While undergoing a transition from the $n=1$ to the $n=2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $\mu$m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n=2$ to the $n=3$ energy level and (b) from the $n=1$ to the $n=3$ energy level?
(c) What is the value of $\sqrt{\left(k^{\prime }/m\right)}$, the angular oscillation frequency of the corresponding Newtonian oscillator?

(a) An electron with initial kinetic energy $32$ eV encoun­ters a square barrier with height $41$ eV and width $0.25$ nm. What is the probability that the electron will tunnel through the barrier?

(b) A proton with the same kinetic energy encounters the same barrier. What is the probability that the proton will tunnel through the barrier?

An electron with initial kinetic energy $6.0$ eV encounters a barrier with height $11.0$ eV. What is the probability of tunneling if the width of the barrier is (a) $0.80$ nm and (b) $0.40$ nm?

An electron is in a box of width 3.0*10^-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the n = 1 level; (b) the n = 2 level; (c) the n = 3 level? In each case how does the wavelength compare to the width of the box?

For the ground level of a harmonic oscillator, $\text{∆}x\text{∆}{p}_x=\text{ħ}/2$. Do a similar analysis for an excited level that has quantum number $$$n$. How does the uncer­tainty product $\text{∆}x\text{∆}{p}_x$ depend on $n$?

A free particle moving in one dimension has wave function ψ(x,t) = A[e^i(kx-ωt) -e^i(2kx-4ωt)] where k and v are positive real constants. (c) Calculate v_av as the distance the maxima have moved divided by the elapsed time.

An electron with initial kinetic energy $5.0$ eV encoun­ters a barrier with height $U_0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U_0=7.0$ eV; (b) $U_0=9.0$ eV; (c) $U_0=13.0$ eV?