Question

The ground state energy of an electron in an infinite well is 5.6 meV. If the
width of the well is doubled, calculate the zero point energy.

Answer 1

Calculate the velocity and kinetic energy of an electron of wavelength 1.66 × 10 –10 m.

Sol: Wavelength of an electron (λ) = 1.66 × 10–10 m

lambda = h/ mv

put the value and u will fing velocity

then put the value of velocity in 1/ mv(square )

then u get kinetic energy .

Answer 2

The Zero point energy of the given electron on doubling the width is 1.4 meV or $2.243 * 10^{-22}$ Joules.

Given :

Ground state energy of electron = 5.6 meV

The width of the infinite well is doubled

To Find :

The zero-point energy of the electron

Solution :

The formula for the ground state energy of an electron with plank's constant "h" and the width of the infinite well "a" is given by :

                  $E = h^{2} / (8m*a^{2})$

Thus, the Zero point energy i.e. E is inversely proportional to the square of the width of the infinite well.

                      $E$ ∝ $a^{-2}$

So, if the width of the well is doubled, the zero point energy decreases by a factor of 4.

The Zero-point energy thus is :

                  $a_{new} = 2*a$

                  $E_{new}= h^{2} / (8m*a_{new} ^{2})$

                            $= h^{2} / (8m*(2*a) ^{2})$

                           $= (1/4) * h^{2} / (8m*a ^{2})$

                           $= E/4$

                           $= 5.6 meV/4$

                           $= 1.4 meV$

Converting zero-point energy to joules from meV.

             $1.4 meV = 1.4 * 10^{-3} eV$

                          $=(1.4 * 10^{-3}) * (1.602 * 10^{-19}) J$

                          $= (1.6022 * 1.4) * 10^{-3-19} J$

                          $= 2.243 * 10^{-22}J$

Hence, the zero point energy on doubling the width is $2.243 * 10^{-22}J$.

 

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