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Chapter 40: Problem 51

A fellow student proposes that a possible wave function for a free particle with mass \(m\) (one for which the potential-energy function \(U(x)\) is zero) is $$ \psi(x)=\left\\{\begin{array}{ll} e^{+\kappa x}, & x<0 \\ e^{-\kappa x}, & x \geq 0 \end{array}\right. $$ where \(\kappa\) is a positive constant. (a) Graph this proposed wave function. (b) Show that the proposed wave function satisfies the Schr枚dinger equation for \(x<0\) if the energy is \(E=-\hbar^{2} \kappa^{2} / 2 m-\) that is, if the energy of the particle is negative. (c) Show that the proposed wave function also satisfies the Schr枚dinger equation for \(x \geq 0\) with the same energy as in part (b). (d) Explain why the proposed wave function is nonetheless not an acceptable solution of the Schr枚dinger equation for a free particle. (Hint: What is the behavior of the function at \(x=0 ?\) ) It is in fact impossible for a free particle (one for which \(U(x)=0\) ) to have an energy less than zero.

Short Answer

Expert verified
The proposed wave function satisfies the Schr枚dinger Equation for both \( x < 0 \) and \( x \geq 0 \). However, the wave function is discontinuous at zero and hence is not a valid solution for Schr枚dinger equation for a free particle.

Step by step solution

01

Graph the proposed Wave Function

Plot two exponential functions \( e^{+\kappa x} \) for \( x<0 \) and \( e^{-\kappa x} \) for \( x \geq 0 \). The graph shows that as \( x \) moves away from 0 in either direction, the output of the function diminishes.
02

Check Schr枚dinger鈥檚 Equation for \( x

Placing the function \( e^{+\kappa x} \) into the time-independent Schr枚dinger equation: \[ -\frac{\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}} + U(x)\psi = E\psi \] invoking the fact that \( U(x) = 0 \) due to zero potential-energy for a free particle, the differential equation simplifies to: \[ E = -\frac{\hbar^{2}}{2m} \frac{d^{2}\psi}{dx^{2}} \] When differentiated twice wrt \( x \), the function \( e^{+\kappa x} \) gives \( \kappa^{2}e^{+\kappa x} \). Substitute and simplify to get the required energy \( E = -\frac{\hbar^{2} \kappa^{2}}{2 m} \) validating the proposed wave function for \( x<0 \).
03

Check Schr枚dinger鈥檚 Equation for \( x\geq0 \)

Follow similar steps as in Step 2 but this time for the function \( e^{-\kappa x} \). Again, the proposed wave function satisfies the Schr枚dinger equation with the same energy value derived in Step 2.
04

Identify the Problem with the Proposed Wave Function

While the function does satisfy the Schr枚dinger Equation for both \( x < 0 \) and \( x \geq 0 \), the function is discontinuous at zero. For a valid wave function, it not only needs to satisfy the Schr枚dinger equation but also be well-behaved. In this case, well-behaved means it must be continuous and have continuous first derivative everywhere. Given the wave function at zero jumps from infinity to zero immediately, it does not meet this criterion and hence is not a valid solution.

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Key Concepts

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

Wave Function
The wave function, generally denoted by the symbol \( \psi \), is a fundamental concept in quantum mechanics, serving as a mathematical tool that describes the quantum state of a particle or system of particles. In the Schr枚dinger equation, the wave function is used to calculate the probability of finding a particle in a particular position and time. A key requirement for any wave function is that it must be continuous and its first derivative must also be continuous. This ensures that the quantum state it describes can be physically realized.

The situation presented in the exercise involves a proposed wave function \( \psi(x) \) for a free particle, which takes different exponential forms based on whether the position \( x \) is less than or greater than zero. While the wave function satisfies the Schr枚dinger equation mathematically, a wave function must also be physically plausible. The proposed \( \psi(x) \) lacks continuity at \( x=0 \), indicating an unphysical solution, as a wave function cannot have a probability that suddenly jumps from one value to another.
Potential Energy Function
The potential energy function, \( U(x) \), is crucial in understanding particle behavior in quantum systems. It represents the potential energy within a system as a function of position and plays a significant role in the Schr枚dinger equation. For a free particle, which is an idealized particle that is not subject to any forces and hence has no potential energy, this function is zero everywhere. That is, \( U(x) = 0 \) regardless of \( x \).

