A particle confined in a rigid one-dimensional box (an infinite well) of length 17.0 fm has an energy level E_n = 24.0 MeV and an adjacent energy level E_n+1= 37.5 MeV. What is the value of the ground state energy? (1 eV = 1.60 × 10^-19 J) A) 1.50 MeV B) 13.5 MeV C) 0.500 MeV D) 4.50 MeV

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The energy levels of a particle confined in a one-dimensional box (an infinite well) can be calculated using the equation: E_n = (n2ℏ2π2)/(2mL2) where n is the quantum number of the energy level, ℏ is the reduced Planck's constant, m is the mass of the particle, and L is the length of the box. We can use this equation to find the difference between the energy levels given: E_n+1 - E_n = [(n+1)2ℏ2π2/(2mL2)] - [n2ℏ2π2/(2mL2)] Simplifying this expression, we get: E_n+1 - E_n = ℏ2π2/(2mL2)[2n+1] We know the values of E_n+1, E_n, and L. We also know that ℏ = 1.055 × 10-34 J·s and that 1 MeV = 1.60 × 10-13 J. Using these values, we can solve for m: (37.5 MeV - 24.0 MeV) = (1.055 × 10-34 J·s)2π2/(2m(17.0 × 10-15 m)2)[2n+1] (13.5 × 1.60 × 10-13 J) = (1.055 × 10-34 J·s)2π2/(2m(17.0 × 10-15 m)2)[2n+1] m = 1.67 × 10-27 kg (mass of a proton) Now we can use the equation for the ground state energy level (n = 1): E_n = (ℏ2π2)/(2mL2) E_n = (1.055 × 10-34 J·s)2π2/(2(1.67 × 10-27 kg)(17.0 × 10-15 m)2) E_n = 1.50 MeV Therefore, the value of the ground state energy is 1.50 MeV, which is choice A.

A Particle Confined in a Rigid One-Dimensional Box (An Infinite