Question

A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2 , t3 Then, answer these questions: What is the period of oscillation of l(, t))2? and 4-3 Describe what is happening to the particle in words (4pts) Compute( r(t)〉 for the above state. What are the minimum and maximum values oft(t)? This is NOT a stationary state... what would ((t)) be for a stationary state? F. G. (2 pts) Suppose you measure the energy of the particle above and happen to get E2. What is the state of the particle after measurement? Write your answer both as an abstract ket AND as a position space wave function. regions depends on the specific value ofthe ene s depe ra. (5.40 goal tion app forbidden reglU in bound states. Particles constrained and so are in unbound Sl t potential energy functions is the chapters. In this chapter, our goal is to study a simple pot ics required for this new wave function approach, enerbe of a paricle thoa is ect of this and d leam the mple potential energy system and lea thi Solving Eq. (5.46) for various important potential e with the simplest example o super ball bouncing between two ture is a ball this system a particle in a box. We observe are consistent with (1) zero force on the ball when betwee fined to a region of space. The classical picture is a walls. We call We begin our journey to energy quantization in a box. three important c e importante walls, (2) the ball is reflected ic ob hese three observations of the motion in Fig. 5.9. The potential encr ball fies fre box no atter how large its energy. These classical system: (1) the bail fies freely between the walls, (2t classical system: t hall remains in it is between the walls, (2) infinite thr The mathematical model that is consistent with these three ticle in a box is given by the potential energy function shown in th t at the walls, and (3) infinite potential energy outside the box.walls, (2) ins Fig. 5.9. The pote for simplicity), and it is infi tential energy Tn within the well (any constant would suffice, but we choose side the well. The discontinuity at the sides of the tion in a piecewise fashion well requires us to write the potential energ in x > L Because of the shape of the potential energy in Fig. 5.9, this system is also square well. Though this model is too simple to accurately represent any real quantum me system, it doesillustrate most of the important features of a particle bound to a limited regionofste referred to as an ie ian V(x) 0o L/2 FIGURE 59 Infinite square potential energy 121 5.4 Infinite Square Well Our goal is to find the energy eigenstates and eigenvalues of the system by solving the energy eigenvalue equation using the potential energy in Eq. (5.47). The potential energy is piecewise, so we ust solve the differential equation (5.46) separately inside and outside the box. Outside the box, the potential energy is infinite and the energy eigenvalue equation is (5.48) (-2mde+oo)фЕ(x) Epe(x), outside box We are looking for solutions with finite energy E, so Eq. (5.48) is satisfied only if the energy eigenstat wave function PE(x) is zero everywhere outside the box. This means that the quantum mechanica particle is excluded from the classically forbidden regions in this example. This correspondence wit the classical situation holds only for the case of infinite potential energy walls on the potential wel. Inside the box, the potential energy is zero and the energy eigenvalue equation is h.d' +0)9a(x) : Ефе(x), inside box (5.4 2m dx2 Thus our task reduces to solving the differential equation inside the box: It is worth reminding ourselves at this point what is known and what is not. The particle has a m m and is confined to a box of size L. These quantities are known, as is h, a fundamental constant. unknowns that we need to find are the energy E and the wave function фЕ(x), which is what it mean solve an eigenvalue problem (now posing as a differential equation). It is convenient to rewrite the differential equation (5.50) as 2mE dx where we have defined a new parameter

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