The ratio of uncertainity in wave length and uncertainity in velocity is equal to

Options

A. `(-lamda)/v`
B. `h/(mv)`
C. `(h^(2))/(mv^(2))`
D. `(lamda^(2))/v`

Explanation

To solve the problem of finding the ratio of uncertainty in wavelength (Δλ) to uncertainty in velocity (Δv), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the de Broglie wavelength formula**:
The de Broglie wavelength (λ) is given by the formula:
\[ \lambda = \frac{h}{mv} \]
where \( h \) is Planck's constant, \( m \) is the mass of the particle, and \( v \) is its velocity.

2. **Differentiate the equation**:
To find the uncertainties, we need to differentiate the de Broglie wavelength with respect to velocity. Since \( h \) is a constant and \( m \) is also constant for a given particle, we can differentiate:
\[ \Delta \lambda = -\frac{h}{m} \cdot \frac{\Delta v}{v^2} \]
This gives us the change in wavelength in terms of the change in velocity.

3. **Express the uncertainty in wavelength**:
Rearranging the above equation, we can express the uncertainty in wavelength in terms of the uncertainty in velocity:
\[ \Delta \lambda = -\frac{h}{mv^2} \Delta v \]

4. **Find the ratio of uncertainties**:
Now, we can find the ratio of the uncertainty in wavelength to the uncertainty in velocity:
\[ \frac{\Delta \lambda}{\Delta v} = -\frac{h}{mv^2} \]

5. **Substitute for λ**:
From the de Broglie wavelength formula, we know that:
\[ \lambda = \frac{h}{mv} \]
We can substitute this into our equation:
\[ \frac{\Delta \lambda}{\Delta v} = -\frac{\lambda}{v} \]

6. **Final result**:
Therefore, the ratio of the uncertainty in wavelength to the uncertainty in velocity is:
\[ \frac{\Delta \lambda}{\Delta v} = -\frac{\lambda}{v} \]

### Conclusion:
The final answer is:
\[ \frac{\Delta \lambda}{\Delta v} = -\frac{\lambda}{v} \]