7.6 The quantum tunneling of particles through potential barriers  (Page 11/22)

The mass of a $\rho$ -meson is measured to be $770\text{MeV}\text{/}{c}^2$ with an uncertainty of $100\text{MeV}\text{/}{c}^2$ . Estimate the lifetime of this meson.

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) $x=0.25L$ ; (b) $x=0.40L$ ; (c) $x=0.75L$ ; and (d) $x=0.90L$ .

A particle in a box [0; L ] is in the third excited state. What are its most probable positions?

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 with the kinetic energy of the ball moving at 2.0 m/s.

Find the expectation value of the position squared when the particle in the box is in its third excited state and the length of the box is L .

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

Consider an infinite square well with wall boundaries $x=0$ and $x=L.$ Explain why the function $\psi (x)=A\text{cos}kx$ is not a solution to the stationary Schrӧdinger equation for the particle in a box.

Atoms in a crystal lattice vibrate in simple harmonic motion. Assuming a lattice atom has a mass of $9.4\times {10}^{\mathrm{-26}}\text{kg}$ , 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?

A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass $5.60\times {10}^{\mathrm{-26}}\text{kg}$ . (a) What is the wavelength of the emitted photon when the molecule makes the transition from the third excited state to the second excited state? (b) Find the ground state energy of vibrations for this diatomic molecule.

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?

A beam of mono-energetic protons with energy 2.0 MeV falls on a potential energy barrier of height 20.0 MeV and of width 1.5 fm. What percentage of the beam is transmitted through 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.

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 $e^{\mathrm{-2}\beta L}$ with $\beta =10.0\text{/nm}$ , 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 Heisenberg’s uncertainty principle to estimate the ground state energy of a particle oscillating on an spring with angular frequency, $\omega =\sqrt{k\text{/}m}$ , where k is the spring constant and m is the mass.

Suppose an infinite square well extends from $\text{−}L\text{/}2$ to $+L\text{/}2$ . 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 $x\text{′}=x-L\text{/}2$ 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 ${\psi }_2(x)$ . (a) Find its average position (which is the expectation value of the position). (b) Where is the particle most likely to be found?