Question

Consider a finite square-well potential well of width 3.1 ✕ 10-15 m that contains a particle of mass 1.8 GeV/c2. How deep does this potential well need to be to contain three energy levels? (Except for the energy levels, this situation approximates a deuteron. Use the infinite square well potential result to approximate the energy levels.)

Answer 1

We can find the energy levels of the particle in the finite square-well potential using the formula for energy of infinite square well.

The formula is given by,

$E_n = n^2 \frac{h^2}{8mL^2}$

Where

Number of levels (n) = 3

Planck constant (h) $= 6.626*10^{-34}J.s$

Mass of the particle is (m) $= 1.88GeV/c^2$

The mass of the particle can be converted to $J/c^2$,

$m=1.88GeV/c^2(\frac{10^9eV}{1GeV})(\frac{1.6*10^{-19}}{1eV})$

$m=3.008*10^{-10}J/c^2$

With all the values we can solve in the first equation, so

$E_3 = (3)^2 \frac{h^2}{8mL^2}$

$E_3 = 9 \frac{h^2c^2}{8mc^2L^2}$

$E_3= \frac{9(6.626*10^{-34})^2(3*10^8)^2}{8(3.008*10^{-10}/c^2)}$(c^2)(3*10^{-15})}

$E_3 = 1.642*10^{-11}J$

We can also convert to eV,

$E_3=1.642*10^{-11}J(\frac{1MeV}{1.6*10^{-13}J})$

$E_3 = 102.62MeV$

Therefore, the depth of the well needed to contain three energy levels is 102.62MeV