The uncertainty in position of an electron in a certain state is `5 xx 10^(-10) m`. The uncertainty in its momentum might be

Options

A. `5.0 xx 10^(-24) kg.m//s`
B. `4.0 xx 10^(-24) kg.m//s`
C. `3.0 xx 10^(-24) kg.m//s`
D. All of the above

Answer

To solve the problem of finding the uncertainty in momentum of an electron given the uncertainty in its position, we will use the Heisenberg Uncertainty Principle. Here are the steps to arrive at the solution:

### Step 1: Understand the Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle states that the product of the uncertainties in position (Δx) and momentum (Δp) of a particle is always greater than or equal to a constant value. Mathematically, it is expressed as:
\[ \Delta x \cdot \Delta p \geq \frac{h}{4\pi} \]
where \( h \) is Planck's constant.

### Step 2: Identify the given values
From the problem, we know:
- The uncertainty in position (Δx) is given as \( 5 \times 10^{-10} \) m.
- Planck's constant \( h \) is approximately \( 6.626 \times 10^{-34} \) J·s.

### Step 3: Rearrange the formula to find Δp
We need to find the uncertainty in momentum (Δp). Rearranging the formula gives:
\[ \Delta p \geq \frac{h}{4\pi \Delta x} \]

### Step 4: Substitute the known values into the equation
Now we will substitute the values of \( h \) and \( Δx \) into the equation:
\[ \Delta p \geq \frac{6.626 \times 10^{-34}}{4\pi \times 5 \times 10^{-10}} \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ 4\pi \times 5 \times 10^{-10} \approx 62.8319 \times 10^{-10} \approx 6.28319 \times 10^{-9} \]

### Step 6: Calculate Δp
Now substituting the denominator back into the equation:
\[ \Delta p \geq \frac{6.626 \times 10^{-34}}{6.28319 \times 10^{-9}} \approx 1.055 \times 10^{-25} \text{ kg m/s} \]

### Step 7: Finalize the answer
Thus, the uncertainty in momentum of the electron is:
\[ \Delta p \geq 1.055 \times 10^{-25} \text{ kg m/s} \]
This means that the uncertainty in momentum could be greater than or equal to this value.