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 gotxi approximation expressed by the exponential function
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- Impurities in solids can be sometimes described by a particle-in-a-box model. Suppose He is substituted for Xe, and assume a particle-in-a-cubic-box model, the length of whose sides is equal to the atomic diameter of Xe (≈ 2.62 Å). Compute the lowest excitation energy for the He atom’s motion. (This is the energy difference between the ground state and the first excited state.)arrow_forwardAn electron is moving along x axis with the speed of 2×106 m/s (known with a precision of 0.50%). Determine the uncertainty in its position.arrow_forwardCalculate the transmission probability for quantum-mechanical tunneling in each of the following cases. (a) An electron with an energy deficit of U - E= 0.010 0 eV is incident on a square barrier of width L = 0.100 nm. (b) An electron with an energy deficit of 1.00 eV is incident on the same barrier. (c) An alpha particle (mass 6.64 × 10-27 kg) with an energy deficit of 1.00 MeV is incident on a square barrier of width 1.00 fm. (d) An 8.00-kg bowling ball withan energy deficit of 1.00 J is incident on a square barrier of width 2.00 cm.arrow_forward
- When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren’t there and thus have transmission coefficients (tunneling probabilities) T equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of T = 1 occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width L of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies E that will result in T = 1. You assume a barrier height of 10 eV and a width of 1.8 x 10-10 m. Calculate the three lowest values of E for which T = 1arrow_forwardQuantum mechanical tunnelling enables chemical reactions to proceed that would be energetically impossible in classical mechanics. Assume that hydrogen (H) and tritium (T) atoms, each with a kinetic energy of 0.9 eV, encounter a potential barrier that is 1.0 eV high and 100 pm broad. Calculate the ratio of probabilities for transmission of the H and T atoms through the barrier. Note: the masses of H and T atoms are 1.674 x 10-27 kg and 5.008 x 10-27 kg, respectively, and 1 eV=1.602x10-19 J.arrow_forwardIn scanning tunnelling microscope the tunnelling current is proportional to the transmission probability T. Suppose the gap potential energy V is greater than the electron energy E by V-E-=4.0 eV. Calculate the ratio of current when the needle is moved from L1=0.20nm to L2=0.35nm from the surface? Please enter your answer with 2 decimals.arrow_forward
- We can approximate an electron moving in a nanowire (a small, thin wire) as a one-dimensional infi nite square-well potential. Let the wire be 2.0 μm long. The nanowire is cooled to a temperature of 13 K, and we assume the electron’s average kinetic energy is that of gas molecules at this temperature ( 3kT/2). (a) What are the three lowest possible energy levels of the electrons? (b) What is the approximate quantum number of electrons moving in the wire?arrow_forwardA thin solid barrier in the xy-plane has a 12.6µm diameter circular hole. An electron traveling in the z-direction with vx 0.00m/s passes through the hole. Afterward, within what range is vx likely to be?arrow_forwardIf the position of an electron in a membrane is measured to an accuracy of 3.58 µm, what is the electron's minimum uncertainty in velocity (in m/s)? a) If the electron has this velocity, what is its kinetic energy in eV? b) What are the implications of this energy, comparing it to typical molecular binding energies?arrow_forward
- Calculate the energy density in the range 650 nm to 655 nm inside a cavity at (a) 25 °C, (b) 3000 °C. For this relatively small range of wavelength it is acceptable to approximate the integral of the energy spectral density ρ(λ,T) between λ1 and λ2 by ρ(λ,T)×(λ2 − λ1).arrow_forwardIn an electron-scattering experiment, an intense reflected beam is found at ϕ = 32° for a crystal with an interatomic distance of 0.23 nm. What is the lattice spacing of the planes responsible for the scattering? Assuming fi rst-order diffraction, what are the wavelength, momentum, kinetic energy, and total energy of the incident electrons?arrow_forwardFor a "particle in a box" of length, L, the wavelength for the nth level is given by An 2L %3D 2п and the wave function is n(x) = A sin (x) = A sin (x). The energy levels are пп %3D n?h? given by En : %3D 8mL2 lPn(x)|2 is the probability of finding the particle at position x in the box. Since the particle must be somewhere in the box, the integral of this function over the length of the box must be equal to 1. This is the normalization condition and ensuring that this is the case is called “normalizing" the wave function. Find the value of A the amplitude of the wave function, that normalizes it. Write the normalized wave function for the nth state of the particle in a box.arrow_forward
- University Physics Volume 3PhysicsISBN:9781938168185Author:William Moebs, Jeff SannyPublisher:OpenStax