In quantum mechanics, the particle in a box is a conceptually simple problem in position space that illustrates the quantum nature of particles by only allowing discrete values of energy. In this problem, we start from the Schrödinger equation, find the energy eigenvalues, and proceed to impose normalization conditions to derive the eigenfunctions associated with those energy levels.
- 1 Begin with the time-independent Schrödinger equation. The Schrödinger equation is one of the fundamental equations in quantum mechanics that describes how quantum states evolve in time. The time-independent equation is an eigenvalue equation, and thus, only certain eigenvalues of energy exist as solutions.
- 2 Substitute the Hamiltonian of a free particle into the Schrödinger equation.
- In the one-dimensional particle in a box scenario, the Hamiltonian is given by the following expression. This is familiar from classical mechanics as the sum of the kinetic and potential energies, but in quantum mechanics, we assume that position and momentum are operators.
- Meanwhile, we let inside the box and everywhere else. Because in the region that we are interested in, we may now write this equation as a linear differential equation with constant coefficients.
- 3 Solve the above equation. This equation is familiar from classical mechanics as the equation describing simple harmonic motion.
- The theory of differential equations tells us that the general solution to the above equation is of the following form, where and are arbitrary complex constants and is the width of the box. We are choosing coordinates such that one end of the box lies at for simplicity of…
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