SKEDSOFT

Introduction to Eigenvalues and eigenfunctions:

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as eigenvalues of the observable and the corresponding wave functions are called eigenfunctions. The eigenfunctions are those eigenfunctions which are definite and single valued. When something is in the condition of being definitely ‘pinned-down’, it is said to possess an eigenvalue. For example, if the position of an electron has been made definite, it is said to have an eigenvalue of position. The term eigen can be roughly translated from German as inherent or as a characteristic). The German word ”eigen” was first used in this context by the mathematician David Hilbert in 1904.

Energy eigen values for a free particle:

The time-independent form of the Schrödinger wave equation in one-dimension is given by,

A free particle is defined as the one which is not acted upon by any external force that modifies its motion. Hence the potential energy U in the Schrödinger equation is taken to be zero. That is,

where E is the total energy of the particle and is purely in the form of kinetic energy. The general solution of such a differential equation is of the form

Its difficult to solve for constants A and B as we cannot impose any boundary conditions on the free particle. Since the solution has not imposed any restriction on the constant √ 2mE /h which we call k, the free particle is permitted to have any value of  energy value

The Schrödinger equation, applied to the free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain.