29.7 Probability: the heisenberg uncertainty principle  (Page 5/11)

What is the Heisenberg uncertainty principle? Does it place limits on what can be known?

(a) If the position of an electron in a membrane is measured to an accuracy of $1\text{.}\text{00 μm}$ , what is the electron’s minimum uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV? (c) What are the implications of this energy, comparing it to typical molecular binding energies?

(a) If the position of a chlorine ion in a membrane is measured to an accuracy of $1\text{.}\text{00 μm}$ , what is its minimum uncertainty in velocity, given its mass is $5\text{.}\text{86}\times {\text{10}}^{-\text{26}}\text{kg}$ ? (b) If the ion has this velocity, what is its kinetic energy in eV, and how does this compare with typical molecular binding energies?

Suppose the velocity of an electron in an atom is known to an accuracy of $2\text{.}0\times {\text{10}}^3\text{m/s}$ (reasonably accurate compared with orbital velocities). What is the electron’s minimum uncertainty in position, and how does this compare with the approximate 0.1-nm size of the atom?

The velocity of a proton in an accelerator is known to an accuracy of 0.250% of the speed of light. (This could be small compared with its velocity.) What is the smallest possible uncertainty in its position?

A relatively long-lived excited state of an atom has a lifetime of 3.00 ms. What is the minimum uncertainty in its energy?

(a) The lifetime of a highly unstable nucleus is ${\text{10}}^{-\text{20}}\text{s}$ . What is the smallest uncertainty in its decay energy? (b) Compare this with the rest energy of an electron.

The decay energy of a short-lived particle has an uncertainty of 1.0 MeV due to its short lifetime. What is the smallest lifetime it can have?

The decay energy of a short-lived nuclear excited state has an uncertainty of 2.0 eV due to its short lifetime. What is the smallest lifetime it can have?

What is the approximate uncertainty in the mass of a muon, as determined from its decay lifetime?

Derive the approximate form of Heisenberg’s uncertainty principle for energy and time, $\mathrm{\Delta }E\mathrm{\Delta }t\approx h$ , using the following arguments: Since the position of a particle is uncertain by $\mathrm{\Delta }x\approx \lambda$ , where $\lambda$ is the wavelength of the photon used to examine it, there is an uncertainty in the time the photon takes to traverse $\mathrm{\Delta }x$ . Furthermore, the photon has an energy related to its wavelength, and it can transfer some or all of this energy to the object being examined. Thus the uncertainty in the energy of the object is also related to $\lambda$ . Find $\mathrm{\Delta }t$ and $\mathrm{\Delta }E$ ; then multiply them to give the approximate uncertainty principle.