Request pdf exact solutions of a particle in a box with a delta function potential. The very first problem you will solve in quantum mechanics is a particle in a box. The factorization method we use the factorization method to find the exact eigenvalues and eigenfunctions. This is a second order ordinary differential equation. The solutions to the problem give possible values of e and. The solution to the finite well particle in a box must be solved numerically, resulting in wave functions that are sine functions inside the quantum well and exponentially decaying functions in the barriers. Particle in a 1d box first we will consider a free particle moving in 1d so vx 0. Pdf the problem of a relativistic spin 12 particle confined to a onedimensional box is solved in a way that resembles closely the solution of the. An example of a problem which has a hamiltonian of the separable form is the particle in a 3d box.
It can also be written in terms of sines and cosines as. Chapter 7 the schroedinger equation in one dimension in classical. Aeikx where k v 2me a general solution of this equation is. Exact solutions of a particle in a box with a delta. The energy of a particle in a box is quantized the particle in the box model system is the simplest nontrivial application of the schrodinger equation, but one which illustrates many of the fundamental concepts of quantum mechanics. This is a second order differential equation which has the solutions exp. Colby college particleinabox and heisenberg uncertainty a2 0. A particle in a rigid box consider a particle of mass m confined in a rigid, one. These are the only values of the energy which allow solution of the schro. The barriers outside a onedimensional box have infinitely large potential, while the interior of the box has a. Therefore, the particles total energy is just its kinetic energy. The potential is zero inside the cube of side and infinite outside.
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. You can solve quantum mechanics classic particle in a box. Timeharmonic solutions to schrodinger equation are of the form. Particle in a 3dimensional box chemistry libretexts. In this video we will discuss in details, all that you should know about particle in 1d box. It is one of the most important example quantum systems in chemistry, because it helps us develop. The particle can move freely between 0 and l at constant speed and thus with constant kinetic energy.