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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
Infinite square wells confine particles entirely within the well due to its theoretically infinite boundaries and only display totally discrete quantized energy levels. However, finite square wells, having a finite potential in the boundaries, permit quantum tunneling, allowing for particles to exist outside the well and present bound as well as unbound energy states.

Step by step solution

01

Understanding the concept of potential wells

In quantum mechanics, a particle can be represented as a wave. The behaviour of this particle (wave) is described by a potential well. There are two types of potential wells 鈥� the infinite potential well (or box) where the walls are of infinite potential energy, and the finite potential well where the walls do not have infinite potential energy.
02

Infinite Square Well Analysis

For the infinite square well, the boundaries are infinitely high, implying that the particle cannot pass through them under any circumstances. The energy levels that the particle can take are quantized and discrete. Because the potential energy outside the well is infinite, the wavefunction representing the particle's state should be and is indeed zero outside the well. Inside the well, the time-independent Schr枚dinger equation applies and can be solved with a sinusoidal solution for the wave function respecting the boundary conditions.
03

Finite square well analysis

For the finite square well, the boundaries are not infinitely high. This means that there is a non-zero probability for the particle to tunnel through the boundaries (quantum tunneling), which cannot happen in an infinite square well. While it maintains energy level quantization, the energy levels are not completely discrete anymore. There are now allowed energy levels higher than the barrier potential, which leads to unbound states that are not confined to the well. Here, the time-independent Schr枚dinger equation applies inside as well as outside the well, being zero beyond done boundary point far enough.
04

Summary of main differences

In essence: 1) While both infinite and finite potential wells display energy-level quantization, only in the infinite well the energy levels are completely discrete and determines where all bound states must exist. 2) Quantum tunneling is a characteristic feature for particles in the finite potential well, not observed in the infinite potential well. This corresponds to the fact that wavefunctions are not zero outside the well region for energy levels that are below or surpass the well's limit.

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Most popular questions from this chapter

The solution to the Schr枚dinger equation for a particular potential is \(\psi=0\) for \(|x|>a\) and \(\psi=A \sin (\pi x / a)\) for \(-a \leq x \leq a\) where \(A\) and \(a\) are constants. In terms of \(a,\) what value of \(A\) is required to normalize \(\psi\) ?

Show explicitly that the difference between adjacent energy levels in an infinite square well becomes arbitrarily small compared with the energy of the upper level, in the limit of large quantum number \(n\)

In terms of de Broglie's matter-wave hypothesis, how does making the sides of a box different lengths remove the degeneracy associated with a particle confined to that box?

The ground-state energy of a harmonic oscillator is 4.0 eV. Find the energy separation between adjacent quantum states.

An infinite square well extends from \(-L / 2\) to \(L / 2 .\) (a) Find expressions for the normalized wave functions for a particle of mass \(m\) in this well, giving separate expressions for even and odd quantum numbers. (b) Find the corresponding energy levels.
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