Solved: The WKB Approximation. It can be a challenge to

Chapter 40, Problem 40.64

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The WKB Approximation. It can be a challenge to solve the Schrdinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the WKB approximation (named for the physicists Gregor Wentzel, Hendrik Kramers, and Lon Brillouin, who pioneered its application to quantum mechanics). The WKB approximation begins from three physical statements: (i) According to de Broglie, the magnitude of momentum p of a quantum-mechanical particle is p = h>l. (ii) The magnitude of momentum is related to the kinetic energy K by the relationship K = p2>2m. (iii) If there are no nonconservative forces, then in Newtonian mechanics the energy E for a particle is constant and equal at each point to the sum of the kinetic and potential energies at that point: E = K + U1x2, where x is the coordinate. (a) Combine these three relationships to show that the wavelength of the particle at a coordinate x can be written as l1x2 = h 22m3E - U1x24 Thus we envision a quantum-mechanical particle in a potential well U1x2 as being like a free particle, but with a wavelength l1x2 that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wavelength? (c) At a point where E = U1x2, Newtonian mechanics says that the particle has zero kinetic energy and must be instantaneously at rest. Such a point is called a classical turning point, since this is where a Newtonian particle must stop its motion and reverse direction. As an example, an object oscillating in simple harmonic motion with amplitude A moves back and forth between the points x = -A and x = +A; each of these is a classical turning point, since there the potential energy 1 2 kx2 equals the total energy 1 2 kA2 . In the WKB expression for l1x2, what is the wavelength at a classical turning point? (d) For a particle in a box with length L, the walls of the box are classical turning points (see Fig. 40.8). Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig. 40.10), so that L = 1n>22l and hence L>l = n>2, where n = 1, 2, 3,c. [Note that this is a restatement of Eq. (40.29).] The WKB scheme for finding the allowed bound-state energy levels of an arbitrary potential well is an extension of these observations. It demands that for an allowed energy E, there must be a half-integer number of wavelengths between the classical turning points for that energy. Since the wavelength in the WKB approximation is not a constant but depends on x, the number of wavelengths between the classical turning points a and b for a given value of the energy is the integral of 1>l1x2 between those points: L b a dx l1x2 = n 2 1n = 1, 2, 3, c2 Using the expression for l1x2 you found in part (a), show that the WKB condition for an allowed bound-state energy can be written as L b a 22m3E - U1x24 dx = nh 2 1n = 1, 2, 3, c2 (e) As a check on the expression in part (d), apply it to a particle in a box with walls at x = 0 and x = L. Evaluate the integral and show that the allowed energy levels according to the WKB approximation are the same as those given by Eq. (40.31). (Hint: Since the walls of the box are infinitely high, the points x = 0 and x = L are classical turning points for any energy E. Inside the box, the potential energy is zero.) (f) For the finite square well shown in Fig. 40.13, show that the WKB expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. (Hint: Assume E 6 U0. Then the classical turning points are at x = 0 and x = L.) This shows that the WKB approximation does a poor job when the potential-energy function changes discontinuously, as for a finite potential well. In the next two problems we consider situations in which the potentialenergy function changes gradually and the WKB approximation is much more useful