If the uncertainties in position and momentum are equal, the uncertainty in the velocity is :

Options

A. `sqrt((h)/(pi))`
B. `sqrt((h)/(2pi))`
C. `(1)/(2m)sqrt((h)/(pi))`
D. none of these

Answer

To solve the problem, we need to follow the steps outlined below:

### Step 1: Understand Heisenberg's Uncertainty Principle
Heisenberg's uncertainty principle states that the product of the uncertainties in position (Δx) and momentum (Δp) cannot be smaller than a certain value:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( h \) is Planck's constant.

### Step 2: Given Condition
The problem states that the uncertainties in position and momentum are equal:
\[ \Delta x = \Delta p \]

### Step 3: Substitute into the Uncertainty Principle
Substituting Δx = Δp into the uncertainty principle gives us:
\[ \Delta x \cdot \Delta x \geq \frac{h}{4\pi} \]
This simplifies to:
\[ (\Delta x)^2 \geq \frac{h}{4\pi} \]

### Step 4: Solve for Δx
Taking the square root of both sides, we find:
\[ \Delta x = \sqrt{\frac{h}{4\pi}} \]

### Step 5: Relate Momentum to Velocity
Momentum (p) is related to velocity (v) by the equation:
\[ p = mv \]
where \( m \) is the mass of the object. The uncertainty in momentum (Δp) can be expressed in terms of the uncertainty in velocity (Δv):
\[ \Delta p = m \Delta v \]

### Step 6: Substitute Δp
Since we have Δp as Δx from the earlier step, we can write:
\[ \Delta p = \sqrt{\frac{h}{4\pi}} = m \Delta v \]

### Step 7: Solve for Δv
Now, we can solve for the uncertainty in velocity (Δv):
\[ \Delta v = \frac{\Delta p}{m} \]
Substituting Δp:
\[ \Delta v = \frac{\sqrt{\frac{h}{4\pi}}}{m} \]

### Step 8: Simplify the Expression
This can be simplified further:
\[ \Delta v = \frac{1}{m} \sqrt{\frac{h}{4\pi}} = \frac{1}{2m} \sqrt{\frac{h}{\pi}} \]

### Final Result
Thus, the uncertainty in velocity is:
\[ \Delta v = \frac{1}{2m} \sqrt{\frac{h}{\pi}} \]

### Conclusion
The correct answer is that the uncertainty in velocity is \( \frac{1}{2m} \sqrt{\frac{h}{\pi}} \).

---