Animated Solution for Chemistry - Atomic Structure: An accelerated electron has a speed of $5\times 10^6{\text{ ms}}^{-1}$ with an uncertainty of $0.02\%$. The uncertainty in finding its location while in motion is $x\times 10^{-9}\text{ m}$. The value of $x$ is ......... . (Nearest integer) [Use mass of electron $=9.1\times 10^{-31}\text{ kg}$, $h=6.63\times 10^{-34}\text{ Js}$, $\pi =3.14$]

Unraveling Heisenberg's Uncertainty Principle

Imagine you are driving a car down the highway. You look at your speedometer, and it reads exactly $60\text{ km/h}$. You also know exactly where you are on the road. In our everyday macroscopic world, knowing both your speed and your position simultaneously is completely normal.

But what if you were an electron? The rules of the universe completely flip. Welcome to the bizarre and fascinating world of quantum mechanics, where certainty is an illusion, and probability rules supreme. In this problem, we are going to explore Heisenberg's Uncertainty Principle by tracking down a speeding electron.

Our electron is zooming through space at a massive speed of $v=5\times 10^6\text{ m/s}$. However, the problem states there is an uncertainty of $0.02\%$ in this speed. This means we don't know the exact speed; it could be slightly faster or slightly slower.

Before we can use any quantum formulas, we need to convert this percentage into an absolute value. How much is $0.02\%$ of $5\times 10^6\text{ m/s}$?

Let's calculate the uncertainty in velocity, denoted as $\Delta v$:

So, our velocity uncertainty is $1000\text{ m/s}$. That might sound like a huge margin of error for a car, but for an electron moving at five million meters per second, it's actually a very tight bound!

Now we bring in the heavy artillery: Heisenberg's Uncertainty Principle. Formulated by Werner Heisenberg in 1927, this principle states that it is fundamentally impossible to know both the exact position and the exact momentum of a particle at the same time. The more precisely you know one, the less precisely you know the other.

Mathematically, it is expressed as:

Where: - $\Delta x$ is the uncertainty in position. - $\Delta p$ is the uncertainty in momentum. - $h$ is Planck's constant ($6.63\times 10^{-34}\text{ Js}$).

Since momentum $p$ is the product of mass and velocity ($p=mv$), and the mass of an electron is constant at non-relativistic speeds, we can rewrite the uncertainty in momentum as $\Delta p=m\Delta v$.

Substituting this into our master equation gives us the working formula for this problem:

This is where many students make silly mistakes. We have a lot of scientific notation to handle. Let's substitute all our known values carefully. We are looking for $\Delta x$, so let's isolate it:

Plugging in the values provided in the question:

Let's group the numbers and the powers of 10 separately to keep things clean:

Notice how the denominator's powers of 10 combine: $10^{-31}\times 10^3=10^{-28}$.

We have our answer, but it's not in the format the question requested. The problem asks for the uncertainty in the form of $x\times 10^{-9}\text{ m}$.

To convert $0.058\times 10^{-6}$ into something times $10^{-9}$, we need to shift the decimal point three places to the right. Mathematically, we are multiplying by $10^3$ and dividing by $10^3$:

Comparing this to the given format $x\times 10^{-9}\text{ m}$, we can clearly see that our integer value is:

Think about what this means. $58\times 10^{-9}\text{ m}$ is 58 nanometers. To a human, that is unimaginably small. But to an electron, which is essentially a point particle, a 58-nanometer region is a massive, fuzzy cloud of probability. This beautiful calculation proves that in the quantum realm, the harder you look at how fast something is going, the more it blurs out of focus!