Physically acceptable wave function The wave function must be single valued. This means that for any given values of x and t , Ψ(x,t) must have a unique value. This is a way of guaranteeing that there is only a single value for the probability of the system being in a given state.

Wave function, in quantum mechanics, variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle’s being there at the time.

A wave function may be used to describe the probability of finding an electron within a matter wave. To do this, the wave function, which may include an imaginary number, is squared to yield a real number solution. Then, the probability of an electron being within a certain area can be assessed.

However, a measurement of x must yield a value lying between −∞ and +∞, because the particle has to be located somewhere. It follows that Px∈−∞:∞=1, or ∫∞−∞|ψ(x,t)|2dx=1, which is generally known as the normalization condition for the wavefunction.

The square of the modulus of the wave function gives the probability density of finding the particle somewhere in space. To fulfill this requirement, the wave function is normalized in such a way that the total probability in the whole space of our consideration is 1 (maximum).

Essentially, normalizing the wave function means you find the exact form of that ensure the probability that the particle is found somewhere in space is equal to 1 (that is, it will be found somewhere); this generally means solving for some constant, subject to the above constraint that the probability is equal to 1.

The probability of finding the particle somewhere in all space is 1, but infinitesimal in any finite interval. As you found, to normalize this solution, the amplitude must be infinitesimal. If you combine two components, the probability distribution is not uniform. The wave functions are waves with phases.

A Free Particle. A free particle is not subjected to any forces, its potential energy is constant. Set U(r,t) = 0, since the origin of the potential energy may be chosen arbitrarily.