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Chapter 37: Problem 7

State whether each of the following statements is true or false. a) The energy of electrons is always discrete. b) The energy of a bound electron is continuous. c) The energy of a free electron is discrete. d) The energy of an electron is discrete when it is bound to an ion.

Short Answer

Expert verified
Explain your reasoning. a) The energy of electrons is always discrete. b) The energy of a bound electron is continuous. c) The energy of a free electron is discrete. d) The energy of an electron is discrete when it is bound to an ion.

Step by step solution

01

Statement a) The energy of electrons is always discrete.

This statement is false. The energy of electrons can be both discrete and continuous, depending on whether the electron is bound or free. In atoms and ions, electrons are bound, and their energy levels are discrete (quantized). However, in a conducting material like a metal, electrons are free to move, and their energy levels form a continuous band. So, the energy of electrons is not always discrete.
02

Statement b) The energy of a bound electron is continuous.

This statement is false. The energy of a bound electron, like the ones in atoms and ions, is discrete (quantized). Bound electrons can only occupy specific energy levels, which are determined by their quantum numbers (n, l, m, and s). The energy levels of bound electrons are not continuous.
03

Statement c) The energy of a free electron is discrete.

This statement is false. Electrons that are free (not bound to atoms or ions) have continuous energy levels. For example, in a metal, the electrons are free to move and can occupy any energy level within a specific range. The energy levels of free electrons are not quantized and can vary continuously.
04

Statement d) The energy of an electron is discrete when it is bound to an ion.

This statement is true. When an electron is bound to an ion, its energy levels are quantized (discrete). These energy levels are determined by the electron's quantum numbers (n, l, m, and s) and are specific to the ion to which the electron is bound. The energy levels of bound electrons in ions are not continuous.

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Key Concepts

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

Quantum Mechanics

At the core of understanding the behavior of electrons lies quantum mechanics, the branch of physics that deals with the motions and interactions of subatomic particles. It is a foundational tenet of this theory that particles such as electrons exhibit both wave-like and particle-like characteristics. A critical aspect of quantum mechanics is the quantization of physical properties, particularly energy levels in atoms and molecules.

  • Quantum mechanics dictates that not all energy values are permissible for electrons bound within atoms, leading to quantized energy levels.
  • This quantization arises from the wave nature of electrons, which can only sustain stable standing waves at certain frequencies within the confines of an atomic nucleus's influence.
  • The permitted energy levels are basically the 'notes' of the atomic 'music' and define the unique electron configurations that underlie the chemistry of elements.

Waves, Particles, and the Uncertainty Principle

Moreover, the quantum mechanical behavior of electrons is governed by the Heisenberg Uncertainty Principle, which states that certain pairs of physical properties, like position and momentum, cannot be simultaneously measured with arbitrary precision. From an educational perspective, quantum mechanics encourages a shift from deterministic to probabilistic thinking – a challenging but essential intellectual leap for students grappling with modern physics.

Quantization of Energy

When delving into the nature of electron energy levels, it's significant to comprehend the concept of quantization of energy. This principle describes how electrons can only possess specific values of energy and nothing in between. The discrete energy levels are akin to the rungs of a ladder: an electron can 'jump' from one rung to another, but it cannot exist halfway between two rungs.

  • Energies of bound electrons in atoms are quantized, which is why they can only absorb or emit photons (light particles) of certain energies—corresponding to the differences between these fixed rungs.
  • Quantization explains a host of phenomena, from the colors emitted by fireworks to the absorption lines in the spectra of stars.
  • It also underlies the understanding of chemical bonding, where electron transitions between quantized energy levels lead to energy absorption or release.

Quantum Numbers and Electron Orbits

Conceptually, quantization can be explored through quantum numbers that define an electron's state. These include the principal quantum number (n), angular momentum quantum number (l), magnetic quantum number (m), and spin quantum number (s), which together specify the discrete energy levels within atoms.

Bound and Free Electrons

The concept of bound and free electrons is central to understanding the differences in energy state behavior. Bound electrons are those attached to an atom or molecule and are confined to specific energy levels as dictated by the laws of quantum mechanics. These electrons can only jump to another level by absorbing or emitting a photon of precise energy. These jumps are what give rise to the spectral lines we see in emission and absorption spectra.

  • In contrast, free electrons, such as those found in metals or in plasma states, are not restricted to discrete energy levels. These electrons have the freedom to move within a continuum of energy states.
  • This distinction between bound and free electrons explains the differing electrical and thermal properties of materials; conductors, semiconductors, and insulators display unique behaviors based on the freedom of their electrons.
  • Understanding the concept of bound and free electrons also clarifies the exercise statements, particularly in comprehending why the energy of free electrons is continuous while that of bound electrons is quantized.

Applications in Technology

The concepts of bound and free electrons are not just theoretical but have practical implications in semiconductor technology, lasers, and even the functioning of solar cells, all of which rely on the manipulation of electron energy states.

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Most popular questions from this chapter

Although quantum systems are frequently characterized by their stationary states or energy eigenstates, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass \(m\) in a one-dimensional potential well with infinite walls (a "box") of width \(a\) is actually in a state with wave function $$ \Psi(x, t)=\frac{1}{\sqrt{2}}\left[\Psi_{1}(x, t)+\Psi_{2}(x, t)\right], $$ where \(\Psi_{1}\) denotes the stationary state with quantum number \(n=1\) and \(\Psi_{2}\) denotes the state with \(n=2 .\) Calculate the probability density distribution for the position \(x\) of the particle in this state.

True or False: The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer.

Suppose \(\psi(x)\) is a properly normalized wave function describing the state of an electron. Consider a second wave function, \(\psi_{\text {new }}(x)=e^{i \phi} \psi(x),\) for some real number \(\phi .\) How does the probability density associated with \(\psi_{\text {new }}\) compare to that associated with \(\psi ?\)

Consider an electron that is confined to the \(x y\) -plane by a two-dimensional rectangular infinite potential well. The width of the well is \(w\) in the \(x\) -direction and \(2 w\) in the \(y\) direction. What is the lowest energy that is shared by more than one distinct state, that is, where two different states have the same energy?

37.2 In an infinite square well, for which of the following states will the particle never be found in the exact center of the well? a) the ground state b) the first excited state c) the second excited state d) any of the above e) none of the above
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