Question #b701d
1 Answer
Explanation:
The idea here is that the Heisenberg Uncertainty Principle tells us that we cannot measure both the position and the momentum of a particle with arbitrarily high precision.
In other words, we will always have a very high uncertainty in measuring the position of a particle if we have a very precise measurement of its momentum.
Similarly, we will always have a very high uncertainty in measuring the momentum of a particle if we have a very precise measurement of its position.
In fact, the uncertainty in position and the uncertainty in momentum must always satisfy the inequality
#color(blue)(ul(color(black)(Deltax * Deltap >= h/(4pi))))#
Here
#Deltax# is the uncertainty in position#Deltap# is the uncertainty in momentum#h# is Planck's constant, equal to#6.626 * 10^(-34)"kg m"^2"s"^(-1)#
Now, the uncertainty in momentum will depend on the mass of the particle,
#color(blue)(ul(color(black)(Deltap = m * Deltav)))#
In your case, you have
#m ~~ 9.10938 * 10^(-31)"kg"#
Plug this back into the inequality to find
#Deltax * m * Deltav >= h/(4pi)#
Rearrange to solve for
#Deltav >= h/(4pi) * 1/(Deltax * m)#
Now, notice that you have a very small uncertainty in position
#497 color(red)(cancel(color(black)("pm"))) * "1 m"/(10^(12)color(red)(cancel(color(black)("pm")))) = 4.97 * 10^(-10)# #"m"#
which means that in order for the Heisenberg Uncertainty Principle to hold, you must have a very high uncertainty in velocity.
Plug in your values to get the uncertainty in velocity
#Deltav = (6.626 * 10^(-34)color(red)(cancel(color(black)("kg"))) "m"^color(red)(cancel(color(black)(2)))"s"^(-1))/(4 * pi) * 1/(479 * 10^(-12)color(red)(cancel(color(black)("m"))) * 9.10938 * 10^(-31)color(red)(cancel(color(black)("kg"))))#
#color(darkgreen)(ul(color(black)(Deltav = 1.21 * 10^(5)color(white)(.)"m s"^(-1))))#
The answer is rounded to three sig figs.
As predicted, you have a very high uncertainty in velocity.
