7.6 The Quantum Tunneling of Particles through Potential Barriers

A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?

What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?

Can a quantum particle ‘escape’ from an infinite potential well like that in a box? Why? Why not?

Show that the wave function in (a) the equation satisfies the equation, and (b) the equation satisfies the equation.

A 5.0-eV electron impacts on a barrier of with 0.60 nm. Find the probability of the electron to tunnel through the barrier if the barrier height is (a) 7.0 eV; (b) 9.0 eV; and (c) 13.0 eV.

A quantum particle with initial kinetic energy 32.0 eV encounters a square barrier with height 41.0 eV and width 0.25 nm. Find probability that the particle tunnels through this barrier if the particle is (a) an electron and, (b) a proton.

A muon, a quantum particle with a mass approximately 200 times that of an electron, is incident on a potential barrier of height 10.0 eV. The kinetic energy of the impacting muon is 5.5 eV and only about 0.10% of the squared amplitude of its incoming wave function filters through the barrier. What is the barrier’s width?

Show that if the uncertainty in the position of a particle is on the order of its de Broglie’s wavelength, then the uncertainty in its momentum is on the order of the value of its momentum.

A particle of mass m is confined to a box of width L. If the particle is in the first excited state, what are the probabilities of finding the particle in a region of width 0.020 L around the given point x: (a) $x = 0.25 L$; (b) $x = 0.40 L$; (c) $x = 0.75 L$; and (d) $x = 0.90 L$.

A 0.20-kg billiard ball bounces back and forth without losing its energy between the cushions of a 1.5 m long table. (a) If the ball is in its ground state, how many years does it need to get from one cushion to the other? You may compare this time interval to the age of the universe. (b) How much energy is required to make the ball go from its ground state to its first excited state? Compare it w…

Consider an infinite square well with wall boundaries $x = 0$ and $x = L .$ Show that the function $\psi \left(x\right) = A \text{sin} k x$ is the solution to the stationary Schrӧdinger equation for the particle in a box only if $k = \sqrt{2 m E} / \hbar .$ Explain why this is an acceptable wave function only if k is an integer multiple of $\pi / L .$

Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of $9.4 \times 10^{−26} \text{kg}$, what is the force constant of the lattice if a lattice atom makes a transition from the ground state to first excited state when it absorbs a $\text{525}- \mu \text{m}$ photon?

An electron with kinetic energy 2.0 MeV encounters a potential energy barrier of height 16.0 MeV and width 2.00 nm. What is the probability that the electron emerges on the other side of the barrier?

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.

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.

Suppose an infinite square well extends from $− L / 2$ to $+ L / 2$. Solve the time-independent Schrӧdinger’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 $x ′ = x - L / 2$ for the solutions obtained for the well extending between 0 and L.