- #1
geoduck
- 258
- 2
In one dimension the normalized momentum eigenstate for a particle with periodic boundary conditions of length L is: [tex]\psi_k(x)=\frac{1}{\sqrt{L}}e^{ikx} [/tex].
Is the completeness relation obvious:
[tex]\Sigma \psi_k(x)\psi_{k}(0)=\frac{1}{L}\Sigma e^{ikx}e^{-ik0}=\frac{1}{L}\Sigma e^{ikx}=\delta(x) [/tex]
where the sum is over discrete eigenstates k?
How would you go about proving that sum?
Is the completeness relation obvious:
[tex]\Sigma \psi_k(x)\psi_{k}(0)=\frac{1}{L}\Sigma e^{ikx}e^{-ik0}=\frac{1}{L}\Sigma e^{ikx}=\delta(x) [/tex]
where the sum is over discrete eigenstates k?
How would you go about proving that sum?