(a) State Heisenbergs uncertainty principle. Give its mathematical expression. Also give its significance.
(b) Calculate the uncertainity in the position of a dust particle with mass equal to 1 mg if the uncertainity in its velocity is `5.5 xx 10^(-20) ms^(-1)` .

Answer

### Step-by-Step Solution

#### (a) Heisenberg's Uncertainty Principle

1. **Statement of the Principle**:
Heisenberg's uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. The more accurately we know the position of a particle, the less accurately we can know its momentum, and vice versa.

2. **Mathematical Expression**:
The principle can be mathematically expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where:
- \(\Delta x\) = uncertainty in position
- \(\Delta p\) = uncertainty in momentum
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{Js}\))

3. **Significance**:
The significance of Heisenberg's uncertainty principle lies in its implication for the behavior of microscopic particles, such as electrons. It highlights the fundamental limits of measurement in quantum mechanics and indicates that classical concepts of position and momentum do not apply at the quantum level. This principle is crucial for understanding phenomena in quantum physics and chemistry.

#### (b) Calculation of Uncertainty in Position

1. **Given Data**:
- Mass of the dust particle, \(m = 1 \, \text{mg} = 1 \times 10^{-6} \, \text{kg}\)
- Uncertainty in velocity, \(\Delta v = 5.5 \times 10^{-20} \, \text{ms}^{-1}\)

2. **Uncertainty in Momentum**:
The uncertainty in momentum (\(\Delta p\)) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
Substituting the values:
\[ \Delta p = (1 \times 10^{-6} \, \text{kg}) \cdot (5.5 \times 10^{-20} \, \text{ms}^{-1}) = 5.5 \times 10^{-26} \, \text{kg m/s} \]

3. **Using Heisenberg's Uncertainty Principle**:
Rearranging the uncertainty principle to find \(\Delta x\):
\[ \Delta x = \frac{h}{4\pi \Delta p} \]
Substituting \(h\) and \(\Delta p\):
\[ \Delta x = \frac{6.626 \times 10^{-34} \, \text{Js}}{4\pi \cdot (5.5 \times 10^{-26} \, \text{kg m/s})} \]

4. **Calculating \(\Delta x\)**:
- First, calculate \(4\pi \cdot (5.5 \times 10^{-26})\):
\[ 4\pi \approx 12.566 \]
\[ 4\pi \cdot (5.5 \times 10^{-26}) \approx 12.566 \cdot 5.5 \times 10^{-26} \approx 6.91 \times 10^{-25} \]
- Now substitute back:
\[ \Delta x \approx \frac{6.626 \times 10^{-34}}{6.91 \times 10^{-25}} \approx 9.59 \times 10^{-10} \, \text{m} \]

5. **Final Result**:
The uncertainty in the position of the dust particle is approximately:
\[ \Delta x \approx 9.59 \times 10^{-10} \, \text{m} \]