Propagator Equation at t=0 Explained

[ehrenfest]

I understand the progator in general but could someone explain this equation for the propagator at t = 0 for me:

$$\delta(x' - x) = K(x',x;0,0) = \sum_m \psi_n(x')\psi_n(x)$$

?

I am confused about the dfiference between x' and x. It seems like the Kronecker would make more sense than Dirac here?

 

[Gokul43201]

x is continuous.

 

[ehrenfest]

x is continuous.

And x' is discrete? What do they represent?

 

[Gokul43201]

No, x and x' are both positions. They (and t, t') are continuously varying parameters; hence the Dirac delta.

The propagator K(x,x';t-t') is the amplitude for a particle initially at (x',t') to be observed at (x,t). With t=t', this is the probability amplitude that a particle at x' is also at x, which is given by the Dirac delta distribution.