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Quantum Analysis in Momentum Space

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 pn being replaced with (h/i)n(∂n/∂xn).

An alternate procedure is to leave p as it is and replace x with −(h/i)(∂ /∂p), with of course, pm being replaced with (h/i)m(∂m/∂pm). This is called quantum analysis in momentum space.

This means that as the energy of the system increases without bound the solution of the time independent Schrödinger's equation asymptotically approaches the spatial average of the time-spent momentum equation for the system.

Source:

David J. Griffiths, Introduction to Quantun Mechanics, Cambridge University Press, (2nd edition) 2017.


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