QUESTION:
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?
SOLUTION:
Let’s suppose:
\Delta E is the uncertainty in energy, and
\Delta t is the uncertainty in time.
Given:
\Delta E = 1.1 \times 10^{ -3}\rm \ eV
To find:
- \Delta t
We know the value:
h = 4.13567 \times 10^{ -15}\rm \ eV\cdot s
From the Heisenberg uncertainty principle, the uncertainty in energy and the uncertainty in time are related by an equation;
\begin{aligned}\Delta E \times \Delta t &\geq \dfrac{h}{4\pi}\\\Rightarrow \Delta t &\geq \dfrac{h}{4\pi\Delta E} \\ &= \dfrac{4.13567 \times 10^{ -15}\rm \ eV\cdot s}{4\pi \times (1.1 \times 10^{ -3}\rm \ eV)} \\&= 2.992 \times 10^{ -13}\rm \ s\\\end{aligned}
Therefore, the smallest possible uncertainty in time is
\begin{aligned}\Delta t &= 2.992 \times 10^{ -13}\rm \ s \\\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) 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?
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?