Question

Calculate the average linear momentum of a particle described by the following wavefunctions: (a) eikx, (b) cos kx, (c) e−ax2 , where in each one x ranges from −[infinity] to +[infinity].

Answer 1

Answer:

a) p=0, b) p=0, c) p= ∞

Explanation:

In quantum mechanics the moment operator is given by

              p = - i h’  d φ / dx

             h’= h / 2π

We apply this equation to the given wave functions

a)  φ = $e^{ikx}$

        .d φ dx = i k $e^{ikx}$

We replace

        p = h’ k $e^{ikx}$

        i i = -1

The exponential is a sine and cosine function, so its measured value is zero, so the average moment is zero

            p = 0

b) φ = cos kx

           p = h’ k sen kx

The average sine function is zero,

          p = 0

c) φ = $e^{-ax^{2} }$

         d φ / dx = -a 2x  $e^{-ax^{2} }$

         .p = i a g ’2x  $e^{-ax^{2} }$

       The average moment is

         p = (p₂ + p₁) / 2

         p = i a h ’(-∞ + ∞)

         p = ∞