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In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar |<\psi_b|V|\psi_a>|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $<a|e^{-iHT}|b>$, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

What would we measure in an experiment to check that the amplitude is $<a|e^{-iHT}|b>$, for a free particle?

In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar \left|\langle\psi_b|V|\psi_a\rangle\right|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $\langle a|e^{-iHT}|b\rangle $, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

What would we measure in an experiment to check that the amplitude is $\langle a|e^{-iHT}|b\rangle$ for a free particle?

In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar |<\psi_b|V|\psi_a>|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $<a|e^{-iHT}|b>$, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

How to set up an experiment to check that the amplitude is $<a|e^{-iHT}|b>$, say for a free particle?

In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar |<\psi_b|V|\psi_a>|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $<a|e^{-iHT}|b>$, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

What would we measure in an experiment to check that the amplitude is $<a|e^{-iHT}|b>$, for a free particle?

In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar |<\psi_b|V|\psi_a>|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $<a|e^{-iHT}|b>$, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

In Fermi's golden rule

$$P_{ab}(t)=2\pi t/\hbar |<\psi_b|V|\psi_a>|^2 \delta(E_f-E_i)$$

for transition probability from state $a$ to $b$, how can the probability grow with time to above 1, how does one interpret a probability above 1? How does one use this formula?

And in this formula for the amplitude to propagate from an position eigenstate a to $b$ in time $T$ $<a|e^{-iHT}|b>$, how does one interpret this physically? We cant have a particle at exact position a, let alone how would we measure it at the exact time $T$? Is there some integral over $T$ or something to get a physical probability?

How to set up an experiment to check that the amplitude is $<a|e^{-iHT}|b>$, say for a free particle?