From: richalivingston@gmail.com   
      
   On Wednesday, August 2, 2023 at 12:11:10=E2=80=AFPM UTC-5, Stefan Ram wrote:   
   > In the third lecture of the series "New Revolutions in   
   > Particle Physics: Basic Concepts" (Fall, 2009) from "The   
   > Theoretical Minimum", Professor Susskind talks about what   
   > a quantum field is and how it is related to particles.   
   >   
   > He considers a circle with a coordinate "x" and an amplitude   
   > "exp( i k x )", and says that the momentum "p" is "hbar k".   
   >   
   > Then he says (about 8 or 9 minutes into the video):   
   >   
   > |"k" is called the "wave number". Uh. It's a wave number.   
   > |And it can be positive or negative? Waves moving to the left,   
   > |moving to the right, corresponding to momenta, going to the   
   > |left or the right.   
   >   
   > How can he talk about "waves moving to the left" if his   
   > amplitude "exp( i k x )" does not depend on the time   
   > (contains no "t")?   
   >   
   > Do you think that "exp( i k x )" without "t" in his model   
   > means that the amplitudes are constant (not depending on   
   > the time) or is "exp( i k x )" a snapshot of the situation   
   > taken at one instant (but actually depending on the time)?   
      
   I haven't seen this lecture, but if he is working in the   
   Heisenberg picture, the wave function is static and   
   all time dependence is in the operators.  The effect   
   wrt your question is that there is an assumed   
   e^(-i \omega t) factor that appears when the position   
   operator is applied.   
      
   Rich L.   
      
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