Question

The WKB Approximation. It can be achallenge to solve the Schrödinger 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 Léon 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 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: $E=K+U(x),$ 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 wavelength? (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 amphtude $A$ 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 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. 40.1$)$ . Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig, $40.3 ),$ so that $L=(n / 2) \lambda$ and hence $L / \lambda=n / 2,$ where $n=1,2,3, \ldots .$ [Note that this is a restatement of Eq. $(40.7) . ]$ 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: $$ \int_{a}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2} \quad(n=1,2,3, \dots) $$ 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 $$ \int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \dots) $$ (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. (40.9). (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. 40.6 , 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.

Name: Problem 57, Chapter 40: University Physics with Modern Physics
Uploaded: 2020-09-13T21:03:41-08:00
Duration: 10 min 1 s
Channel: Guilherme Barros

   The WKB Approximation. It can be achallenge to solve the Schrödinger 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 Léon 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 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: $E=K+U(x),$ 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 wavelength? (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 amphtude $A$ 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 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. 40.1$)$ . Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig, $40.3 ),$ so that $L=(n / 2) \lambda$ and hence $L / \lambda=n / 2,$ where $n=1,2,3, \ldots .$ [Note that this is a restatement of Eq. $(40.7) . ]$ 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:
$$
\int_{a}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2} \quad(n=1,2,3, \dots)
$$
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
$$
\int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \dots)
$$
(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. (40.9). (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. 40.6 , 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.

University Physics with Modern Physics

Hugh D. Young, Roger… 12th Edition

Chapter 40, Problem 57

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Solved by Expert Guilherme Barros

09/13/2020

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Step 1: We start with the de Broglie equation which relates the momentum of a particle with its wavelength: $p=h / \lambda$. Show more…

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The WKB Approximation. It can be achallenge to solve the Schrödinger 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 Léon 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 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: $E=K+U(x),$ 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 wavelength? (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 amphtude $A$ 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 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. 40.1$)$ . Furthermore, the number of wavelengths that fit within the box must be a half-integer (see Fig, $40.3 ),$ so that $L=(n / 2) \lambda$ and hence $L / \lambda=n / 2,$ where $n=1,2,3, \ldots .$ [Note that this is a restatement of Eq. $(40.7) . ]$ 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: $$ \int_{a}^{b} \frac{d x}{\lambda(x)}=\frac{n}{2} \quad(n=1,2,3, \dots) $$ 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 $$ \int_{a}^{b} \sqrt{2 m[E-U(x)]} d x=\frac{n h}{2} \quad(n=1,2,3, \dots) $$ (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. (40.9). (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. 40.6 , 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.

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Key Concepts

WKB Approximation

A semi-classical method used to approximate solutions of the Schrödinger equation when an exact solution is difficult or impossible to obtain. It relies on the idea that, in regions where the potential varies gradually, a quantum particle behaves similarly to a classical one with a spatially varying wavelength. This approach allows for quantization conditions based on the phase accumulated between classical turning points.

de Broglie Hypothesis

The fundamental concept that particles exhibit wave-like properties, with the wavelength related to momentum through the relation ? = h/p. This idea provides the basis for associating a position-dependent wavelength to particles in a quantum system, which is a key step in applying the WKB approximation.

Momentum-Energy Relationship

In both classical and quantum mechanics, the kinetic energy of a particle is related to its momentum by the formula K = p²/(2m). This relation is crucial when connecting the de Broglie wavelength to the energy of the system, as it provides a bridge between the particle’s wave characteristics and its dynamical behavior.

Classical Turning Points

Points in a potential where the particle’s kinetic energy becomes zero, meaning that the total energy equals the potential energy. These points mark the boundaries within which a particle is allowed to move classically, and in the WKB method, they serve as the limits for integrating the phase to achieve quantization.

Quantization Condition

A condition derived from the requirement that the phase accumulated by the particle's wavefunction between classical turning points must be an integer (or half-integer) multiple of ? or a related constant. In the WKB approximation, this leads to an integral expression that quantizes the bound-state energy levels, ensuring that the wavefunction matches the necessary boundary conditions.

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