The quantum mechanical treatment of a two-nucleon bound system in the "S" state involves describing the relative motion and spin interactions of the two nucleons. The "S" state refers to a specific configuration where the total angular momentum of the system is zero.
In nuclear physics, the interaction between nucleons (protons and neutrons) is often described using the nuclear potential, which incorporates both the strong nuclear force and other electromagnetic forces. The nuclear potential represents the effective potential energy between the two nucleons.
To solve the quantum mechanical problem for the two-nucleon bound system in the "S" state, one typically starts with the Schrödinger equation. The Schrödinger equation describes the time-independent behavior of the system and is given by:
Hψ = Eψ
where H is the Hamiltonian operator, ψ is the wave function of the system, E represents the energy of the system, and ℏ is the reduced Planck's constant.
The Hamiltonian operator includes the kinetic energy of the nucleons and the nuclear potential energy. The form of the nuclear potential depends on the specific model used to describe the interaction between nucleons. Examples of commonly used nuclear potentials include the Yukawa potential, the Reid potential, and the Argonne v18 potential.
Solving the Schrödinger equation for the two-nucleon system involves finding the wave function ψ that satisfies the equation. This can be a challenging task and often requires approximations and numerical methods.
Once the wave function is determined, various properties of the system can be calculated, such as the probability density distribution, expectation values of observables, and energy eigenvalues. These calculations provide insights into the behavior and properties of the two-nucleon bound system in the "S" state, such as the binding energy, spatial distribution, and spin correlations between the nucleons.
It's important to note that the description provided here is a simplified overview, and the actual calculations and details involved in the quantum mechanical treatment of a two-nucleon bound system can be quite complex and involve advanced techniques and models.