Question

Find the wavefunction of ground state ψ1(x) and first excited state ψ2(x) of the infinite square well of width l and plot them as a function of position for 0≤x≥l. What are the energies e1, e2 and oscillation frequencies Ω1, Ω2 of each state, respectively? Leave answers in terms of m, „, and l.

Answer 1

The ground state wavefunction ψ1(x) for an infinite square well is a sine wave with one half-wavelength within the well, while the first excited state ψ2(x) has two half-wavelengths. Their energies are given by specific formulae involving Planck's constant, mass, and the width of the well, with the first excited state having four times the energy of the ground state.

Step-by-step explanation:

The ground state wavefunction ψ1(x) for a particle in an infinite square well of width l can be derived from the stationary Schrödinger equation. It has the form ψ1(x) = √(2/l) sin(πx/l) for 0 ≤ x ≤ l, and ψ1(x) = 0 elsewhere. The first excited state wavefunction ψ2(x) is ψ2(x) = √(2/l) sin(2πx/l) for 0 ≤ x ≤ l, and ψ2(x) = 0 elsewhere. These wavefunctions are sine waves that exhibit one and two half-wavelengths within the well, respectively.

The energies associated with these states are given by e1 = (π2ħ2)/(2ml2) for the ground state and e2 = 4e1 for the first excited state. The oscillation frequencies Ω1 and Ω2 can be found using the relation Ω = e/ħ, yielding Ω1 = e1/ħ and Ω2 = e2/ħ.