If the uncertainty in the position of a particle is equal to its de-Broglie wavelength, the minimum uncertainty in its velocity should be

Options

A. `1/(4 pi)`
B. `v/(4pi)`
C. `v/(4pi m)`
D. `(mv)/(4pi)`

Answer

To solve the problem, we need to follow these steps:

### Step 1: Understand the given information
We are given that the uncertainty in the position of a particle (\( \Delta x \)) is equal to its de Broglie wavelength (\( \lambda \)).

### Step 2: Write the de Broglie wavelength formula
The de Broglie wavelength (\( \lambda \)) is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the particle. The momentum \( p \) can also be expressed in terms of mass (\( m \)) and velocity (\( v \)):
\[ p = mv \]
Thus, we can rewrite the de Broglie wavelength as:
\[ \lambda = \frac{h}{mv} \]

### Step 3: Use the uncertainty principle
According to the Heisenberg uncertainty principle, the product of the uncertainties in position and momentum is given by:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
Here, \( \Delta p \) can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where \( \Delta v \) is the uncertainty in velocity.

### Step 4: Substitute \( \Delta x \) and \( \Delta p \) in the uncertainty principle
Substituting \( \Delta x = \lambda \) and \( \Delta p = m \cdot \Delta v \) into the uncertainty principle gives:
\[ \lambda \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

### Step 5: Substitute for \( \lambda \)
Now, substituting \( \lambda = \frac{h}{mv} \) into the equation:
\[ \frac{h}{mv} \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

### Step 6: Simplify the equation
This simplifies to:
\[ \frac{h \Delta v}{v} \geq \frac{h}{4\pi} \]
Now, we can cancel \( h \) from both sides (assuming \( h \neq 0 \)):
\[ \frac{\Delta v}{v} \geq \frac{1}{4\pi} \]

### Step 7: Solve for \( \Delta v \)
Multiplying both sides by \( v \) gives:
\[ \Delta v \geq \frac{v}{4\pi} \]

### Final Answer
Thus, the minimum uncertainty in the velocity (\( \Delta v \)) is:
\[ \Delta v = \frac{v}{4\pi} \]

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