During one of the Einstein-Bohr debates, Einstein proposed a thought experiment that would prove that one could measure time and energy simultaneously. It's known as the Einstein box:(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/Bohr–Einstein_debates#Einstein.27s_second_criticism

Then Bohr came up with a resolution (invoking the gravitational time dilation!) which is described e.g. there on Wikipedia. I also have it in my book, but I don't understand it. I have two questions:

1. Why isn't the explaination as simple as:

If the emission time of the photon is measured with uncertainty Δt, the change in vertical momentum that could have been introduced to the photon by gravity is maximally:

Δp = mg Δt

(where [itex] m = E/c^2 [/itex])

On the other hand the uncertainty in the vertical position Δz of the box (and hence, of the photon), corresponds to the uncertainty in the potential energy:

ΔE = mg Δz

Then since: [itex] Δz Δp ≥ \hbar [/itex] then also:

[itex] (\frac{ΔE}{mg})(mg Δt) ≥ \hbar → ΔE Δt ≥ \hbar [/itex]

2. This must be wrong somehow, otherwise it wouldn't take Bohr a whole day to come up with a solution! Could you explain to me Bohr's argument in more detail? It seems to me like the gravitational time dilation would be a second order uncertainty, since it's an uncertainty in the uncertainty of the emission time.

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# Bohr's response to the Einstein box

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