/designe mal forces on each other, as well as frictional forces that resist their slipping relative to contact forces arise from a complex interplay between the electrostatic forces between lons in the objects and the laws of quantum mechanics. As two surfaces are pushed ces increase exponentially over an atomic distance scale, easily becoming strong the buik material in the objects ifthey approach too close. In everyday experience, limited by the deformation or acceleration of the objects, rather than by the atomic forces. Hence, we can conclude the following contact forces is determined by EF = mä, that is, by the other forces on, and e contacting bodies. The only exception is that the trictional forces cannot exceed jun n be smaler than this or even zero) and f paralel to the plane of contact. Kinetic friction when surfaces slide When one surface is siding past the other experiments show three things about the friction force (denoted f a 1. The frictional force opposes the relative motion at the point of contact. 2 f is proportional to the nomal force, and 3. the ratio of the magnitude of the frictional force to that of the normal force is fairty constant over a wide range of speeds The constant of proportionality is called the coefficient of kinetic triction, often designated a As long as the sliding continues, the frictional force is then = Pan (valid when the surfaces side by each other). Static friction when surfaces don't slide When there is no relative motion of the surfaces, the trictional force can assume any value from zero up to a maximum n where is the coeficient of static triction. Invariably. is larger than a in agreen is large enough that something breaks loose and starts to slide, it offen accelerates. The frictional force for surfaces with no relative motion is therefore f

НА • Two copper nanowires are insulated by a copper oxide nano-layer that provides a 10.0- eV potential barrier. Estimate the tunnelling probability between the nanowires by 7.00-eV electrons through a 5.00-nm thick oxide layer. What if the thickness of the layer were reduced to just 1.00 nm? What if the energy of electrons were increased to 9.00 eV? ess Page T 17 System properties 6. EN - ) I hp

124. For a 1-kg ball in a l-m² square two-dimensional box (i.e., L = 1 m), what is the lowest quantum energy allowed? If the ball had this kinetic energy, how long would it take to travel from one side of the box to the other? Is this time something that you think you could measure?

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1. Consider a quantum system in state |) = ¢1) +l¢2)+l¢3) expressed in terms of the orthonormal eigenvectors {|Øn}} of operator, Q, where its spectrum of allowed measured values is given by Qn = n². If many different experiments were performed, what is the expected average outcome of quantity, Q?

Wave Interference 22. A 2-slit experiment involves electrons that are accelerated with bias equal to 1,050 V, with a slit separation of 2 nm and slit-screen separation of 15 cm. (a) Calculate the kinetic energy of the electrons in units of eV. (b) Calculate the velocity of the electrons in m/s. (c) Calculate the wavelength of the electrons in nm. (d) Calculate the distance between the fringes in cm.