Give the mathematical expression of uncertainty principle.

Answer

To derive the mathematical expression of the uncertainty principle, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Uncertainty Principle**:
The uncertainty principle is a fundamental concept in quantum mechanics that states there is a limit to how precisely we can know certain pairs of physical properties of a particle, such as position and momentum.

2. **Mathematical Expression**:
The mathematical expression for the uncertainty principle can be written 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 (\(6.626 \times 10^{-34} \, \text{Js}\)).

3. **Relating Momentum to Velocity**:
Momentum (\(p\)) can be expressed in terms of mass (\(m\)) and velocity (\(v\)):
\[ p = m \cdot v \]
Therefore, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where \(\Delta v\) is the uncertainty in velocity.

4. **Substituting Momentum into the Expression**:
By substituting \(\Delta p\) into the uncertainty principle expression, we get:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

5. **Rearranging the Expression**:
We can rearrange the expression to isolate the uncertainties in position and velocity:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]
This shows the relationship between the uncertainties in position and velocity, dependent on the mass of the particle.

### Final Expression:
Thus, the final mathematical expression of the uncertainty principle can be summarized as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \quad \text{or} \quad \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]