... (a) Find the energy of the ground state ($n = 1$) and the first two excited states of a neutron in a one-dimensional box of length $L = 1.00 \times 10^{-15} m = 1.00$ fm (about the diameter of an atomic nucleus). Make an energy-level diagram for the system. Calculate the wavelength of electromagnetic radiation emitted when the neutron makes a transition from (b) $n = 2$ to $n = 1$, (c) $n = 3$ to $n = 2$, and (d) $n = 3$ to $n = 1$.
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Step 1: The energy levels of a particle in a one-dimensional box are given by: $E_n = \frac{n^2 h^2}{8mL^2}$ where $n$ is the quantum number, $h$ is Planck's constant, $m$ is the mass of the particle, and $L$ is the length of the box. Show more…
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(a) Use the results of Problem $5-46$ to find the energy of the ground state $(n=1)$ and the first two excited stated of a proton in a one-dimensional box of length $L=10^{-15} \mathrm{~m}=1 \mathrm{fm} .$ (These are of the order of magnitude of nuclear energies.) Calculate the wavelength of electromagnetic radiation emitted when the proton makes a transition from $(b) n=2$ to $n=1,(c) n=3$ to $n=2,$ and $(d) n=3$ to $n=1$
Let's model a neutron in a nucleus as a particle of mass 1.675 × 10⁻²⁷ kg in an infinitely deep 3-dimensional potential well in the form of a cubic box of side-length L = 8.96 fm in (x, y, z), so that its wave function has the form ψ(x, y, z) = A sin (ℓπx/L) sin (mπy/L) sin (nπz/L). Note that "fm" is a femtometer, 10⁻¹⁵ m, and that (ℓ, m, n) are integers, each starting at 1 and moving upward to 2, then 3, etc. In this box the potential energy is U = 0. Use Schroedinger's equation in (x, y, z) to find a formula for the energy levels of the neutron, E(ℓ, m, n), and use your formula to find the energy in MeV of the photon (gamma ray) that is emitted when the neutron drops from its first excited state to the ground state. MeV (± 0.05 MeV)
Ameer S.
Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional box of width 10.0 fm. (a) What are the energies of the proton when it is in the states corresponding to $n=1, n=2,$ and $n=3$ ? (b) What are the energies of the photons emitted when the proton makes the transitions from the first and second excited states to the ground state?
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