Chapter 4: Waves and Particles II: Matter Behaving as Waves

(a) Experiment X is carried out nine times identically, and the value 5is obtained all nine times. Calculate the mean by definition (4-12). Then the standard deviation by definition (4-13). (b) Experiment Yis carried out nine times identically, and the integers 1 through 9 are each obtained once. Repeat the calculations of part (a) for this experiment. (c) For nine repetitions of the experimentZ.The tally is that 1, 5, and 9 are each obtained three times. Repeat the calculations. (d) Explain any differences between the results in parts (b) and (c). Is standard deviation a reasonable measure of spread?

Verify the claim made in a section 4.4 that if all results of a repeated experiment are equal, the standard deviation, equation (4.13) Will be0.

In Figure $\mathbf{4}\mathbf{.}\mathbf{6}\mathbf{.}$Rays are shown scattering off atoms that are lined up in columns, with the atoms in one atomic plane exactly above those in the plane below. Actually, the atoms in different planes are usually not aligned this way, they might, for instance, be aligned as in the accompanying figure. Does this affect the validity of the Bragg law $\mathbf{(}\mathbf{4}\mathbf{-}\mathbf{1}\mathbf{)}$? Explain your answer.

A visual inspection of an ant of mass 0.5 mg verifies that it to within an uncertainty of $\mathbf{0}\mathbf{.}\mathbf{7}\mathbf{}\mathit{\mu }\mathit{m}$of a given point, apparently stationary. How fast might the ant actually be moving?

The uncertainty in the position of a baseball of mass$\mathbf{0}\mathbf{.}\mathbf{145}\mathbf{}\mathit{k}\mathit{g}\mathbf{}\mathit{i}\mathit{s}\mathbf{}\mathit{\mu }\mathit{m}$.What is the minimum uncertainty in its speed?

A mosquito of mass $\mathbf{\text{0.15mg}}$is found to be flying at a speed of $\mathbf{\text{50cm/s}}$within an uncertainty of $\mathbf{\text{0.5mm/s}}$.(a) How precisely may its position be known? (b) Does this inherent uncertainty present any hindrance to the application of classical mechanics?

The position of a neutron in a nucleus is known within an uncertainty of -5x10^-13 m. At what speeds might we expect to find it moving?

To how small a region must an electron be confined for borderline relativistic speeds say$\mathbf{\text{0.05}}$to become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near${\mathbf{\text{10}}}^{\mathbf{\text{-10}}}\mathbf{\text{m}}$? For a lead atom "inner-shell" electron of orbit radius${\mathbf{\text{10}}}^{\mathbf{\text{-12}}}\mathbf{\text{m}}$?

Aman walks at$\mathbf{1}\mathit{m}\mathbf{/}\mathit{s}$, known to within an uncertainty (unrealistically small) of$\mathbf{\text{1}}\mathbf{}\mathit{\mu }\mathbf{\text{m/h}}$.

(a) Compare the minimum uncertainty in his position to his actual physical dimension in his direction of motion $\mathbf{25}\mathit{c}\mathit{m}$,from front to back.

(b) Is it sensible to apply the uncertainty principle to the man?

One of the cornerstones of quantum mechanics is that bound particles cannot be stationary-even at zero absolute temperature! A "bound" particle is one that is confined in some finite region of space. as is an atom in a solid. There is a nonzero lower limit on the kinetic energy of such a particle. Suppose minimum kinetic energy of width $\mathbf{L}$. Obtain an approximate formula for its minimum kinetic energy.