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The Thomson's Lamp paradox:

A mad scientist owns a desk lamp. It begins in the toggled on position. The scientist toggles the lamp off after one minute, then on after another half-minute. After a quarter-minute the lamp is toggled off, then the scientist waits an eigth-minute and turns the lamp on again. The scientist continues toggling the lamp, waiting one-half of the previously waited time between toggles. After a total sum of two minutes of toggling, what is the state of the lamp (on or off)?

The Wiki article states that supertasks are impossible and the lamp is neither on nor off after the two minutes. This does not make sense to me, as this would mean that the lamp is in a superposition of two states.

Proof (1 is on, 0 is off):

$S = \sum \limits_{i=0}^n {(-1)^i}$

$S=1-1+1-1+1...$

$S=1-(1-1+1-1+1...)$

$S=1-S$

$S=\frac1 2$

How can a macroscopic object like a lamp exist in such a state?

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4  
The state is only specified for $t \in [0,2)$. Your sum doesn't make sense. –  Umberto P. Jul 17 '13 at 14:35
3  
In reference to your question at the end: How can a macroscopic object like a lamp change states in an infinitely short amount of time? Without the answer to this question, there's not much point in coming up with a physically realizable answer to your question. –  Foo Barrigno Jul 17 '13 at 14:38
    
Theoretically, If you were to consider such a situation, it would appear as if the lamp is both on and off at the same time. Physically, you cannot switch the lamp on and off instantaneously. After a certain point, the lamp will switch states slower than the mad scientist pressing the switch, so you would never get to infinity. –  udiboy1209 Jul 17 '13 at 14:38
    
@Umberto P.: You should consider making that a reply. –  Raskolnikov Jul 17 '13 at 15:07
    
The light will be on twice as long as it is off, so obviously the probability that it will be on at $t=2$ will be 2/3. <grin> –  Rick Decker Jul 17 '13 at 15:36

4 Answers 4

The series $S = \sum \limits_{i=0}^\infty {(-1)^i}$ does not converge, so your calculation makes no sense to begin with (its Cesaro sum does converge to $0.5$, though).

Furthermore, a physical object such as lamp can't be turned off and on at arbitrarily short time intervals. As soon as you reach Planck time (approximately $5\cdot 10^{-44}$ seconds) really weird things are to be expected, though you'd probably be better off asking about it at physics.se. And that's completely ignoring the fact that it takes some time for the lamp to turn on (or off).

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After Planck time, physical abd infromational laws don't apply. –  Torsten Hĕrculĕ Cärlemän Jul 17 '13 at 15:00

The following tries to avoid physical arguments:

The lamp $L$ has a well defined state $s\in\{0,1\}$ at time $t$ only if there is an $\epsilon>0$ (which may depend on $t$) such that $L$ is in state $s$ during the complete interval $\ ]t-\epsilon,t+\epsilon[\ $. It follows that the lamp is not in a well defined state at any of the switching times $t_n=2-2^{-n}$ $\>(n\geq0)$. Why then should we expect it to be in a well defined state at time $t_\infty=2\ $?

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Did you mean to specify a relationship between $\epsilon$ and $n$, for example $n>-\log_2\epsilon$ instead of $\>(n\geq0)$? –  John Bentin Jul 17 '13 at 17:36
    
@John Bentin: The instant $t$ is given first. When $t_n<t<t_{n+1}$ for some $n$ you can find an $\epsilon>0$ such that $s$ is constant on $\ ]t-\epsilon,t+\epsilon[\ $. When $t=t_n$ for some $n$ then there is no such $\epsilon$, since the states of the lamp differ immediately to the left and the right of $t$. –  Christian Blatter Jul 17 '13 at 18:08

Consider that at the moment when the mad scientist begins to flip the switch infinitely quickly, the switch itself will be moving at an infinite speed back and forth. This is physically impossible since the switch would be unable to travel faster than the speed of light.

If it were somehow possible, the lamp would probably just dim a bit since the light itself would heat up during the on phases, but would not completely turn off during the off-phases.

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Besides, the switch would break since it would be subjected to unbounded acceleration.

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