Question

A particle is described by the wavefunction (x)=sin(2x) −/ħ, for0≤x≤ Show that the wavefunction (x) is a solution of the Schrödinger equation and obtain from that energy of the particle in this state.

Answer 1

In order to show that the wavefunction \(\psi(x) = \sin(2x)\) is a solution to the Schrödinger equation, one must substitute it into the equation and verify that it satisfies the differential equation. This process demonstrates the wavefunction's role in quantum mechanics for a quantum particle in a box scenario, where the particle's energy states are quantized.

Step-by-step explanation:

The subject of the student's question is Physics, specifically quantum mechanics. The student is asking to demonstrate that a given wavefunction is a solution to the Schrödinger equation and to determine the energy of the particle when described by this wavefunction. This relates to the concept of a quantum particle in a box, where the energy levels are quantized due to the boundary conditions imposed on the wavefunction.

The time-independent Schrödinger equation for a particle with mass m in a potential V(x) is given by:

-\(\frac{\hbar^2}{2m}\)\frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E\psi(x)

For a particle in a box, V(x) = 0 inside the box, which simplifies the equation to:

-\(\frac{\hbar^2}{2m}\)\frac{d^2\psi(x)}{dx^2} = E\psi(x)

To show the given wavefunction \(\psi(x) = \sin(2x) is a solution, we need to substitute it into the Schrödinger equation and verify if it satisfies the equation and calculate the energy E of the particle.