The neutrons in a parallel beam, each having kinetic energy \(1 / 40 \mathrm{eV}\) (which is approximately corresponding to “room temperature"), are directed through two slits \(0.50 \mathrm{~mm}\) apart. How far apart will the interference peaks be on a screen \(1.5 \mathrm{~m}\) away?

Given the kinetic energy of the neutrons, we can find their momentum and subsequently their de Broglie wavelength using the following formula: $$ \lambda = \frac{h}{p} $$ Where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant (\(6.626 \times 10^{-34} \mathrm{Js}\)), and \(p\) is the momentum of the neutron. To find the momentum, we need the mass and velocity of the neutron. We can find the velocity using the kinetic energy formula: $$ K = \frac{1}{2}mv^2 $$ Where \(K\) is the kinetic energy, \(m\) is the mass, and \(v\) is the velocity. For a neutron, the mass is approximately \(m = 1.675 \times 10^{-27} \mathrm{kg}\), and the given kinetic energy is \(K = \frac{1}{40} \mathrm{eV} = \frac{1}{40} \times 1.6 \times 10^{-19} \mathrm{J}\).

The formula for double-slit interference is given by: $$ d \sin{\theta} = m\lambda $$ Where \(d\) is the separation between the two slits, \(m\) is an integer representing the peak order, and \(\theta\) is the angle of the interference peak from the central maximum. Given the small angle approximation, we can use the following relationship: $$ \tan{\theta} \approx \sin{\theta} \approx \theta \approx \frac{y}{L} $$ Where \(y\) is the distance between the interference peaks on the screen, and \(L\) is the distance between the slits and the screen. In this problem, \(d = 0.50 \mathrm{mm} = 0.50 \times 10^{-3} \mathrm{m}\), and \(L=1.5 \mathrm{m}\). We can now find \(y\) using the formula \(y = L \tan{\theta} = L \frac{\lambda}{d}\), where \(\lambda\) is the de Broglie wavelength we calculated in Step 1.

Now that we have all the variables, we can plug them into the formula: $$ y = L \frac{\lambda}{d} $$ After substituting the values for \(L\), \(\lambda\), and \(d\), we will find the distance \(y\) between the interference peaks on the screen.