Question

Andrea, whose mass is $50\text{kg}$, thinks she's sitting at rest in her 5.0-m-long dorm room as she does her physics homework. Can Andrea be sure she's at rest? If not, within what range is her velocity likely to be?

Short answer

She cannot be sure that she is at rest and her velocity is likely to be within the range of$1.3\times {10}^{-36}\text{m}/\text{s}$.

Step-by-step solution

Step.1.

Heisenberg uncertainty principle states that it is not possible to make a simultaneous determination of the position and the momentum of a particle with unlimited precision. The mathematical representation of Heisenberg uncertainty principle is as follows:

$\Delta x\Delta p_x\ge \frac{h}{2}$

Here, $\Delta x$is the uncertainty in position, $\Delta p_x$is the uncertainty in momentum, and$h$ is the Planck's constant.

The uncertainty in momentum of a particle is also given by the following relation:

$\Delta p_x=m\Delta v_x$

Here,$\Delta v_x$ is the uncertainty in the velocity of the particle.

Step.2.

Rearrange the equation $\Delta x\Delta p_x\ge \frac{h}{2}$for $\Delta p_x$.

$\Delta p_x\ge \frac{h}{2\Delta x}$

Substitute $m\Delta v_x$for $\Delta p_x$in the above equation and rearrange the equation for $\Delta v_x$.

$\begin{array}{l}m\Delta v_x\ge \frac{h}{2\Delta x}\\ \Delta v_x\ge \frac{h}{2m\Delta x}\end{array}$

Substitute $6.626\times {10}^{-34}\text{J}\cdot \text{s}$for $h,50\text{kg}$for $m$and $5.0\text{m}$for $\Delta x$in the above equation and calculate the range of velocity.

$\begin{array}{l}\Delta v_x=\frac{h}{2m\Delta x}\\ =\frac{\left(6.626\times {10}^{-34}\text{J}\cdot \text{s}\right)}{2(50\text{kg})(5.0\text{m})}\\ =1.3\times {10}^{-36}\text{m}/\text{s}\end{array}$

Hence, she cannot be sure that she is at rest and her velocity is likely to be within the range of $1.3\times {10}^{-36}\text{m}/\text{s}$.