Question

1. an effective treatment for some cancerous tumors involves irradiation with \fast\ neutrons. the neutrons from one treatment source have an average velocity of 3.1×10^7 m/s. if the velocities of individual neutrons are known to within 2% of this value, what is the uncertainty in the position of one of them? (mass of a neutron = 1.6749×10^-27 kg) a) 1.0 × 10^-3 pm b) 1.6 × 10^6 pm c) 3.6 × 10^-1 pm d) 5.1 × 10^-2 pm e) 4.7 × 10^2 pm

Explanation:

Step1: Calculate the uncertainty in velocity

The average velocity \(v = 3.1\times10^{7}\text{ m/s}\), and the uncertainty in velocity \(\Delta v=0.02v\).
So \(\Delta v = 0.02\times3.1\times 10^{7}\text{ m/s}=6.2\times10^{5}\text{ m/s}\)

Step2: Use the Heisenberg uncertainty principle

The Heisenberg uncertainty principle is \(\Delta x\Delta p\geq\frac{h}{4\pi}\), and since \(p = mv\), \(\Delta p=m\Delta v\) (for non - relativistic case, \(m = 1.6749\times 10^{-27}\text{ kg}\)). Then \(\Delta x\geq\frac{h}{4\pi m\Delta v}\), where \(h = 6.626\times 10^{-34}\text{ J}\cdot\text{s}\)

Substitute the values:

$$
LATEXBLOCK0
$$

$$
LATEXBLOCK1
$$

Since \(1\text{ pm}=10^{-12}\text{ m}\), \(\Delta x = 5.1\times 10^{-2}\text{ pm}\)

Answer:

d) \(5.1\times 10^{-2}\text{ pm}\)