Question

(a) Why does one have to ensure that a function is properly normalized?

(b) Normalize the function e^-r over the interval 0≤ r ≤ inf; 0≤ θ ≤ π; 0≤ ϕ ≤2π

Answer 1

a. Wavefunction can be find out by applying schrodinger equation. Generally, wave functions are the complex function in space and time. Square of the amplitude of the function gives the probability of a particle present in a given region. Normalization of a function is somewhat related to its probability. Normalization is basically the probability of a particle present in entire space. From the theorems of probability, we know that the total probability of any event should be 1. So, we normalize the function over space to assure that the particle is present somewhere in the space.

b. Normalization is given be

where are the wave function and complex conjugate respectively.

and

Given, because is real.

Let A be the normalization constant that we need to find

Solving above integral with respect to will give . So our equation will be

Solving above equation will give

above equation will give

or

So the wavefunction will be