Question 41.10: Science Fiction As we saw in Chapter 39’s CASE STUDY (page 1......
Science Fiction
As we saw in Chapter 39’s CASE STUDY (page 1270), science fiction writers often use real physics in their stories. Let’s test an idea a writer had. Feeding people on deep space missions is challenging. In a sense, the mission would need to somehow pack all the nutrition the crew needs for the entire voyage. The writer’s solution to this problem is based on Heisenberg’s uncertainty principle. According to the principle, the conservation of energy can be violated for a brief period of time given by \Delta E \Delta t \approx h. So a meal with energy ΔE could be created “out of nothing” and last a short time Δt. The real trick is that the person would need to eat the meal in that short time. Estimate Δt.
INTERPRET and ANTICIPATE
The key to this problem is estimating the energy created out of nothing. This estimate comes from Einstein’s famous equation E = mc^2 (Eq. 39.44). We find the mass of a typical meal, then the energy that must be created.
SOLVE
Let’s say the meal is a quarter-pound hamburger. The mass including the bun, French fries, and drink may be about 0.2 kg. We find the energy that must be created.
According to Heisenberg’s uncertainty principle, the mass can be created out of nothing as long as it disappears again in a short time given by Equation 41.36.
\begin{aligned}& \Delta E \Delta t \approx h \quad \quad (41.36)\\& \Delta t \approx \frac{h}{\Delta E}=\frac{6.63 \times 10^{-34} J \cdot s }{1.8 \times 10^{16} J }=3.68 \times 10^{-50} s \\& \Delta t \approx 4 \times 10^{-50} s\end{aligned}CHECK and THINK
This is a very short time. As a comparison, it takes about 300 milliseconds to blink your eye. So in a blink of an eye about 10^{45} such meals could come into existence and vanish. The writer should try to think of another solution for feeding the fictional crew. In case you are worried that we haven’t done a good job of making our estimate because the person really only needs the chemical energy stored in the food (the calories), and not the mass, try Problem 41.80.