1. PH 401 Dr. Cecilia Vogel

2. Review Outline  Sx, Sy, Sz eigenstates  spinors, matrix representation  states and operators as matrices  multiplying them  Resuscitating Schrödinger's cat  Pauli Exclusion Principle  EPR Paradox

3. Eigenstates of Spin Components  Let the eigenstates of Sz be represented by  |+>  and |->  Sz|+> =  /2 |+>  Sz|-> = -  /2 |->  Then the eigenstates of Sx and Sy  are not eigenstates of Sz  but rather linear combinations of |+> and |->

4. Eigenstates of Sx and Sy  The eigenstates of Sx and Sy  are linear combinations of |+> and |->  |S x =+  /2> =  |S x =-  /2> =  |S y =+  /2> =  |S y =-  /2> =  Note that the amplitude of |+> and |-> in these equations for the eigenstates, when we take the absolute square, gives us the probability …  that we will find that value of Sz, given that the particle is in that eigenstate of Sx (or Sy)

5. Matrix notation  Instead of writing all that out,  we can write each as a matrix, listing the amplitudes in the linear combo:  |+> =  |-> =  |S x =+  /2> =  |S x =-  /2> =  |S y =+  /2> =  |S y =-  /2> =

6. Overlap  The overlap of two states |a> and |b>  is written  bra…ket  tells you how much state |a> is like state |b>  can be calculated by multiplying matrices  ket matrix is the column matrix we just saw  bra matrix is the transpose (turn row into column) and complex conjugate  The absolute square of the overlap  is the probability of finding state |b> when observing a particle known to be in state |a>

7. Overlap  Let’s calculate the overlap of |Sy=+  /2> and |->  =  If we take the absolute square of this, we get the probability that a particle known to have Sy=+  /2 will be found to have Sz=-  /2. That prob=1/2.  Likewise the probability that a particle known to have Sx=+  /2 will be found to have Sy=-  /2 can be calculated: = Prob = (Note: you’ll get ½ any time the two dir’s are ┴ ) note complex conj

8. Overlap  Let’s calculate the overlap of |Sy=+  /2> and | Sy=+  /2 >  =  If we take the absolute square of this, we get the probability that a particle known to have Sy=+  /2 will be found to have Sy=+  /2.  That prob better be 1, or 100%  If you know it has Sy=+  /2, then it is in an eigenstate of Sy, and you will find that value 100% of the time  generally: = 1  if state is normalized

9. Matrix notation  In matrix notation  states are written as column vectors,  operators are written as square matrices.  Operating with an operator on a state  means multiplying a square matrix by a column matrix  … the result is a column matrix,  as it should be:  when you operate on a state, you should get a (probably un-normalized) state

10. Matrix notation  In matrix notation  the observable operators corresponding to each component of the spin are given by these matrices:  S x =  S y =  S z =

11. Eigenstates  We can verify that the eigenstates ?I gave earlier are indeed eigenstates with the stated eigenvalue  For example, is |+> and eigenstate of Sz?  S z |+>=  =  /2 times the original state  so it is an eigenstate with eigenvalue  /2  Similarly S y | Sy=-  /2> ==  =-  /2 times the original state  so it is an eigenstate with eigenvalue -  /2

12. PAL