An electron is in a box of width $3.0\times {10}^{-10}$ 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?

(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?

Recall that $\left(\mid \psi {\mid }^2\right)dx$ is the probability of finding the par­ticle that has normalized wave function $\psi \left(x\right)$ in the interval $x$ to $x+dx$. Consider a particle in a box with rigid walls at $x=0$ and $x=L$. Let the particle be in the ground level and use ${\psi }_n$ as given in Eq. ($40.35$) ${\psi }_n\left(x\right)=\sqrt{\frac{2}{L}}sin\left[\right(\frac{n\pi x}{L}\left)\right]$ where $n=1,2,3,\dots$.

(a) For which values of $x$, if any, in the range from $0$ to $L$ is the probability of finding the particle zero?

(b) For which values of $x$ is the probability highest?

(c) In parts (a) and (b) are your answers consistent with Fig. $40.12$? Explain.

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box of width $0.360$ nm.

(b) The electron makes a transition from the $n=1$ to $n=4$ level by absorbing a photon. Calculate the wave­length of this photon.

An electron in a one­-dimensional box has ground ­state energy $2.00$ eV. What is the wavelength of the photon absorbed when the electron makes a transition to the second excited state?

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?