To find the position of a particle in a bounded state, you would typically solve the appropriate quantum mechanical equation that describes the system. The most common equation used for describing bound states is the Schrödinger equation.
The Schrödinger equation for a particle in a potential energy well (such as a particle confined within a box or a harmonic oscillator potential) is given by:
Ĥψ = Eψ
Here, Ĥ is the Hamiltonian operator, ψ is the wave function of the particle, E represents the energy of the particle in the given state, and the equation is in its time-independent form.
Solving the Schrödinger equation involves finding the eigenvalues (allowed energy values) and the corresponding eigenfunctions (wave functions) that satisfy the equation. The eigenfunctions provide information about the spatial distribution of the particle.
Once you have the wave function ψ, the probability density of finding the particle at a particular position x is given by |ψ(x)|², where |ψ(x)| represents the magnitude of the wave function at position x. The probability density is a real, positive quantity that indicates the likelihood of finding the particle at a specific location.
In practice, solving the Schrödinger equation for complex systems may require various mathematical techniques, such as separation of variables, numerical methods, or approximation methods. The specific approach depends on the nature of the potential and the complexity of the system.
It's worth noting that the Schrödinger equation provides a probabilistic description of the particle's behavior, where the square of the wave function gives the probability density. The exact position of the particle in a bound state is not determined with certainty but rather described in terms of probabilities.