Question #277d6

1 Answer
Jan 31, 2018

Heisenberg Uncertainty Principle states that the position and the velocity of a particle cannot be measured with accuracy and this accuracy decreases as the size of the particle decreases

A formal definition of the Heisenberg Uncertainty Principle states that the product of the position and momentum uncertainties #sigma_vecx# and #sigma_(vecp_x)# is no less than #ℏ//2#, where #h# is Planck's constant, #6.626xx10^(-34) "J"cdot"s"#, and #ℏ = h//2pi# is the reduced Planck's constant.

That is,

#\mathbf(sigma_(vecx)sigma_(vecp_x) >= ℏ//2)#

(By the way, this happens to hold for any direction, not just the #x# direction, so this extends to our three dimensions.)

Mathematically, this is expressed using the following 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

The uncertainty in momentum will depend on the mass of the particle, #m#, and on its uncertainty in velocity, #Deltav#

#color(blue)(ul(color(black)(Deltap = m * Deltav)))#

But first we'll calculate

#p = mv#

#m = 9.11*10^(-31)kg#
#v = 2.2xx10^6m"/"s#

#p = 9.11*10^(-31)kg xx 2.2xx10^6m"/"s#

#p = 2.0042xx10^-24#

But since we have been given that the electron velocity is known to be within 10% of 2.2 X 10^6 m/s

#Deltap = 10/100p#

#Deltap = 2.0042xx10^-24 * 0.1 = 2.0042xx10^-25#

#Deltax>=( 6.626*10^(-34)"kg" *" m"^2"/"s)/((4pi)*2.0042xx10^-25(kg*m)/s)#

#Deltax >= 2.63xx10^(-10)"m"#

Answer B

#2.63xx10^(-10)m >= "Diameter of H"#

The uncertainty is higher as the speed of the electron is less in a hydrogen nucleus than the give situation.

The proton also plays a major role in decreasing the uncertainty and this uncertainty is equal to the diameter of the hydrogen