00:01
So in this question we have a hamiltonian, i'll just put the epsilon in, nought i -e -epsilon, nought minus i, epsilon, nought, nought, nought minus epsilon.
00:15
And we want to find the eigenvalues and eigenvectors of this.
00:19
So to find the eigenvalues, we're going to use the determinant of this matrix is equal to zero.
00:34
And if the determinant of that's equal to zero, then the determinants of minus that matrix will also be zero.
00:42
So now let's take this determinant and i'm going to expand along this row here.
00:47
So i'm going to get lambda times lambda times epsilon plus lambda.
00:54
So that's going to be this times the determinant of this matrix.
00:57
And i'm going to get minus this times the determinant of these four elements as a matrix.
01:06
And then this bit gives me zero.
01:09
So that's equal to zero.
01:11
So we get epsilon plus lambda, lambda squared minus epsilon squared equals zero.
01:16
So lambda 1, lambda 2, lambda 3 are going to be equal to minus epsilon, minus epsilon, and plus epsilon.
01:34
So we have a 2d subspace with a eigenvalue minus epsilon and a 1d subspace with an eigenvalue plus epsilon.
01:44
So let's first have a look at the plus epsilon eigenvector.
01:48
So we have the h lambda 3 equals epsilon lambda 3.
01:57
So this is going to tell us that we have nought i -e -e -a -e -a -minexel, nought -0 -0 -0 -0 -minus a epsilon, times, sorry, this should be v3, v3, v3, v3, v3 -3 -x, v3 -3 -x, v3 -3 -6, v3 -3 -6, v3 -3 -3 -6, v3 -3 -3 -6, v3 -3 -3 -3 -3 -6.
02:27
Z is equal to epsilon v3x, epsilon v3y, epsilon v3y, epsilon v3 z.
02:35
So this is a, this is three equations now.
02:40
We've got one, i -e -e -epsilon v3y equals epsilon v3x.
02:45
We can divide out by epsilon here, of course.
02:48
So on both sides we're dividing by epsilon.
02:53
And then two tells us that minus i v3x equals v3.
03:02
And we've got minus v3 z equals v3 z so this tells us that v3 z equals zero.
03:12
Putting these together, we're going to get the v3x equals v3x.
03:18
So v3x is going to be unrestricted.
03:28
And v3y, so i'm going to call this a, and v3y is minus ia.
03:39
So we have the v3 is a, a times 1 minus i 0.
03:50
And then to make this normalized, i'm just going to take a is 1 over root 2.
03:55
So this is one eigenvector.
04:00
And then the other two -dimensional eigen space is just perpendicular to v3.
04:06
So all vectors perpendicular to v3 are going to be in that space.
04:11
So if i take v equals v1, v2, v3, and i dot it with v3, this, so then i'm going to get this is 1 minus i 0, 1 over root 2.
04:28
I get this is 0 if v is an eigenvector with eigenvalue minus epsilon.
04:34
So if v1 minus i v2 equals 0, then we get that v is an eigenvector.
04:50
So i'm going to take two perpendicular eigenvectors, so i'm just going to take the eigenvector, so i'm just going to take the eigenvector v1 equals nought -0 -1, and v2 is, so iv1 is iv2, so it's going to be 1 i0, 1 over root 2.
05:17
So now we have three eigenvectors.
05:21
V1, v2, v3 equals 0 -01, 1 over root 2 i1, i1, and 1 over root 2 1 minus i -0.
05:37
With corresponding eigenvectors, lambda 1, lambda 2, lambda 3, equals minus epsilon, minus epsilon, epsilon.
05:52
So now, a, what values will we measure, will we obtain when measuring the particles energy? and that's going to be, so either minus epsilon or epsilon...