again the inner products shown here. So in the two dimensional column vector notation, these co options become the elements of the column vector shown here. So the bra, which is dual of these, cat, alpha, is then the same vector in a row vector format as shown here, the two component column vectors are called the two components spinner or just spinner. And so you can represent an arbitrary state as a two dimensional column vector has shown here. The elements of this vector are given by the inner product of the state, cat alpha with the two eigenkets or eigenbras in this case. So if we represent these inner product as c+ and c- then you can write these arbitrary state factor chi as a superposition of or linear combination of the basis factors chi +, and chi- 1, 0 and 0, 1 column vectors. And the mission at joint would then be given by taking a complex conjugate and the and transpose as is defined in linear algebra, so it will give you this row vector. And in the linear combination of the basis bras in this case or in this case the row vectors you would represent it like this, now we can represent the x, y, z component to spin operators in terms of these basis cats and bras of the s z, and then you can represent that in the 2 by 2 matrices as shown here. These matrices here after factoring out these constant h bar over to in all cases the remaining 2 by 2 matrices are called the Polish spin matrices and typically represented by the symbol sigma, so sigma x is here, sigma y is here, and sigma z shown here. You can show that the power early spin matrices satisfy this commutation relation, absolutely j, k once again, it's the Levee city, it's a symbol. And we can define anti commutation relations, these braces denote the anti computation of these two operators define as the sigma phi sigma times sigma j plus sigma j times sigma i. And that if you do the calculation using the matrix is given in the previous slide, you will see that it satisfied two times delta, i, j. Other properties of the polish matrices are it's our mission traces are zero and the determinant is all -1. You can also treat these sigma as a three dimensional vector and this time each vector component is a 2 by x 2 matrix. And so if you take a dot product with any regular vector with three components a, then you use the usual definition of the dot product between 23 dimensional vectors. And then you can write it as a result as a two dimensional matrix as shown here. Now, let's represent a rotation operator in a two component matrix formalism. The definition of the rotation operator is shown here. So once again the generator is these angular momentum operator in this case we are talking about spin angular momentum here and factor represents the axis and fee is the angle of rotation. So this is a standard definition of the rotation operation, now in terms of the powerless matrix, you can rewrite this as this. Okay, now we're going to express this exponential expression as an infinite series and we collect all of the even powered terms of sigma dot n here in this first bracket, and all of the odd power terms in the second bracket. And then we use this special properties of spin public spin matrix, so if you take an end power of sigma dot a unit factor in for an arbitrary unit factor then for all even even power, you simply get the identity matrix, whereas for all the odd power it simply is equal to this. So you all of these even power term here in the first bracket simply gives you this cosine of phi over two. And then this second bracket, all the add power term simply turns out to give you this sigma dot n, and then the remaining term will give you a sine of phi over two. Explicitly, you can write these exponential function now in a two dimensional matrix form and the matrix elements are given in terms of the ends of z and x and y are the x, y, z component of this unit vector and unit vector and here or here, and then the sine and cosine of the rotation angle phi divided by two. Now we can use this matrix to rotate a cat, so rotation of a ket is in our test space, rotation operator is defined by this exponential operator containing this angular momentum operator. Now that is represented, two by two matrix form using this exponential function containing the Powley matrices. And we have just derived the explicit form 2 by 2 matrix representation of this exponential operator. So we can rotate any cat In a two dimensional cat space describing the spin state of a spin one half system using this operator. So let us find an eigenvector of sigma dot n with an eigen value plus one. So the item value equation is given by this, right, so this is the operator and this is the ket, and the I am values plus one. So right hand side is simply just eigen ket itself eigrenvector itself, now this is a two by two matrix eigenvalue equation. So you can just brute force solve it, or since we have just derived a rotation operator in the two dimensional ket space, what we can do is to rotate these z plus cat by a certain angle. Okay so this here z + is an eigen ket we don't Eigen value plus one corresponding to z-axis. So this is the eigen value equation in terms of the polish spinor representation, so this is the sigma z Paoli matrix for z component and the x+ the chi +. I'm sorry is the spinor representation for the z + ket, okay so now what we can do is we can transform this eigenvalue equation by performing a proper rotation two, and turn it into this eigenvalue equation that we want to solve. So what we do is we basically to this item value equation, we apply the appropriate rotation R which turns the z-axis into this axis specified by this unit vector n. And then so we apply this D of R to both sides of these eigen value equations shown here. And then also we insert these D inverse D operator, which is simply an identity operators. So we can do this in between we we insert is this product between sigma z and chi +. So the task now becomes to find the appropriate rotation D of R that rotates the z-axis into and access specify this unit vector, and that is given by a shown in this figure here, so this is the z-axis. And to turn this axis into this n-axis, what you need to do is to first rotate the axis by an angle beta about y-axis. And then you rotate that rotated vector by an angle alpha about z-axis, that's what you need to do, rotation by beta about y-axis, and then rotated by alpha about z-axis. But we know what these rotations are already, so rotation first rotation by an angle beta about why that's this exponential operator. And then the second rotation about z-axis by alpha, given by this exponential operator, and we have the 2 by 2 matrix representation for these two exponential operators. And they are shown here, this is the first exponential and this is the second exponential and chi + of course is just a simple column vector 1, 0. So if you do this matrix computation, you get this column vector, this is the Eigen vector with an eigenvalue +1, and about an axis represented by an angle alpha and beta, as shown here. So instead of solving the Eigen value equation, brute force, you can simply apply a proper rotation to unknown State 1,0 you get the result.
