From: ram@zedat.fu-berlin.de   
      
   I'm just watching Lecture 2 of "The Theoretical Minimum:   
     Quantum Mechanics" by Leonard Susskind, about 1 hour and   
     10 minutes in.   
      
     A: IIRC, the state |spin up> is orthogonal to the state |spin down>,   
     because if one prepares a source for |spin up>, one never measures   
     |spin down>.   
      
     B: If one prepares a source of |spin up>, one always measures   
     |spin up>. But if one rotates the measurement device by 180 degrees   
     so that it is upside down, one always measures |spin down>.   
      
     So, the state |spin down> in the Hilbert space is orthogonal   
     to the state |spin up> (A). Usually, in the normal two- or three-   
     dimensional spaces I imagine that "orthogonal" means "90 degree".   
     But to get from |spin up> to |spin down> the measurement device has   
     to be rotated by "180 degrees" (B). It's as if the angle in the   
     Hilbert space of states (90 degrees) is /half the angle/ (180 degrees)   
     in locational space.   
      
     Then, Susskind talks about the states |spin left> and |spin right>   
     one measures by rotating the measurement device by 90 degrees.   
     He explains that |spin right> is (1/sqrt(2))|spin up>+   
     (1/sqrt(2))|spin down>. But I know that (1/sqrt(2))(1,1) are the   
     coordinates of a unit vector that encloses an angle of 45 degrees   
     with the x axis. So a rotation of 90 degrees in locational space   
     now corresponds to a rotation of 45 degrees in state space, again   
     a half of the angle of 90 degrees.   
      
     Finally, I remember vaguely that there is a situation where   
     the state is restored only after a rotation by 720 degrees   
     in the locational space, which by a bisection would correspond   
     to a rotation by 360 degrees (i.e., identity) in space state.   
     (It is difficult to imagine that after a rotation of 360 degrees   
     in locational space not everything is the same again!)   
      
     So, have I made a mistake in my description or has this been   
     observed and discussed before that sometimes a rotation in   
     locational space corresponds to half that rotation in state space?   
      
   --- SoupGate-Win32 v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)