Ch 6, Mastering Tutorial 6c (6.4) questions

[Zellmer, Robert]

I got the following question from a student concerning question #2 from
Tutorial 6e (section 6.4) of the Mastering.  It deals with the Heisenberg
Uncertainty Principle.

A student is examining a bacterium under the microscope. The E. coli bacterial
cell has a mass of m = 0.400 fg (where a femtogram, fg,  is 10−15g) and is
swimming at a velocity of v = 8.00 μm/s , with an uncertainty in the velocity
of 3.00 % . E. coli bacterial cells are around 1 μm ( 10−6 m) in length. The
student is supposed to observe the bacterium and make a drawing. However,
the student, having just learned about the Heisenberg uncertainty principle in
physics class, complains that she cannot make the drawing. She claims that the
uncertainty of the bacterium's position is greater than the microscope's viewing
field, and the bacterium is thus impossible to locate.

What is the uncertainty of the position of the bacterium?

This is Heisenberg Uncertainty Principle:

delta(x) * delta(p) > or = h/4*pi

h = Planck’s constant
delta(x) = uncertainty in position, x
delta(p) = uncertainty in momentum, p

p = m*v           m = mass        v = velocity
since there’s no uncertainty in the mass we can write this as:

delta(x) * m* delta(v) > or = h/4*pi

delta(v) = uncertainty in velocity, v

Make sure you use SI units for everything,
kg for mass
meters for distances

For this particular problem:

m = 0.400 fg (0.400 femtograms) = 0.400 x 10^-15 g = 0.400 x 10^-18 kg

the uncertainty in the velocity is 3.00 %.  The velocity is 8.00 um/s (micrometers).

delta(v) = (0.03)(8.00 x 10^-6 m/s) = 2.4 x 10^-7 m/s

So, solving the HUP eqn for uncertainty in the position, delta(x),

                          h                                           6.626 x 10^-34 J*s
delta(x) = ----------------------  = -------------------------------------------------------------------
                 m*delta(v)*4*pi       (0.400 x 10^-18 kg)( 2.4 x 10^-7 m/s)(4)(3.14159)

delta(x) = 5.59 x 10^-10 m

E. coli cells are around 1 micrometer in length (1 x 10^-6 m).  This uncertainty
is much smaller than the size of the cell so the student is wrong, they should be
able to see the bacterium under the microscope.

Dr. Zellmer

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