Question

An electron confined on one-dimensional infinite potential well has an energy of 180ev. What its wavelength?

Answer 1

The wavelength is 91.5 pm ( 91.5 Pico meter).

Explanation:

The formula can be expressed below for electron’s energy,

            $\text {Energy of electron}=\frac{p^{2}}{2 m}$

Where,

p = momentum

m= mass of electron

We know, mass of electron =  $9.1 \times 10^{-31} \mathrm{kg}$

Energy of electron, $1 e V=1.6 \times 10^{-19} \mathrm{J}$

Therefore, $\text { energy of electron, 180 eV }=180 \times 1.6 \times 10^{-19} J$

By substituting the known values in the equation, we get,

          $180 \times 1.6 \times 10^{-19}=\frac{p^{2}}{2 \times 9.1 \times 10^{-31}}$

          $p^{2}=180 \times 1.6 \times 10^{-19} \times 2 \times 9.1 \times 10^{-31}$

          $p^{2}=5241.6 \times 10^{-50}$

Taking square root, we get

          $\text {Momentum, } p=72.399 \times 10^{-25} \mathrm{kg} . \mathrm{m} / \mathrm{s}$

We know,

               $\lambda=\frac{h}{p}$

Here, h – Planck constant = $6.626 \times 10^{-34} \mathrm{J.s}$

So, the wavelength would be,

              $\lambda=\frac{6.626 \times 10^{-34}}{72.399 \times 10^{-25}}=0.0915 \times 10^{-34+25}=0.0915 \times 10^{-9} \mathrm{m}$

Adding $10^{-3}$  in both numerator and denominator we get the value as

              $\lambda=0.0915 \times 10^{-9} \times \frac{10^{-3}}{10^{-3}}=0.0915 \times 10^{3} \times 10^{-12}=91.5 \mathrm{pm}$

Where, pm – Pico meter - $10^{-12}$