For a free particle, we can derive the following well-known relation:

$$\langle k|k'\rangle = \frac{1}{2\pi} \int_{-\infty}^{+\infty} e^{-i(k-k')x} \, dx = \delta(k-k'). \tag{1.10.33}$$

Reference: R Shankar - Principles of Quantum Mechanics - Eqn. (1.10.33)

Is this relation valid only for a free particle or for all particles? If it is applicable for all other particles, then how do we generalize this relation for all particles?

  • 2
    $\begingroup$ Particles? Which particles? This is a formal relation among momentum eigenstates. You don't have to think of them as plane-wave wavefunctions. $\endgroup$ – Cosmas Zachos Jun 24 '17 at 0:18

This relation is just the orthogonality relation for the functions of the exponential form $\frac{1}{\sqrt{2\pi}}e^{-ikx}$. In the context you're referring to, these functions are the eigenfunctions of the operator $K=-i\frac{d}{dx}$, expressed in the $|x\rangle $ basis.
Therefore, in general, this mathematical identity doesn't have anything to do with any specific physical system.


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