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The most detailed explanation I've found runs something like this:

Let |ψ

_{0}> be the initial quantum state of a system at time 0 and let |ψ

_{t}> be its state at some later time t.

The dynamical evolution of the system is described by a unitary operator U(t) that is a complex function of the initial system's Hamiltonian: U(t) = e

^{-iHt}. Thus: |ψ

_{t}> = U(t)|ψ

_{0}>.

The "survival" probability P

_{s}that the system will still be in the initial state at t is given by:

P

_{s}= |<ψ

_{0}|ψ

_{t}>|

^{2}= |<ψ

_{0}|e

^{-iHt}|ψ

_{0}>|

^{2}

So far so good. But now standard explanations assert that:

P

_{s}= |<ψ

_{0}|e

^{-iHt}|ψ

_{0}>|

^{2}= 1 - (ΔH)

^{2}t

^{2}

(Where (ΔH)

^{2}= <ψ

_{0}|H

^{2}|ψ

_{0}> - (<ψ

_{0}|H|ψ

_{0}>)

^{2})

Where does that come from? Is it meant to be obvious that 1 - (ΔH)

^{2}t

^{2}follows from the left hand side?

At any rate, we can now define the Zeno time Z = 1/ΔH so that:

P

_{s}= 1 - [itex]\frac{t^{2}}{Z^{2}}[/itex]

Presumably this shows that as t gets smaller the probability tends to 1 so that the faster we measure the system after time = 0 the more probable it will be found in its initial state.

Now for the final bit. If we consider N measurements then we can understand the survival probability given those N measurements as:

P[itex]^{N}_{s}[/itex] = (1 - [itex]\frac{t^{2}}{N^{2}Z^{2}}[/itex])

^{N}

...so that in the limit of continuous measurements where N → ∞ we get:

[itex]\stackrel{Lim}{N→∞}[/itex] P[itex]^{N}_{s}[/itex] = 1

I just don't see how this final bit follows. After all, if t is large then increasing N won't bring on the QZE. Surely we also need t → 0 but I don't see how the above accounts for this.

Any help would be most appreciated, thanks.