Question #c0def

Start by writing down the equation for the de Broglie wavelength

#color(blue)(ul(color(black)(lamda_ "matter" = h/(m * v))))#

Here

• #lamda_ "matter"# is its de Broglie wavelength
• #h# is Planck's constant, equal to #6.626 * 10^(-34)"J s"#
• #m# is the mass of the particle
• #v# is its velocity

Now, the Heisenberg Uncertainty Principle states that it's impossible for us to measure both the position and the momentum of a particle with arbitrarily high precision.

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)))#

This means that you can rewrite the inequality that describes Heisenberg's Uncertainty Principle as

#Deltax * m* Deltav >= h/(4pi)" "color(darkorange)("(*)")#

Now, the problem tells you that this particle is moving with a velocity of #4 * 10^5# #"m s"^(-1)# and that its uncertainty in position is equal to its de Broglie wavelength.

This means that you have

#Deltax = lamda_"matter"#

or

#Deltax = h/(m * v)#

Plug this into #color(darkorange)("(*)")# to get

#color(red)(cancel(color(black)(h)))/(color(red)(cancel(color(black)(m))) * v) * color(red)(cancel(color(black)(m))) * Deltav >= color(red)(cancel(color(black)(h)))/(4pi)#

Since # v > 0#, you can multiply both sides by #v# without having to change the sign of the inequality, so

#Deltav >= v/(4pi)#

Plug in the value you have for the velocity of the particle to get

#Deltav >= (color(red)(cancel(color(black)(4))) * 10^5color(white)(.)"m s"^(-1))/(color(red)(cancel(color(black)(4))) * pi)#

#color(darkgreen)(ul(color(black)(Deltav >= 3.2 * 10^4color(white)(.)"m s"^(-1))))#

The answer is rounded to two sig figs, the number of sig figs you have for the velocity of the particle.