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An observable Q for a physical system is a function Q(p, x) of its momentum p and its location x. The usual procedure
for obtaining its operator Q^ for use in Schrödinger's equations is to leave x as it is and replace p with (h/i)(∂/∂x).
This involves p^{n} being replaced with (h/i)^{n}(∂^{n}/∂x^{n}).
An alternate procedure is to leave p as it is and replace x with −(h/i)(∂ /∂p), with of course,
x^{m} being replaced with (h/i)^{m}(∂^{m}/∂p^{m}). This is called
quantum analysis in momentum space.
This means that as the energy of the system increases without bound the spatial average of the solution of the time independent Schrödinger's equation asymptotically approaches of the timespent momentum equation for the system.
Source:
David J. Griffiths, Introduction to Quantun Mechanics, Cambridge University Press, (2nd edition) 2017.
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