Question

An electron has the following wave function Ae^-x(1-e^-x) determine the normalisation constant A and the expected value of x

Answer 1

Given : Wave function  psi (x) =  Ae^-x(1-e^-x)

To find: Determine the normalization constant A and the expected value of x

Solution:

• Now we have given the wave function, so to determine the normalization constant A, we need to integrate the wave function.

• So,

             1=∫  psi* (x) psi (x) dx,                           limits: from - infinity to + infinity

               1=∫ (A²) x (e^-2x) x  (1−e^-x ) x 2 dx               limits: from 0 to infinity

              1=A² ( 1/12 )

              A² =12

• Now, the value of x can be:

              <  A (cap)>=∫ psi∗ (x)  A (cap) psi (x)dx     limits: from - infinity to + infinity

              <x> = 12 x ∫   xe^−2x  (1−e^−x  )2 dx              limits : from 0 to infinity.

              <x> =  13 /12

Answer:

             The normalization constant A is √12 and the expected value of x is 13/12.