Question

Find the initial momentum of the electron if the momentum of the electron is changed by ${P_m}$ and the de Broglie wavelength associated with it changes by $0.50\% $ .

Answer 1

Hint: Here we have to apply the de Broglie formula to get the answer.
As per wave-particle duality, the De Broglie wavelength is a wavelength shown in all the items in quantum mechanics which decides the probability of finding the article at a given point. The wavelength of the de Broglie particle is not directly proportional to its momentum.

Complete step by step answer:
On account of electrons going around and around the nuclei in atoms, the de Broglie waves exist as a closed loop, with the end goal that they can exist just as standing waves, and fit uniformly around the loop. Due to this necessity, the electrons in atoms circle the centre in specific arrangements, or states, which are called stationary orbits.
De Broglie contemplated that matter additionally can show wave-particle duality, much the same as light, since light has properties both of a wave (it tends to be diffracted and it has a wavelength) and of a particle (it contains energy $h\nu $ ). And furthermore contemplated that matter would follow a similar condition for wavelength as light in particular,$\lambda = \dfrac{h}{P}$

Where
$\lambda $ is the wavelength
$h$ is the Planck’s constant
$P$ is the linear momentum
According to question,
$
  P - {P_m} = \dfrac{h}
{{\lambda + \dfrac{{0.5}}
{{100}}\lambda }} \\
 \Rightarrow P - {P_m} = \dfrac{P}
{{\dfrac{{1005}}
{{1000}}}} \\
 \Rightarrow P = \dfrac{{1005{P_m}}}
{5} \\
 \Rightarrow P = 200{P_m} \\
$
Additional information: Linear momentum: Linear momentum is characterized as the result of a system's mass times its velocity. Linear momentum is mathematically expressed as $P = mv$ . Momentum is directly proportional to the object's mass and furthermore its velocity. Accordingly the higher an object's mass or the higher is its velocity, the greater is the momentum.


Note: We have to pay attention as to what is asked in the question. In confusion we may subtract $P$ from ${P_m}$ which is wrong as it is mentioned in the question that the momentum of the electron is changed by ${P_m}$ .