a) you measure the position of a bacterium cell in amicroscope accirate to about

Question

a) you measure the position of a bacterium cell in amicroscope accirate to about 1 micro - meter(x). assume acell volume fo 10 (micro meter)3 and a density of thecell of approx. 1g/cm3. estimate the assocoatedminimum velocity of the cell due to heisenberg uncertainty. convertthe result to micro-meter/second. Will the cell disappeatfrom under your microscope within a second because of thisvelocity? hint : you need to callculate the mass of the cell,too. Pay attention to the units! b) suppose we had an "X-ray microscope" that we could ise tomeasure the position of an electron in an atomic orbit to within0.1 Angstrom(1 angstrom = 10-10m is roughly an atomicdismeter). what is the minimun velocity of the electron causedby locationg it to within 0.1 Angstrom, due to heisenberg'suncertainty? c) the chatacteristic time for an electron orbit in an atom is1attosecond (10-15s). Express the velocity of part b)not in m/s but in atomic diameters per attosecond. You will noticethat the result id huge compared to what we are teying tomeasure.

Explanation / Answer

Given x = 1 micro metre = 1 * 10^-6 m Cell Volume , V = 10 (micro meter)3                         = 10 ( 10 ^ -6 ) ^3 m^3                         = 10 * 10 ^ -18 m^3                         = 10 ^ -17 m ^3                         = 10 ^ -17 * ( 10 ^ 2 ) ^3 cm ^3                         = 10 ^ -11 cm^3 density of the cell ,d = 1g/cm3 We know density , d = mass / Volume So, mass of the cell ,m = d * V                                   = 1 * 10^-11 g                                   = 1 * 10^ -14 Kg moment , p = m* v v is the velocity of the cell Also x * p = h / 4         x * mv =h / 4                   v = h / 4mx                      = 6.625 * 10^-34 / ( 4 * 10 ^ -14 * 10^-6)                      = 0.5272 * 10 *-14 m / s                      = 0.5272 * 10^-14 * 10^-6 * 10^6 m/s                      = 0.5272 * 10^-8 micro metre / s (b)   x * p = h / 4         x * mv =h / 4                   v = h / 4mx                      = 6.625 * 10^-34 / ( 4 * 9.1 * 10 ^ -31 * 0.1 *10^-10)                      = 0.57934 * 10 ^ 7 m / s We know density , d = mass / Volume So, mass of the cell ,m = d * V                                   = 1 * 10^-11 g                                   = 1 * 10^ -14 Kg moment , p = m* v v is the velocity of the cell Also x * p = h / 4         x * mv =h / 4                   v = h / 4mx                      = 6.625 * 10^-34 / ( 4 * 10 ^ -14 * 10^-6)                      = 0.5272 * 10 *-14 m / s                      = 0.5272 * 10^-14 * 10^-6 * 10^6 m/s                      = 0.5272 * 10^-8 micro metre / s (b)   x * p = h / 4         x * mv =h / 4                   v = h / 4mx                      = 6.625 * 10^-34 / ( 4 * 9.1 * 10 ^ -31 * 0.1 *10^-10)                      = 0.57934 * 10 ^ 7 m / s         x * mv =h / 4                   v = h / 4mx                      = 6.625 * 10^-34 / ( 4 * 9.1 * 10 ^ -31 * 0.1 *10^-10)                      = 0.57934 * 10 ^ 7 m / s

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