I try to read Sakurai's Modern Quantum Mechanics but I stuck at this point,

$$\delta(x^{'}-x^{''})=|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr)$$

This is an expression for finding normalization constant ,After this step he shows

$$|N|^{2}\int dp^{'}\exp\Biggl({ip^{'}(x^{'}-x^{''})2\pi\over h}\Biggr) =h|N|^{2}\delta(x^{'}-x^{''})$$

I don't get this step , Please help me to move more

  • 2
    $\begingroup$ Check Dirac Delta definition $\endgroup$ – user183962 May 21 at 8:32

The first equation expresses the orthogonality property of the momentum eigenfunctions. There is the well-known formula for the Dirac delta function: $$ \delta(x) = \frac1{2\pi}\int\! dk\ e^{ikx}. $$ If you will make the change of variable $$ p' = k\frac{h}{2\pi} $$ in the integral, then you would obtain your second formula. Together two formulas give $$ |N|^2 = \frac1{h} $$

  • $\begingroup$ Thanks for your precious help Gec @ Gec $\endgroup$ – Robin Raj May 21 at 14:35
  • $\begingroup$ @Robin Raj , Why wouldn't you accept the answer, then? $\endgroup$ – Cosmas Zachos Jul 10 at 14:54

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