Before we proceed into a formal derivation, we must try to understand what #Deltax# and #Deltap# actually mean.
They are infact the standard deviations in measurements of #x# and #p# respectively and may be defined as in probability theory as,
#Delta x = sqrt( < (x - < x> )^2> )#
#Delta p = sqrt( < (p - < p> )^2>)#
Now, for two complex valued functions #f# and #g#, there holds the following inequality,
#int |f(x)|^2dx int |g(x)|^2dx >= # #1/4 [int (dot fg + dot gf) dx]^2#
where #dot f# and #dot g# are complex conjugates.
For simplicity, assume that #< x> = 0# and #< p> = 0#
then for, #f = hat ppsi = -ibarh (delpsi)/(delx)# and #g = ibar hpsi#
we have, #int dotffdx = bar h^2 int (deldotpsi)/(delx)(delpsi)/(delx)dx#
#implies int dotffdx = bar h^2 [-int dotpsi(del^2psi)/(delpsi)^2dx]#
In the last step, integration by parts has been done and the first term is put equal to zero since wave functions go to zero when #x# does to infinity.
#int dotffdx = int dotpsi (-ibarhdel/(delx))^2psidx#
#implies int dotffdx = < p^2># where #hat p = -ibarhdel/(delx)# is the one dimensional momentum operator.
Now, #int dotggdx = int dotpsix^2psidx#
#implies int dotgg dx = < x^2>#
and #int (dotfg + dotgf)dx = -bar h int(deldotpsi)/(delx)xpsidx - barh int (delpsi)/(delx)xdotpsidx#
#implies int (dotfg + dotgf)dx = -barh int del/(delx)(dotpsixpsi)dx + bar h int dotpsipsi dx#
Now, the first term when integrated shall go to zero because wavefunctions go to zero as x goes to infinity.
#implies int (dotfg + dotgf)dx = barh#
Therefore, using the inequality stated previously,
#int |f(x)|^2dx int |g(x)|^2dx >= # #1/4 [int (dot fg + dot gf) dx]^2#
#implies (Deltax)^2*(Deltap)^2 >= 1/4 (bar h^2)#
Thus, #Deltax*Deltap >= bar h/2#
Which is the uncertainty principle.
This may also be derived for the more general case that #< x># and #< p># not zero by taking,
#f = ( p - < p>)psi# and #g = (x - < x>)psi#.
The evaluation of the result would be somewhat laborious so I skipped doing it myself.
References -
1) Quantum Mechanics : Theory and Applications by A Ghatak and S Lokanathan
2) Introduction to Quantum Mechanics by DJ Griffiths
