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Physics for Scientists and Engineers, 6e. Chapter 41 - Quantum Mechanics. Consider the wave function for the free particle, Equation 41.3. At what value of x is the particle most likely to be found at a given time?. at x = 0 at small nonzero values of x at large values of x

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## Physics for Scientists and Engineers, 6e

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**Physics for Scientists and Engineers, 6e**Chapter 41 - Quantum Mechanics**Consider the wave function for the free particle, Equation**41.3. At what value of x is the particle most likely to be found at a given time? • at x = 0 • at small nonzero values of x • at large values of x • anywhere along the x axis**The probability density for this wave function is |ψ|2 =**ψ*ψ = (Ae–ikx)(Aeikx) = A2, which is independent of x. Consequently, the particle is equally likely to be found at any value of x. This is consistent with the uncertainty principle—if the wavelength is known precisely (based on a specific value of k in Equation 41.3), we have no knowledge of the position of the particle.**A particle is in a box of length L. Suddenly, the length of**the box is increased to 2L. What happens to the energy levels shown in the figure? • Nothing—they are unaffected. • They move farther apart. • They move closer together.**According to Equation 41.12, if L is increased, all**quantized energies become smaller. Thus, the energy levels move closer together. As L becomes macroscopic, the energy levels are so close together that we do not observe the quantized behavior.**Which of the following will exhibit quantized energy**levels? • an atom in a crystal • an electron and a proton in a hydrogen atom • a proton in the nucleus of a heavy atom • all of the above • none of the above**The particles in all three parts 1, 2, and 3 are part of a**bound system.**Consider an electron, a proton, and an alpha particle (a**helium nucleus), each trapped separately in identical infinite square wells. Which particle corresponds to the highest zero-point energy? • the electron • the proton • the alpha particle • The zero-point energy is the same in all three cases.**In Equation 41.15, we set n = 1 for the zero-point energy**and see that the energy is inversely proportional to the particle mass.**Consider the three particles in question 5 again. Which**particle has the longest wavelength when the system is in the ground state? • the electron • the proton • the alpha particle • All three particles have the same wavelength.**The wavelengths of the wave functions in Figure 41.8 are**longer than those in Figure 41.4 because the wave function spreads out into the classically forbidden region. For an infinite and a finite square well of the same length L, the quantized energies of the particle in a finite well are • the same as those for a particle in an infinite well • higher than those for a particle in an infinite well • lower than those for a particle in an infinite well • impossible to determine**The longer wavelength results in a smaller value of the**momentum from the de Broglie relationship. If the momentum of the particle is decreased, the energy also decreases.**For a particle undergoing simple harmonic motion in the n =**0 state, the most probable value of x for the particle according to quantum mechanics is • x = 0 • x = ±A • All values of x are equally likely.**The ground state represents the largest deviation from**classical behavior. Classically, the most probable values of x are x = ±A because the particle is moving the slowest near these points. As the top graph in Figure 41.15 shows, quantum mechanics predicts that the maximum probability density in the ground state is at x = 0

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