Final Answer:
The allowed energy levels for the particle in a one-dimensional box are determined by the equation:
where n is a positive integer, hbar is the reduced Planck's constant, m is the particle's mass, and L is the length of the box.
Step-by-step explanation:
The wavefunction provided, ψ(x) = N * sin(pi * x / L), is a solution to the Schrödinger equation for a particle in a one-dimensional box with a potential energy of zero within the box. To find the allowed energy levels, we apply the boundary conditions.
The wavefunction must vanish at the boundaries of the box, i.e., ψ(0) = ψ(L) = 0. This condition is satisfied when n is a positive integer, and the wavefunction becomes zero at x = 0 and x = L. The quantization condition arises from the requirement that the wavelength of the particle must fit an integer number of times within the box.
The general form of the wavefunction is a sine function, and the normalization constant N is determined by ensuring that the total probability of finding the particle within the box is equal to 1. This normalization condition leads to N = sqrt(2 / L).
Substituting the wavefunction into the time-independent Schrödinger equation and solving for energy (E) yields the allowed energy levels En. These levels depend on the quantum number n and are quantized, meaning only certain discrete energy values are allowed for the particle in the box. The final expression for the allowed energy levels is given by
where n is a positive integer.
Complete Question:
"A particle is in a one-dimensional box of length L (i.e., the box spans from x= 0 to x = L). The wavefunction for this particle can be described as: ψ(x) = N * sin(pi * x / L). Assuming that the potential energy of this particle is 0 everywhere within the box (i.e., V(x) = 0), find the allowed energy levels for this particle."