The speed of a certain proton is 4.5×10^5m/s. If the uncertainty in its momentum is to be reduced to 0.1%. Calculate the uncertainty in its position

To calculate the uncertainty in position, we need to use the Heisenberg's uncertainty principle, which states that the product of the uncertainty in position (Δx) and the uncertainty in momentum (Δp) must be greater than or equal to h (Planck's constant) divided by 4π.

The equation for the uncertainty principle is given by:

Δx * Δp >= h / (4π)

To calculate the uncertainty in position, Δx, we can rearrange the equation as:

Δx >= (h / (4π)) / Δp

Now, let's plug in the values we have:

h = 6.626 x 10^-34 J·s (Planck's constant)
Δp = (0.1 / 100) * p (momentum uncertainty as a percentage of the momentum)
p = mass * velocity (momentum of the proton)

We have the speed of the proton (velocity) given as 4.5 × 10^5 m/s. To calculate its momentum, we need to know the mass of the proton, which is approximately 1.67 x 10^-27 kg.

So, the momentum (p) of the proton is:

p = (1.67 x 10^-27 kg) * (4.5 x 10^5 m/s)

Now, substitute the values into the equation:

Δx >= (6.626 x 10^-34 J·s / (4π)) / ((0.1 / 100) * (1.67 x 10^-27 kg) * (4.5 x 10^5 m/s))

Calculating this will give us the minimum uncertainty in the position of the proton.