Final answer:
To determine the eigenvalues and eigenfunctions for a one-dimensional quantum system, the Schrö;dinger equation is solved using the Hamiltonian operator and the wavefunction, considering the required boundary conditions and potential energy function of the system.
Step-by-step explanation:
To find the eigenvalues and eigenfunctions of the Hamiltonian operator (И) for a one-dimensional quantum system, we use the Schrödinger equation:
Иy = Ey
where И is the Hamiltonian, representing the total energy (kinetic + potential) of a quantum particle, y is the wavefunction depicting the quantum state of the particle, and E is the eigenvalue corresponding to the total energy of the particle. In a one-dimensional system, the potential energy (U) is typically provided, and the task is to solve this differential equation to find the permitted energy levels (eigenvalues) and their corresponding wavefunctions (eigenfunctions).
The solution of the equation involves finding a function y(x) that remains unchanged except for a multiplicative scaling factor (the eigenvalue E) when the Hamiltonian operator acts on it. This often requires utilizing boundary conditions such as the wave function being normalizable and symmetric about certain points (for instance, the bottom of a potential well).
An example of an eigenfunction for a free particle in one dimension can be given by a complex exponential wave:
Y(x, t) = Aei(kx-wt)
Where A is the amplitude, k is the wave number, and ω is the angular frequency. This complex function is not directly observable, but its absolute square |Y(x, t)|² gives the real probability density for the particle's location.
For a particle in a potential well, the eigenfunctions might look like standing waves (e.g., sin(kx) or cos(kx)), but these must be evaluated with respect to the particular potential energy function of the system.