Zeno walks into a bar.



"I have a problem," he said.



"What is it?" said the bartender.



"Well, it has to do with the movement of physical bodies," said Zeno.



"Talk to my friend Max," said the bartender. He gestured toward a German man wearing round spectacles.



"Sir," said Zeno, "I wonder if you could help me with a problem."



"What's the problem?" said Max.



"Suppose I shoot an arrow from point A to point B," said Zeno. "Before it reaches point B it must first reach a point C1 midway between points A and B."



"Naturally," said Max.



"And before the arrow reaches point C1 it must reach a point C2 midway between points C1 and A," continued Zeno.



"I see," said Max.



"And before the arrow reaches point C2 it must reach a point C3 midway between points C2 and A," continued Zeno.



"Wait a minute," said Max. "How far apart are points A and B?"



"10 meters," said Zeno.



"Then yes," said Max. "I understand your situation."



"And before the arrow reaches point C3 it must reach a point C4 midway between points C3 and A," continued Zeno. "Do you see the impasse?"



"Nope," said Max, "I think we're getting somewhere. How long is the arrow?"



"One meter," said Zeno.



"The distance between points C3 and C4 is five eighths of a meter," said Max. "A one-meter-long arrow can be at point C3 and C4 at the same time."



"Let's consider the tip of the arrow then," said Zeno. "Before the tip of the arrow reaches point C3 it must reach a point C4 midway between points C3 and A."



They talked deep into the night.



"And before the high-energy particle reaches point C118 it must reach a point C119 midway between points C118 and A," continued Zeno.



"Hold on," said Max. "How far apart are points A and C119?"



"1.5×10−35 meters" said Zeno.



"That's shorter than 1.6×10−35 meters," said Max. "The uncertainty in the position of a particle must always exceed 1.6×10−35 meters, because of space-time equivalence and the quantum-mechanical velocity operator's non-commutation with position. Even theoretically, the wave function of a particle can't ever occupy a space smaller than 1.6×10−35 meters."



"Thanks," said Zeno.



"By the way," said Max, "What brought you to this question in the first place?"



"I wanted to know how to define the momentum of a particle at an instantaneous moment of time," said Zeno.



"You could have just asked," said Max. "The probability distribution of a particle's momentum is determined by the instantaneous phase and magnitude of its wave."

