The nuclear potential energy that binds protons and neutrons in a nucleus is often approximated by a square well. Imagine a proton confined in an infinitely high square well of length 11.8 fm, a typical nuclear diameter. Assuming the proton makes a transition from the n = 3 state to the ground state, calculate the following.
(a) the energy of the emitted photon.
(b) the wavelength of the emitted photon in fm.
(c) Identify the region of the electromagnetic spectrum to which this wavelength belongs.

a. gamma-ray
b. X-ray
c. ultraviolet
d. infrared
e. microwave

(A). 4.72*10⁷J (b). 4.2*10^-13m (c). The wavelength belongs to gamma-ray

Explanation:

L = 11.8fm = 11.8*10^-15m

h = Planck's constant

c = 3.0*10⁸m/s

M = 6.67*10^-27kg

∇E = hc / λ

∇E = E₂ - E₁ = [(h².3² / 8mL²) - (h².1² / 8mL²)]

∇E = h²/8mL² .[9-1] = 8h²/8mL²

∇E = (6.626*10^-24)² / [6.67*10^-27 *(11.8*10^-15)²]

∇E = 4.72 * 10⁷J

B. h² / mL² = hc / λ

λ = hcmL² / h² = cmL² / h

λ = [3.0*10⁸ * 6.626*10^-27 * (11.8*10^-15)²] / 6.626 * 10^-24

λ = 4.20 * 10^-23

λ = 4.20*10^-23 / (1.0*10^-10)m = 4.2* 10^-13m

C. From value of the wavelength, it lies in gamma ray region of electromagnetic spectrum.

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