The WKB Approximation. It can be a challenge to solve the Schrodinger 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 $W K B$ approximation (named for the physicists Gregor Wentzel,

Hendrik Kramers, and Leon 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 $p=h / \lambda$ (ii) The magnitude of momentum is related to the kinetic energy $K$ by the relationship $K=p^{2} / 2 m$ . (iii) If there are no non-conservative 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: $E=K+U(x),$ 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 $$\lambda(x)=\frac{h}{\sqrt{2 m[E-U(x)]}}$$ Thus we envision a quantum-mechanical particle in a potential well $U(x)$ as being like a free particle, but with a wavelength $\lambda(x)$ that is a function of position. (b) When the particle moves into a region of increasing potential energy, what happens to its wave-length? (c) At a point where $E=U(x),$ 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 $\frac{1}{2} k^{\prime} x^{2}$ equals the total energy $\frac{1}{2} k^{\prime} A^{2} .$ In the WKB expression for $\lambda(x),$ what is the wave-length 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 $L=(n / 2) \lambda$ and hence $L / \lambda=n / 2,$ where $n=1,2,3, \ldots . .$ INote 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$/ \lambda(x)$ between those points:

$$\int_{a}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2} \quad(n=1,2,3, \ldots)$$

Using the expression for $\lambda(x)$ you found in part (a), show that the $W K B$ condition for an allowed bound-state energy can be written as

$$\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \ldots)$$

(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 $E<U_{0}$ . 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 potential-energy function changes gradually and the WKB approximation is much more useful.