SM_chapter41

SM_chapter41 - 41 Quantum Mechanics Note: In chapters 39,...

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41 Quantum Mechanics CHAPTER OUTLINE 41.1 An Interpretation of Quantum Mechanics 41.2 The Quantum Particle under Boundary Conditions 41.3 The Schrödinger Equation 41.4 A Particle in a Well of Finite Height 41.5 Tunneling Through a Potential Energy Barrier 41.6 Applications of Tunneling 41.7 The Simple Harmonic Oscillator ANSWERS TO QUESTIONS Q41.1 A particle’s wave function represents its state, contain- ing all the information there is about its location and motion. The squared absolute value of its wave function tells where we would classically think of the particle as spending most its time. Ψ 2 is the probability distribution function for the position of the particle. *Q41.2 For the squared wave function to be the probability per length of ± nding the particle, we require ψψ 2 048 74 016 = == .. nm nm nm and 0.4/ nm (i) Answer (e). (ii) Answer (e). *Q41.3 (i) For a photon a and b are true, c false, d, e, f, and g true, h false, i and j true. (ii) For an electron a is true, b false, c, d, e, f true, g false, h, i and j true. Note that statements a, d, e, f, i, and j are true for both. *Q41.4 We consider the quantity h 2 n 2 /8 mL 2 . In (a) it is h 2 1/8 m 1 (3 nm) 2 = h 2 /72 m 1 nm 2 . In (b) it is h 2 4/8 m 1 (3 nm) 2 = h 2 /18 m 1 nm 2 . In (c) it is h 2 1/16 m 1 (3 nm) 2 = h 2 /144 m 1 nm 2 . In (d) it is h 2 1/8 m 1 (6 nm) 2 = h 2 /288 m 1 nm 2 . In (e) it is 0 2 1/8 m 1 (3 nm) 2 = 0. The ranking is then b > a > c > d > e. Q41.5 The motion of the quantum particle does not consist of moving through successive points. The particle has no de± nite position. It can sometimes be found on one side of a node and sometimes on the other side, but never at the node itself. There is no contradiction here, for the quantum particle is moving as a wave. It is not a classical particle. In particular, the particle does not speed up to in± nite speed to cross the node. 463 Note : In chapters 39, 40, and 41 we use u to represent the speed of a particle with mass, reserving v for the speeds associated with reference frames, wave functions, and photons.
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464 Chapter 41 Q41.6 Consider a particle bound to a restricted region of space. If its minimum energy were zero, then the particle could have zero momentum and zero uncertainty in its momentum. At the same time, the uncertainty in its position would not be inF nite, but equal to the width of the region. In such a case, the uncertainty product ∆∆ xp x would be zero, violating the uncertainty principle. This contradiction proves that the minimum energy of the particle is not zero. *Q41.7 Compare ±igures 41.4 and 41.7 in the text. In the square well with inF nitely high walls, the particle’s simplest wave function has strict nodes separated by the length L of the well. The particle’s wavelength is 2 L , its momentum h L 2 , and its energy p m h mL 22 2 28 = . Now in the well with walls of only F nite height, the wave function has nonzero amplitude at the walls. In this F nite-depth well … (i) The particle’s wavelength is longer, answer (a).
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SM_chapter41 - 41 Quantum Mechanics Note: In chapters 39,...

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