A particle in an infinite square well of length L has a zero probability of being found at a point L/2.What is the lowest possible energy this particle can have? A)h²/(2ma²) B)h²/(ma²) C)2h²/(ma²) D)3h²/(ma²) E)4h²/(ma²)

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Quizplus · Accepted Answer

The lowest possible energy state for a particle in an infinite square well is the ground state, which has a wavefunction that is symmetric about the center of the well. The wavefunction for the ground state is given by:$$\psi(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right)$$The probability of finding the particle at a point x is given by:$$|\psi(x)|^2 = \frac{2}{L} \sin^2\left(\frac{\pi x}{L}\right)$$The probability of finding the particle at a point L/2 is:$$|\psi(L/2)|^2 = \frac{2}{L} \sin^2\left(\frac{\pi L/2}{L}\right) = 0$$Therefore, the lowest possible energy this particle can have is the ground state energy, which is given by:$$E_1 = \frac{h^2}{8mL^2}$$

A Particle in an Infinite Square Well of Length L