Every general explanation of the quantum zeno effect I've found is (from my perspective) so full of gaps that I cannot understand the explanation. I am wondering if anyone can help me fill in the gaps here?(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Quantum Zeno Effect: What is the argument?

Loading...

**Physics Forums | Science Articles, Homework Help, Discussion**