Summary of Spin 1/2

00:00:00 - 00:30:00

The video discusses the operators S plus and S minus, and their matrix definitions. Additionally, the poly spin matrices and the equations for s squared and SX and sy are discussed. The video is a transcript of a lecture on matrix eigenvalue problems.

• 00:00:00 In this video, the basic structure of a spin-one-half system is introduced, as well as the operators that act on the two state vectors. It is shown that the square of the spin angular momentum operator, s, has two eigenvalues, H bar squared and s plus 1, and that these are the eigenstates of the operator. Additionally, it is shown that the minus case of s has an eigenvalue of -H bar squared, and that the plus case has an eigenvalue of H bar squared plus 1.
• 00:05:00 This video explains how to calculate the spin angular momentum (S) of a system of particles, given its spin state, operator, and basis. The video also explains how to calculate the spin angular momentum equivalents of the raising and lowering operators.
• 00:10:00 In this video, the authors discuss the operators S plus and S minus, and their matrix definitions. They also discuss the poly spin matrices, which are matrices that represent the spin of a particle in terms of its X and Y component angular momentum operators. Finally, they provide an equation for s squared and an equation for SX and sy.
• 00:15:00 The video is a transcript of a lecture on matrix eigenvalue problems. The professor reviews the basics of the problem, explains how to solve it, and provides an example. He then goes on to discuss the eigenvalues and eigenvectors associated with the plus and minus H-bar eigenvalues, which are equal to 1 and -1, respectively.
• 00:20:00 In a spin 1/2 system, the actual normalized eigenvector is 1 over root 2. These 1 over root 2s are things that you see a lot when you're dealing with two-state quantum systems. Griffiths calls this chi plus with a superscript x to signify that it is the eigenvector associated with the x component of the spin angular momentum with the plus sign in the area in the eigenvalue if we're working with the minus h-bar over 2 eigenvalue we're going to get the opposite signs here so if I said 1 minus 1 well I'm going to get something very similar when I normalize it 1 over root 2 minus 1 over root 2. Chi minus with the superscript X so these are our eigen values and eigen vectors for S sub X what that means is that these are the states with definite X component of angular momentum so this is a little bit strange if I combine two states with definite Z angular momentum in this very specific superposition I end up with a state of definite X angular momentum and depending on the sign of the superposition here whether I add them together or subtract them from each other the plus Z component or the minus Z component the spin up case or the spin down case if I
• 00:25:00 In quantum mechanics, a quantum system can be in an unknown state, and the outcomes of a measurement of one of its components (Z or X) can be either h-bar over 2 or minus h-bar over 2. Knowing the angular momentum in the Z direction doesn't provide information about the angular momentum in the X direction. This is surprising because if you measure the Z component to be down, measure its X component to be negative, and measure its Z component again, you have forgotten about what the Z component was.
• 00:30:00 The video explains how to find the eigenvalues and eigenvectors of a two-dimensional system of spin and momentum. Depending on the eigenvalues and eigenvectors found, different outcomes can occur.