QUESTION:
Suppose the speed of electrons present in the first shell of an atom is 50% of the speed of light. If the uncertainty in velocity is 1000\rm \ m/s, what is the uncertainty in the position of this electron?
SOLUTION:
Let’s suppose:
m_e is the mass of the electron, and
V is the speed of the electron.
Given:
V = \dfrac{c}{2}, and
\Delta V = 1000 \ m/s.
To find:
\Delta x
We know the value:
h = 6.626 \times 10^{ -34}\rm \ J \cdot s, and
Rest mass of electron, m_e = 9.11\times 10^{-31}\rm \ kg.
Here the speed of the electron is comparable to the speed of light so the mass of the electron at this speed will be quite more than the rest mass of the electron. So, let’s calculate it.
The relativistic mass of the electron when its speed is 50% of the speed of light is given by
\begin{aligned}m &= m_e \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}\\&= (9.11\times 10^{-31}\rm \ kg) \times \dfrac{1}{\sqrt{1-\frac{1}{4}}}\\&= 12.147 \times 10^{-31}\rm \ kg
\\\end{aligned}
The uncertainty in the momentum is;
\begin{aligned}\Delta P &= m\Delta V\\&= (12.147 \times 10^{-31}\rm \ kg )\times (1000 \ m/s )\\&= 12.147 \times 10^{-27}\rm \ kg \cdot \ m/s\\\end{aligned}
From the Heisenberg uncertainty principle, the uncertainty in position and the uncertainty in momentum are related by an equation;
\begin{aligned}\Delta x \times \Delta P &\geq \dfrac{h}{4\pi}\\\Rightarrow \Delta x &\geq \dfrac{h}{4\pi\Delta P}\\ &= \dfrac{6.626 \times 10^{ -34}\rm \ : J \cdot s}{4\pi \times (12.147 \times 10^{-27}\rm \ kg \cdot \ m/s)}\\&= 4.34 \times 10^{ -9} \rm \ m\\\end{aligned}
Therefore, the minimum uncertainty in the particle’s position is
\begin{aligned}\Delta x &= 4.34 \times 10^{ -9} \rm \ m \\\end{aligned}
Similar Problems based on Heisenberg’s Uncertainty Principle
The location of a particle is measured with an uncertainty of 2.145\rm \ nm . a) What will be the resulting minimum uncertainty in the particle’s momentum? b) If the mass of the particle is 4.734 \times 10^{ -27}\rm \ kg then what will be the minimum uncertainty in the velocity measurement?
The uncertainty in an electron’s position is 10.0\rm \ pm. a) Find the minimum uncertainty in its momentum. b) Find the minimum uncertainty in its velocity. c) What will be the kinetic energy of the electron if the momentum is equal to the uncertainty in momentum?
A particle’s energy is measured with an uncertainty of 1.1 \times 10^{ -3}\rm \ eV. What will be the smallest possible uncertainty in our knowledge of when the particle had the given uncertainty of energy?
a) If an electron’s position can be measured to a precision of 2.0 \times 10^{ -8} \rm \ m, what will be the uncertainty in its momentum? b) If its momentum is equal to its uncertainty then what will be the electron’s wavelength?