Heisenberg's uncertainty principal rules out the exact simultaneous measurement of

Options

A. Probability and intensity
B. Energy and relocity
C. Charge density and radius
D. Position and velocity

Answer

**Step-by-Step Solution:**

1. **Understanding Heisenberg's Uncertainty Principle**: The principle states that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with arbitrary precision. This means that the more accurately we know one of these properties, the less accurately we can know the other.

2. **Mathematical Representation**: 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.

3. **Substituting Momentum**: Since momentum (\(p\)) is defined as the product of mass (\(m\)) and velocity (\(v\)), we can express uncertainty in momentum as:
\[ \Delta p = m \cdot \Delta v \]
Thus, the uncertainty principle can also be written as:
\[ \Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi} \]

4. **Conclusion on Measurement**: From the above equations, it is clear that the uncertainties in position (\(\Delta x\)) and velocity (\(\Delta v\)) cannot both be minimized at the same time. Therefore, if we measure the position of a particle very accurately, the uncertainty in its velocity will increase, and vice versa.

5. **Final Answer**: Thus, Heisenberg's Uncertainty Principle rules out the exact simultaneous measurement of **position and velocity** of a particle.

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