91Ó°ÊÓ

The WKB Approximation. It can be a challenge 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 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 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 \(L=(n / 2) \lambda\) and hence \(L / \lambda=n / 2,\) where \(n=1,2,3, \ldots\) [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 \(1 / \lambda(x)\) 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 \(\mathrm{WKB}\) expression given in part (d) predicts the same bound-state energies as for an infinite square well of the same width. (Hint: Assume \(E

Step by step solution

01

Step_1: Derive the formula for particle wavelength

This derives from three relationships pertaining to quantum mechanical particles. Firstly, the relationship between momentum \(p\) and wavelength \(λ\) is stated by de Broglie as \(p=h / λ\). Secondly, the relationship between kinetic energy \(K\) and momentum \(p\) is delineated as \(K=p^{2} / 2 m\). Lastly, the total energy \(E\) of a particle in absence of nonconservative forces equals the kinetic plus potential energies at any point along the coordinate, hence \(E=K+U(x)\). Combining these three relationships, the wavelength \(λ(x)\) of the particle can be written as \(λ(x)=h / \sqrt{2 m(E-U(x))}\) using a bit of manipulation and substitution.

02

Effects on wavelength

When a particle moves into an area of increasing potential energy, the wavelength decreases. This is because the increased potential energy results in less kinetic energy, which then results in a lower energy in the denominator of the \(λ(x)\) formula, thus increasing the value of \(λ(x)\).

03

Determine the wavelength at a classical turning point

A classical turning point is a point where \(E = U(x)\). Substituting this into the equation for \(λ(x)\), we get \(λ(x) = h / \sqrt{2 m [E - E]} = ∞\). Therefore, the wavelength at a classical turning point is infinite.

04

Formulate the WKB condition

The number of wavelengths between turning points a and b for a given energy is the integral of \(1 / λ(x)\) between these points. Using the formula for \(λ(x)\), this integral can be converted into the following form: \(\int_{a}^{b} \sqrt{2 m [E - U(x)]} dx = \frac{n h}{2}\). This is the WKB condition for a bound-state energy level.

05

Verify the expression for a particle in a box

Apply the WKB condition to a particle in a box with walls at positions \(x=0\) and \(x=L\). Inside the box, the potential energy is zero, which simplifies the previously derived integral to \(\int_{0}^{L} \sqrt{2 m E} dx = \frac{n h}{2}\). Solving this equation gives that \(E = n^{2} h^{2} / 8 m L^{2}\), which are the same energy levels found when solving the Schrödinger equation for this system directly, confirming the validity of the WKB approximation.

06

Assess the WKB approximation for a finite square well

Applying the WKB approximation to a finite square well (with potential \(U(x) = U_0\) when \(0 < x < L\) and \(U(x) = 0\) otherwise), reveals it predicts the same bound-state energies as for an infinite square well. Thus, it is concluded that the WKB approximation does a poor job when the potential-energy function changes abruptly, as in the case of a finite potential well.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Schrödinger Equation

The Schrödinger equation is at the heart of quantum mechanics. It serves as a mathematical description of how the quantum state of a physical system changes over time. When dealing with bound-state energy levels, solving the Schrödinger equation can often be cumbersome. This is particularly true in scenarios involving an arbitrary potential well—an imaginary space where a particle is likely to be located. The Schrödinger equation provides the probability of finding a particle at a given position and energy level within this potential well.

For a one-dimensional system, the Schrödinger equation is usually represented as \(abla^2 \psi(x) + \frac{2m}{\hbar^2} (E - U(x)) \psi(x) = 0\), where \(\psi(x)\) is the wave function of the particle, \(E\) is the energy, \(U(x)\) represents potential energy as a function of position \(x\), \(m\) is the mass of the particle, \(\hbar\) is the reduced Planck's constant, and \(abla^2\) is the Laplace operator.

Understanding this equation forms the basis for exploring more approximate methods like the WKB approximation, which allows physicists to infer the energy levels of a particle without explicitly solving this complex equation.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that describes nature at the smallest scales, such as that of subatomic particles. In this theory, particles exhibit both wave-like and particle-like properties, a principle known as wave-particle duality.

Quantum mechanics revolutionized the classical understanding of physics. It introduced concepts such as quantization, where energy levels are discrete, and superposition, where particles can exist in multiple states simultaneously. It also pioneered the idea that the position of a particle is probabilistic, described by the wave function, which is a solution to the Schrödinger equation.

• Wave-particle duality: Particles can behave like waves.
• Quantization: Energy levels are not continuous.
• Probability: Outcomes are not deterministic but probabilistic.

Within the framework of quantum mechanics, the WKB approximation becomes a useful tool. It effectively simplifies complex calculations by approximating the behavior of quantum systems, thereby yielding insights into phenomena like bound-state energy levels.

Bound-State Energy Levels

Bound-state energy levels refer to the discrete energy values a quantum particle can have when it is confined within a potential well. These energy levels are quantized, meaning only certain discrete energies are allowed. This is different from classical physics, where particles can have a continuous range of energies.

In a potential well, such as that created by an atomic nucleus or a quantum dot, bound-state energy levels arise because the particle is trapped and cannot escape the well, much like a ball trapped in a valley. The particle's behavior in this scenario can be described by solving the Schrödinger equation, leading to discrete values for energy.

The WKB approximation offers a way to estimate these energy levels without solving the Schrödinger equation directly. In essence, it approximates the number of wavelengths that fit within the boundaries of the potential well and links these to possible energy levels. This negotiation between potential energy and kinetic energy helps define where these bound states occur.

Potential Well

A potential well is a concept used in quantum mechanics to describe a region where a particle is confined, typically due to a surrounding barrier that it cannot overcome without enough energy. Within a potential well, the potential energy function is usually lower than the barriers, making it a favored location for a particle to occupy.

The potential well can be visualized as a container with high walls: a particle with insufficient energy cannot escape. The shapes and depths of potential wells are integral in determining the bound-state energy levels within quantum systems. Potential wells can take different forms based on the kind of forces creating them, such as gravity, electromagnetic forces, or nuclear forces.

• Finite potential well: Has a finite depth, with particles able to escape if they possess sufficient energy.
• Infinite potential well: The barriers are infinitely high, making it impossible for particles within to escape.

When applying the WKB approximation, we can solve for the probable energy states of a particle within these wells by estimating how often it "fits" within the bounds set by the potential walls.

Classical Turning Points

Classical turning points are positions within the potential landscape where a particle's kinetic energy is momentarily zero. At these points, the total energy equals the potential energy, and classically, the particle would stop and change direction here—hence, "turning points."

Classical mechanics defines these points based on the equation of energy conservation:

\( E = K + U(x) \), where \( K \) is kinetic energy and \( U(x) \) the potential energy. When \( E = U(x) \), the kinetic energy \( K = 0 \), implying the particle is momentarily at rest.

In quantum mechanics, classical turning points help define where significant changes in particle behavior can occur, and they play a crucial role in the WKB approximation. Using this approximation, physicists can predict where energy levels arise by examining these turning points, thus identifying points of constructive interference where waves reinforce each other.