### Challenge Problems

An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the emitted photon when the electron makes a transition from the first excited state to the ground state and from the second excited state to the first excited state.

In STM, an elevation of the tip above the surface being scanned can be determined with a great precision, because the tunneling-electron current between surface atoms and the atoms of the tip is extremely sensitive to the variation of the separation gap between them from point to point along the surface. Assuming that the tunneling-electron current is in direct proportion to the tunneling probability and that the tunneling probability is to a good approximation expressed by the exponential function ${e}^{\mathrm{\xe2\u02c6\u20192}\mathrm{\xce\xb2}\phantom{\rule{0.2em}{0ex}}L}$ with $\mathrm{\xce\xb2}=10.0\text{/nm}$, determine the ratio of the tunneling current when the tip is 0.500 nm above the surface to the current when the tip is 0.515 nm above the surface.

If STM is to detect surface features with local heights of about 0.00200 nm, what percent change in tunneling-electron current must the STM electronics be able to detect? Assume that the tunneling-electron current has characteristics given in the preceding problem.

Use Heisenbergâ€™s uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, $\mathrm{\xcf\u2030}=\sqrt{k\text{/}m}$, where *k* is the spring constant and *m* is the mass.

Suppose an infinite square well extends from $\text{\xe2\u02c6\u2019}L\text{/}2$ to $+L\text{/}2$. Solve the time-independent SchrÓ§dingerâ€™s equation to find the allowed energies and stationary states of a particle with mass *m* that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation $x\text{\xe2\u20ac\xb2}=x\xe2\u02c6\u2019L\text{/}2$ for the solutions obtained for the well extending between 0 and *L*.

A particle of mass *m* confined to a box of width *L* is in its first excited state ${\mathrm{\xcf\u02c6}}_{2}(x)$. (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to be found?