Final answer:
The Schrödinger equation is used to describe the behavior of a quantum particle. For an energy eigenfunction, we can solve the time-independent Schrödinger equation to find the allowed energy states. In the case of a particle in a box, the wave function terminates at the box wall, leading to quantized energy levels.
Step-by-step explanation:
The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum particle. When the initial wave function is an energy eigenfunction, we can solve the time-independent Schrödinger equation to determine the allowed energy states of the particle.
In the case of a particle in a box, the wave function must terminate at the box wall, which leads to quantized energy levels. The solutions to the time-independent Schrödinger equation for a particle in a box are given by the stationary states, and each stationary state corresponds to a specific energy level.
To find the time-dependent wave function, we can use the solutions of the time-independent Schrödinger equation and apply the time modulation factor. This allows us to determine the complete wave function that describes the particle's behavior in time.