When \( U(x) \) is zero, the time-independent Schr枚dinger equation simplifies and reflects the kinetic energy of the particle only. This often allows for easier analysis of a system, as seen in the exercise where the potential energy function being zero streamlined the steps to check if the proposed wave function satisfied the Schr枚dinger equation. However, one must remember that even if \( U(x) \) simplifies the equation, a physically meaningful solution must be sought, which is not the case with the function proposed in the exercise.
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the scale of atoms and subatomic particles. It diverges from classical mechanics on the atomic and subatomic scale due to the quantization of certain variables. Quantum mechanics provides a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter.

In the realm of quantum mechanics, particles are often described using wave functions, and their behavior is governed by equations such as the Schr枚dinger equation. This equation is the cornerstone of quantum mechanics, allowing physicists to calculate how wave functions evolve over time, and thereby predicting the behavior of quantum systems. The exercise presented highlights a situation where insight from quantum mechanics is applied to analyze a proposed wave function. This demonstrates how critical principles of continuity and physical plausibility are in not just solving equations, but also in determining the legitimacy of solutions according to the established laws of quantum mechanics.

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Most popular questions from this chapter

Consider a wave function given by \(\psi(x)=A \sin k x,\) where \(k=2 \pi / \lambda\) and \(A\) is a real constant. (a) For what values of \(x\) is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of \(x\) is the probability zero? Explain.

A particle moving in one dimension (the \(x\) -axis) is described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} A e^{-b x}, & \text { for } x \geq 0 \\ A e^{b x}, & \text { for } x<0 \end{array}\right. $$ where \(b=2.00 \mathrm{~m}^{-1}, A>0,\) and the \(+x\) -axis points toward the right. (a) Determine \(A\) so that the wave function is normalized. (b) Sketch the graph of the wave function. (c) Find the probability of finding this particle in each of the following regions: (i) within \(50.0 \mathrm{~cm}\) of the origin, (ii) on the left side of the origin (can you first guess the answer by looking at the graph of the wave function?), (iii) between \(x=0.500 \mathrm{~m}\) and \(x=1.00 \mathrm{~m}\).

Quantum Dots. A quantum dot is a type of crystal so small that quantum effects are significant. One application of quantum dots is in fluorescence imaging, in which a quantum dot is bound to a molecule or structure of interest. When the quantum dot is illuminated with light, it absorbs photons and then re- emits photons at a different wavelength. This phenomenon is called fluorescence. The wavelength that a quantum dot emits when stimulated with light depends on the dot's size, so the synthesis of quantum dots with different photon absorption and emission properties may be possible. We can understand many quantum-dot properties via a model in which a particle of mass \(M\) (roughly the mass of the electron) is confined to a two-dimensional rigid square box of sides \(L\). In this model, the quantumdot energy levels are given by \(E_{m, n}=\left(m^{2}+n^{2}\right)\left(\pi^{2} \hbar^{2}\right) / 2 M L^{2},\) where \(m\) and \(n\) are integers \(1,2,3, \ldots\) According to this model, which statement is true about the energy-level spacing of dots of different sizes? (a) Smaller dots have equally spaced levels, but larger dots have energy levels that get farther apart as the energy increases. (b) Larger dots have greater spacing between energy levels than do smaller dots. (c) Smaller dots have greater spacing between energy levels than do larger dots. (d) The spacing between energy levels is independent of the dot size.

When low-energy electrons pass through an ionized gas, electrons of certain energies pass through the gas as if the gas atoms weren't there and thus have transmission coefficients (tunneling probabilities) \(T\) equal to unity. The gas ions can be modeled approximately as a rectangular barrier. The value of \(T=1\) occurs when an integral or half-integral number of de Broglie wavelengths of the electron as it passes over the barrier equal the width \(L\) of the barrier. You are planning an experiment to measure this effect. To assist you in designing the necessary apparatus, you estimate the electron energies \(E\) that will result in \(T=1\). You assume a barrier height of \(10 \mathrm{eV}\) and a width of \(1.8 \times 10^{-10} \mathrm{~m} .\) Calculate the three lowest values of \(E\) for which \(T=1\)

A proton is bound in a square well of width \(4.0 \mathrm{fm}=\) \(4.0 \times 10^{-15} \mathrm{~m} .\) The depth of the well is six times the ground-level energy \(E_{1-\text { IDW }}\) of the corresponding infinite well. If the proton makes a transition from the level with energy \(E_{1}\) to the level with energy \(E_{3}\) by absorbing a photon, find the wavelength of the photon.
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