Normalization

Step 1: The given orthonormal basis set is: For the functions to be orthonormal, the third function needs to be normalized. We focus on the third function: To normalize this function, we compute the following integral: Step 2: Expand the integrand: Now expand : So the integral becomes: Step 3: Compute the individual integrals: Substitute these values into the integral: Step 4: Now, substitute this value into the normalization equation: Thus, the value of is .

Step 1: The given orthonormal basis set is: For the functions to be orthonormal, the third function needs to be normalized. We focus on the third function: To normalize this function, we compute the following integral: Step 2: Expand the integrand: Now expand : So the integral becomes: Step 3: Compute the individual integrals: Substitute these values into the integral: Step 4: Now, substitute this value into the normalization equation: Thus, the value of is .