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In quantum mechanics, the hydrogenic wave function is essential for describing the state of an electron in a hydrogen-like ion. A hydrogen-like ion is essentially a nucleus with one electron orbiting it. The wave function for this system in its ground state state quantifies the probability of finding the electron at a particular point around the nucleus. For a quantum state characterized by quantum numbers \( n=1, l=0 \), with \( Z \) representing the nuclear charge and \( a_0 \) standing for the Bohr radius, we represent it mathematically as:\[ \psi_{n=1, l=0}(\mathbf{x}) = \frac{1}{\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2} e^{-Z r / a_0}. \]This function shows that the electron is most likely found near the nucleus when in its lowest energy state (ground state), depending on the charge of the nucleus (and thus adjusting how sharply the probability density peaks around the center). Initially, for the triton, we have \( Z = 1 \). In this case, as the problem describes a sudden change due to beta decay, the nuclear charge changes to\( Z = 2 \), affecting the wave function correspondingly.

The sudden approximation in quantum mechanics refers to a situation where an external influence (like a beta decay) acts on an atomic system so quickly that the system does not have sufficient time to change its initial state. This rapid transformation influences the system to become a superposition of its new potential final states.Upon the occurrence of the beta decay process, the atomic nucleus in the exercise changes its charge instantaneously from \( Z = 1 \) to \( Z = 2 \).When applying the sudden approximation, the primary aspect to consider is time. The time scale of the external action should be much shorter than the natural time scale of the system—this home's internal oscillations/movements. Once this sudden change occurs, the wave function does not evolve smoothly but is abruptly thrust into a new scenario dictated by the new conditions, like the change in nuclear charge in our problem. The electron's initial wave function instantly corresponds to the new ionic state, allowing the calculation of probabilities for state survival post-change.

Beta decay is a type of radioactive decay where a beta particle (electron or positron) and an antineutrino (or neutrino) are emitted from a nucleus. This decay occurs because the nucleus has an imbalance of neutrons and protons, thus undergoing a transformation to reach a more stable state.In the context of this problem, when tritium \((^3\text{H})\) undergoes beta decay, it emits an electron and an antineutrino, thereby converting a neutron into a proton. This increases the atomic number by one, transforming the triton nucleus into a helium ion \((Z = 2; ^3\text{He})\). The energy released in tritium beta decay is about \( 18 \text{ keV} \), which provides the energy to propel the reaction forward swiftly. The movement from tritium to helium occurs suddenly, allowing us to apply the sudden approximation. In physics problems like this one, understanding beta decay helps explain how nuclear transformations affect atomic states —highlighting the interrelation between nuclear physics and quantum mechanical models.