- #1
Loro
- 80
- 1
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:
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 explanation 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.
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 explanation 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.