Relation of Measure of Time Evolution 1- Integral dt (0,T) dt' (0,T) <W(x,t)|W(x,t')><W(x,t)|W(x,t')> and Infinite Well Momentum Wavefunction
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Correction: March 31, 2023 We remove the form D = 1-a(p)a(p) in the note as it does not equal the time evolution measure of (1). We wish to only point out that sinc(T/2 |(En-Em)|) in the expression for D has an analogue in the spatial behaviour of a particle in an infinite well where:
sinc(T/2 |En-Em\) from D --> a(k) sqrt[3.14/(hbar L)] [n3.14 + kL] / (n3.14) i.e. is proportional to the momentum wavefunction a(k) multiplied by a linear function in k. We point out that even though this analogy exists, the summations involved in D do not map over to the single particle case. Thus there is not a complete analogy. In particular, one could sum over k, but would not generally sum over n, the energy level.
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In (1) a measure to determine time evolution of a quantum system is given as:
D = 1- Integral dt (0,T) dt’ (0,T) <W(x,t)|W(x,t’)><W(x,t)|W(x,t’)>.
Using W(x,t) = Sum over n cn exp(-iEnt) Wn(x) (1) shows:
D= 1 - Sum (n,m) |cncm||cncm| sinc[wnm T/2] sinc[wnm T/2]
The authors argue that this measure may be applied to qubit systems. Their result is based on the form exp(-iEnt)Wn(x) and the idea of a fixed time interval T.
Here we argue that an identical form appears in the spatial realm for a system which instead of a finite time interval contains a finite length. The relevant system with a finite length is that of an infinite well potential with length L because the wavefunction is 0 outside this length. In such a case, instead of exp(-iEnt)Wn(x) we relate exp(-ipx)Wn(x) to a(p), the momentum wavefunction, and show that measure of time evolution is essentially of the same mathematical form as: 1- a(p)a(p) for an infinite well of length L. 1- a(p)a(p) has the interpretation of the probability not to find a particle in an infinite well in a momentum state of p for a given energy level with average momentum of n 3.14/L.