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Chapter 35: Problem 5

What's the essential difference between the energy-level structures of infinite and finite square wells?

Short Answer

Expert verified
The main difference between the energy level structures of infinite and finite square wells is that for infinite wells, energy levels are only discrete with the particle strictly confined within the well, whereas for finite wells, there are both discrete and continuous energy levels with the particle having the possibility to exist outside of the well.

Step by step solution

01

Understanding Infinite Square Wells

In an infinite square well, known as a particle in a box, the potential energy within the box is zero and is infinite everywhere else. Thus, the particle is completely confined to the region of the box and cannot leave. As a result, energy levels are quantized, meaning that they only occur at discrete values. Each energy state corresponds to a unique wave function solution which describes the quantum state of the particle.
02

Understanding Finite Square Wells

In a finite square well, the potential energy within the well is less than outside, but not zero. So, the particle can theoretically exist outside the well. Here, the energy levels are still quantized for bound states. However, there are also states with energy levels that are continuous, but higher than that of the well depth. In these cases, the particle behaves more like a free particle.
03

Contrasting Infinite and Finite Square Wells

The essential difference between infinite and finite square wells refers to the presence of continuous energy states for finite wells, and the possibility of the particle existing outside of the well. In contrast, in an infinite square well, the energy levels are all discrete and the particle is strictly confined to the well.

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Most popular questions from this chapter

A particle is confined to a two-dimensional box whose sides are in the ratio \(1: 2 .\) Are any of its energy levels degenerate? If so, give an example. If not, why not?

Your roommate is taking Newtonian physics, while you've moved on to quantum mechanics. He claims that QM can't be right, because he didn't see any evidence of quantized energy levels in a mass-spring harmonic oscillator experiment. You reply by calculating the spacing between energy levels of this system, which consists of a \(1-\mathrm{g}\) mass on a spring with \(k=80 \mathrm{N} / \mathrm{m}\) What is that spacing, and how does this help your argument?

What did Einstein mean by his remark, loosely paraphrased, that "God does not play dice"?

A particle is confined to a 1.0 -nm-wide infinite square well. If the energy difference between the ground state and the first excited state is \(1.13 \mathrm{eV},\) is the particle an electron or a proton?

The ground-state energy for an electron in infinite square well \(A\) is equal to the energy of the first excited state for an electron in well B. How do the wells' widths compare?
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