Quantum Zeno Effect and Evolution Operator Properties

[Markus Kahn]

Homework Statement

Let $U_t = e^{-iHt/\hbar}$ be the evolution operator associated with the Hamiltonian $H$, and let $P=\vert\phi\rangle\langle \phi\vert$ be the projector on some normalized state vector $\vert \phi\rangle$.

Show that
$$\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n\text{ times}}P = \langle \phi\vert U_{t/n}\vert\phi\rangle ^n P$$
and that
$$\langle\phi\vert U_{t/n}\vert\phi\rangle ^n = e^{-i\langle H\rangle t/\hbar}(1+\mathcal{O}(t^2/n))$$

Homework Equations

All given above.

The Attempt at a Solution

For the first eq. I wrote out
$$\begin{align*}\langle \phi\vert U_{t/n}\vert\phi\rangle^n &= \langle \phi\vert U_{t/n}\underbrace{\vert\phi\rangle \langle\phi\vert}_{=P} U_{t/n}\vert\phi\rangle \dots \langle \phi\vert U_{t/n}\vert\phi\rangle\\
&= \langle \phi\vert U_{t/n} \underbrace{P\dots PU_{t/n}}_{n-1\text{ times}}\vert\phi\rangle .\end{align*}$$
I'm not sure how this is supposed to add up since the last ket in the last eq. is just not there in the expression that we want.

For the second equation we can expand
$$U_{t/n} = e^{-iHt/n\hbar} = 1 -\frac{it}{n\hbar} H+ \mathcal{O}(t^2/n^2).$$
The problem arises when I try to calculate the expectation value:
$$\begin{align*}\langle\phi\vert U_{t/n}\vert \phi\rangle &= \left\langle \phi\Bigg\vert 1 -\frac{it}{n\hbar} H+ \mathcal{O}(t^2/n^2)\Bigg\vert\phi\right\rangle\\
&=1-\frac{it}{n\hbar} \langle H\rangle+ \langle\mathcal{O}(t^2/n^2)\rangle ,\end{align*}$$
but this seems quite wrong. I'm not really sure where the error exactly happened...

Hope someone can help a bit.
Thanks

 

[Markus Kahn]

I think I was able to figure out how to do the first part of the question:
We want to prove that for $U_{t/n}$ defined as above
$$
\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n\text{ times}}P = \langle \phi\vert U_{t/n}\vert\phi\rangle ^n P
$$
holds.

Proof by induction:

• $n=1$: $PU_{t/n}P=\vert\phi\rangle \langle\phi\vert U_{t/n} \vert\phi\rangle\langle\phi\vert = \langle\phi\vert U_{t/n}\vert\phi\rangle \vert\phi\rangle\langle\phi\vert = \langle\phi\vert U_{t/n}\vert\phi\rangle P$
• Assume that the statement is true for $n\in\mathbb{N}$.
• $n+1$: $$\begin{align*}\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n+1\text{ times}}P &= PU_{t/n}\underbrace{PU_{t/n}P\dots PU_{t/n}}_{n\text{ times}}P = PU_{t/n}\langle\phi\vert U_{t/n}\vert\phi\rangle ^n P\\ &=\langle\phi\vert U_{t/n}\vert\phi\rangle ^nPU_{t/n}P = \langle\phi\vert U_{t/n}\vert\phi\rangle ^{n+1} P\end{align*}$$

I hope this works... Regarding the second part of the question, well I'm still completely clueless.

 

[Markus Kahn]

Got a solution to the problem (not my own):
The first part of the problem works just as I showed above. For the second part one can proceed like this:
$$\begin{aligned} \left\langle \phi \left| U _ { t / n } \right| \phi \right\rangle ^ { n } & = \left( \exp \left( - \frac { i ( t / n ) } { \hbar } \langle H \rangle \right) \left( 1 + \mathcal { O } \left( t ^ { 2 } / n ^ { 2 } \right) \right) \right) ^ { n } \\ & = \exp \left( - \frac { i t } { \hbar } \langle H \rangle \right) \left( 1 + \mathcal { O } \left( t ^ { 2 } / n ^ { 2 } \right) \right) ^ { n } \end{aligned}$$
Now one can expand $\left( 1 + \mathcal { O } \left( t ^ { 2 } / n ^ { 2 } \right) \right) ^ { n } = 1 + n \mathcal { O } \left( t ^ { 2 } / n ^ { 2 } \right) = 1 + \mathcal { O } \left( t ^ { 2 } / n \right)$, so we find
$$\left\langle \phi \left| U _ { t / n } \right| \phi \right\rangle ^ { n } = \exp \left( - \frac { i t } { \hbar } \langle H \rangle \right) \left( 1 + \mathcal { O } \left( t ^ { 2 } / n \right) \right) .$$