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?

The energy levels of a proton are a fundamental concept in quantum mechanics, especially in the context of nuclear physics. When we confine a proton to a one-dimensional box, such as the nucleus of an atom, it can only occupy certain defined energy levels, much like a ladder with distinct rungs. These levels are determined by quantum mechanics and are quantified by a principal quantum number, denoted by `n`. The proton cannot exist at energy levels between these defined states, a strange concept when compared with the continuous energy variation we're used to in the classical world.

Considering a proton trapped in a box with a size of 10.0 femtometers (10^-15 meters), we calculate the energy levels using the formula \[E_n = \frac{h^2n^2}{8mL^2}\], where `h` is Planck's constant, `m` is the mass of the proton, `L` is the width of the box, and `n` is the quantum number. The resulting energies are found to be significantly spaced apart, illustrating that protons in smaller nuclei exist at relatively high and discrete energy levels. Students should also appreciate how these energy levels change with the quantum number `n` and how they become more closely spaced as the value of `n` increases.

In the microscopic realm governed by the laws of quantum mechanics, particles like protons have specific quantum states. These states are defined by quantum numbers which, in the context of energy, is represented as `n` in the formula used to calculate energy levels. Each quantum state has a unique set of properties and energy. The ground state, represented by `n=1`, is the lowest possible energy state. As the quantum number increases, the energy levels rise, signifying that the proton possesses more energy and is in an 'excited' state.

The concept of quantized states is instrumental in understanding phenomena like spectral lines and the stability of atomic nuclei. It's important for students to understand that quantum states are not just theoretical constructs but have very real consequences in the behavior of particles at the nuclear level. These states dictate transitions that ultimately lead to particle interactions, radioactive decay, and the emission of photons during such transitions.

When a proton transitions from a higher to a lower energy state, the energy difference is not simply lost; it is emitted as a photon. This is what is known as photon emission, and the energy of the photon equals the energy difference between the two states. It is fascinating that this singular photon carries away precisely the amount of energy lost by the proton, a direct result of energy conservation at the quantum level.

To calculate the energy of emitted photons when a proton moves from an excited state to the ground state, we find the difference in energies: for example, \[E_{photon} = E_{initial} - E_{final}\]. The energy of the photon is then expressed in joules \(J\), which can be converted to frequencies or wavelengths for spectroscopic analysis. Understanding photon emission is crucial for interpreting nuclear reactions and the spectra of stars, among other phenomena, giving students a peek into how energy is transferred and conserved at the smallest scales of the universe.