Question

When the uncertainty in momentum is zero then the uncertainty in the position of a particle is

• A. \( \dfrac{h}{4\pi} \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( \dfrac{1}{2} \)
• E. infinity

Hint:

Remember the uncertainty relation: \[ \Delta x \Delta p \geq \frac{h}{4\pi} \] If one uncertainty becomes extremely small, the other must become extremely large.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
This question is based on Heisenberg's Uncertainty Principle, a fundamental concept in quantum mechanics. The principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle, such as position and momentum, can be known simultaneously.
Step 2: Key Formula or Approach:
The mathematical expression for Heisenberg's Uncertainty Principle relating position and momentum is:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \] where:
\(\Delta x\) is the uncertainty in position.
\(\Delta p\) is the uncertainty in momentum.
\(h\) is Planck's constant.
Step 3: Detailed Explanation:
We are given that the uncertainty in momentum is zero.
\[ \Delta p = 0 \] Let's substitute this into the uncertainty principle inequality:
\[ \Delta x \cdot 0 \geq \frac{h}{4\pi} \] This simplifies to:
\[ 0 \geq \frac{h}{4\pi} \] This statement is mathematically false, as \(h\) and \(\pi\) are positive constants, so their ratio is positive. To properly interpret the principle, we should rearrange the formula to solve for \(\Delta x\):
\[ \Delta x \geq \frac{h}{4\pi \cdot \Delta p} \] Now, let's consider what happens as \(\Delta p\) approaches zero:
\[ \Delta x \geq \lim_{\Delta p \to 0} \frac{h}{4\pi \cdot \Delta p} \] As the denominator (\(\Delta p\)) approaches zero, the value of the fraction approaches infinity.
\[ \Delta x \geq \infty \] Therefore, the uncertainty in the position (\(\Delta x\)) must be infinite. This means if we know the momentum of a particle exactly (\(\Delta p = 0\)), we have absolutely no information about its position.
Step 4: Final Answer:
If the uncertainty in momentum is zero, the uncertainty in position must be infinite to satisfy Heisenberg's Uncertainty Principle. This corresponds to option (E).