Question

If uncertainties in the measurement of position and momentum of an electron are equal, calculate uncertainty in the measurement of velocity.

Answer 1

Hint: The uncertainty principle says that the position and momentum of the object cannot be measured precisely at same time duration. The uncertainty in velocity is given by the formula which says that momentum is equal to mass multiplied by the velocity.

Complete step by step answer:
It is given that the measurement of position and momentum of an electron are equal.
So, the equation is given as shown below.
\[\Delta x=\Delta p\]
Where,
$\Delta x$ is uncertainty in position
$\Delta p$ is uncertainty is momentum
The uncertainty principle says that the position of the object and velocity of the object cannot be determined at the same time.
The uncertainty principle is given by the equation as shown below.
$\Delta x\times \Delta p\ge \dfrac{h}{4\pi }$
Where,
h is the planck's constant
As both the position and momentum are equal, then the equation can be written as shown below.
$ \Rightarrow {(\Delta p)^2} \geqslant \dfrac{h}{{4\pi }}$
$ \Rightarrow \Delta p \geqslant \sqrt {\dfrac{h}{{4\pi }}} $
As we know that momentum is equal to mass multiplied by the velocity
The equation to calculate momentum is shown below.
\[p = m \times v\]
So the uncertainty in momentum is given by mass multiplied by uncertainty in velocity.
The equation to calculate uncertainty in momentum is shown below.
$\Delta p = m \times \Delta v$
Thus, the uncertainty in velocity is given as shown below.
\[\Delta v \geqslant \dfrac{1}{m}\sqrt {\dfrac{h}{{4\pi }}} \]
The value of m is $9.1 \times {10^{ - 31}}$ kg
The value of planck's constant h is $6.626 \times {10^{ - 34}}$
Substitute the values in the above equation to calculate the velocity.
$ \Rightarrow \Delta v \geqslant \dfrac{1}{{9.1 \times {{10}^{ - 31}}}}\sqrt {\dfrac{{6.626 \times {{10}^{ - 34}}}}{{4 \times 3.14}}} $
\[ \Rightarrow \Delta v = 7.98 \times {10^{12}}m/s\]
Therefore, uncertainty in the measurement of velocity is $7.98 \times {10^{12}}$ m/s

Note:
The uncertainty principle is also known as the Heisenberg uncertainty principle or indeterminacy principle. The functions which are non commutative follow this principle.