What would be the uncertainty in momentum of an electron whose position is known with absolute certainty ?

Explanation

To solve the problem of finding the uncertainty in momentum of an electron whose position is known with absolute certainty, we can use the Heisenberg Uncertainty Principle. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the position and momentum of a particle with absolute precision. The principle can be mathematically expressed as:

\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]

where:
- \(\Delta x\) is the uncertainty in position,
- \(\Delta p\) is the uncertainty in momentum,
- \(h\) is Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\)).

### Step 2: Analyze the Given Condition
In the problem, it is stated that the position of the electron is known with absolute certainty. This means that the uncertainty in position (\(\Delta x\)) is zero:

\[ \Delta x = 0 \]

### Step 3: Substitute into the Uncertainty Principle
Substituting \(\Delta x = 0\) into the uncertainty principle equation gives:

\[ 0 \cdot \Delta p \geq \frac{h}{4\pi} \]

### Step 4: Interpret the Result
Since any number multiplied by zero is zero, the left side of the inequality becomes zero:

\[ 0 \geq \frac{h}{4\pi} \]

This implies that the uncertainty in momentum (\(\Delta p\)) must tend towards infinity:

\[ \Delta p \rightarrow \infty \]

### Conclusion
Therefore, if the position of the electron is known with absolute certainty, the uncertainty in its momentum becomes infinite. This means that we cannot measure the momentum (or velocity) of the electron accurately.

### Summary of the Solution
- The Heisenberg Uncertainty Principle states that \(\Delta x \cdot \Delta p \geq \frac{h}{4\pi}\).
- If \(\Delta x = 0\) (position known with absolute certainty), then \(\Delta p\) must tend to infinity.
- Thus, the uncertainty in momentum of the electron is infinite.