Calculate the product of uncertainty in position and velocity for an electron of mass `9.1 xx 10^(-31)kg` according to Heisenberg uncertainty principle

Explanation

To solve the problem of calculating the product of uncertainty in position and velocity for an electron according to Heisenberg's Uncertainty Principle, we will follow these steps:

### Step 1: Understand Heisenberg's Uncertainty Principle
Heisenberg's Uncertainty Principle states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) is at least as large as a certain constant. Mathematically, it can be 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

### Step 2: Relate momentum to velocity
Momentum (p) is defined as:
\[ p = m \cdot v \]
where:
- \( m \) = mass of the electron
- \( v \) = velocity of the electron

Thus, the uncertainty in momentum can be expressed as:
\[ \Delta p = m \cdot \Delta v \]
where \( \Delta v \) is the uncertainty in velocity.

### Step 3: Substitute into the uncertainty principle
Substituting the expression for Δp into the uncertainty principle gives:
\[ \Delta x \cdot (m \cdot \Delta v) \geq \frac{h}{4\pi} \]

### Step 4: Rearranging the equation
Rearranging the equation to solve for the product of uncertainty in position and velocity:
\[ \Delta x \cdot \Delta v \geq \frac{h}{4\pi m} \]

### Step 5: Insert known values
Now, we will insert the known values:
- Planck's constant \( h = 6.626 \times 10^{-34} \, \text{Joule second} \)
- Mass of the electron \( m = 9.1 \times 10^{-31} \, \text{kg} \)
- Value of \( \pi \approx 3.14 \)

### Step 6: Calculate the product
Now we can calculate:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{4 \cdot 3.14 \cdot 9.1 \times 10^{-31}} \]

Calculating the denominator:
\[ 4 \cdot 3.14 \cdot 9.1 \times 10^{-31} \approx 1.136 \times 10^{-29} \]

Now, calculating the fraction:
\[ \Delta x \cdot \Delta v \geq \frac{6.626 \times 10^{-34}}{1.136 \times 10^{-29}} \approx 5.83 \times 10^{-5} \]

### Final Answer
Thus, the product of uncertainty in position and velocity for the electron is approximately:
\[ \Delta x \cdot \Delta v \approx 5.83 \times 10^{-5} \, \text{m}^2/\text{s} \]

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