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Chapter 40: Problem 22

(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of 0.125 \(\mathrm{nm}\) . (b) The electron makes a transition from the \(n=1\) to \(n=4\) level by absorbing a photon. Calculate the wave-length of this photon.

Short Answer

Expert verified
Excitation energy: 12.1 eV; photon wavelength: 103 nm.

Step by step solution

01

Understand the Problem

We need to find two things: (a) the energy difference between the ground state and the third excited state for an electron in a box, and (b) the wavelength of a photon that enables the electronic transition from \(n=1\) to \(n=4\) levels by absorbing energy.
02

Energy in a Quantum Box

For an electron confined in a one-dimensional box, the energy levels are given by the formula \[ E_n = \frac{n^2 h^2}{8mL^2} \] where \(n\) is the quantum number, \(h\) is Planck's constant \(6.626 \times 10^{-34} \ \mathrm{Js}\), \(m\) is the mass of the electron \(9.109 \times 10^{-31} \ \mathrm{kg}\), and \(L\) is the width of the box \(0.125 \ \mathrm{nm} = 0.125 \times 10^{-9} \ \mathrm{m}\).
03

Calculate Energy for Ground and Third Excited Level

The ground level \(n=1\) has energy \( E_1 \) and the third excited level \(n=4\) has energy \( E_4 \).Calculate \( E_1 = \frac{1^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot 9.109 \times 10^{-31} \cdot (0.125 \times 10^{-9})^2} \).Calculate \( E_4 = \frac{4^2 \cdot (6.626 \times 10^{-34})^2}{8 \cdot 9.109 \times 10^{-31} \cdot (0.125 \times 10^{-9})^2} \).
04

Calculate Excitation Energy

The excitation energy from \(n=1\) to \(n=4\) is \(\Delta E = E_4 - E_1\). Calculate this difference using the previously obtained energy values.
05

Find Wavelength of Photon

The energy of the absorbed photon \(\Delta E\) can be related to its wavelength \(\lambda\) using the formula \[ \Delta E = \frac{hc}{\lambda} \] where \(c = 3 \times 10^8 \ \mathrm{m/s}\) is the speed of light. Rearrange to find \(\lambda = \frac{hc}{\Delta E}\). Calculate \(\lambda\) using \(\Delta E\) obtained in Step 4.

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Key Concepts

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

Energy Levels
Energy levels in quantum mechanics are the distinct levels of energy that an electron can have within a system. These levels are quantized, meaning that electrons can only occupy specific energy amounts. In a quantum box, these energy levels are influenced by the properties of the system, such as its size and the inherent physical constants.
  • For an electron in a one-dimensional quantum box, the allowed energy levels are determined by the formula: \[ E_n = \frac{n^2 h^2}{8mL^2} \ \]
  • Here, \( n \) is the quantum number, showing the level of excitement (with \( n = 1 \) being the ground level).
  • \( h \) is Planck's constant, \( m \) is the electron's mass, and \( L \) represents the width of the box.
These specific quantized energy levels dictate how electrons behave in confined systems, and this is essential for understanding electron transitions within those systems.
Quantum Box
A quantum box, or quantum well, is a model in physics where a particle is confined to move in a restricted area. You can think of it as a tiny "cage" for electrons, limiting the dimensions in which they can move. The box's dimensions greatly impact the energy levels available to the particle.
  • The width of the box \( L \) is crucial, as even small changes can result in significant changes in energy levels.
  • The concept of a quantum box simplifies the analysis of electron behavior in nanoparticles, quantum dots, and similar phenomena in nanotechnology.
  • By understanding the principles of a quantum box, one can predict how particles behave under spatial confinement.
These boxes are models used to explore fundamental quantum mechanics and their practical implementations across various technologies.
Photon Wavelength
A photon is a particle of light, and its wavelength determines its energy. When electrons transition between different energy levels, they absorb or emit photons of particular wavelengths. The energy \( \Delta E \) associated with these photons is related to their wavelength \( \lambda \) as follows:
  • The formula to convert energy to wavelength is: \[ \Delta E = \frac{hc}{\lambda} \ \]
  • Where \( h \) is the Planck's constant and \( c \) is the speed of light.
  • This relationship shows that shorter wavelengths correspond to higher energy photons.
Understanding this relationship helps predict the characteristics of light involved in quantum transitions, relevant both in theory and practical applications such as spectroscopy.
Electronic Transition
Electronic transitions occur when electrons move between different energy levels within an atom or molecule. In the context of a quantum box, these transitions are quantized, meaning that electrons can only jump between specific levels.
  • An electron absorbs or emits a photon to move from one energy level to another.
  • In the given problem, the transition from level \( n=1 \) to \( n=4 \) involves absorbing a photon with a certain wavelength.
  • The energy difference between these two levels determines the photon's energy and, consequently, its wavelength.
This process is foundational in understanding chemical reactions, photon emission (like how lasers work), and the electronic properties of materials. It underscores the interplay between light and matter at a microscopic scale.

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Most popular questions from this chapter

Normalization of the Wave Function. Consider a particle moving in one dimension, which we shall call the \(x\) -axis. (a) What does it mean for the wave function of this particle to be normalized? (b) Is the wave function \(\psi(x)=e^{a x},\) where \(a\) is a positive real number, normalized? Could this be a valid wave function? (c) If the particle described by the wave function \(\psi(x)=A e^{b x},\) where \(A\) and \(b\) are positive real numbers, is confined to the range \(x \geq 0\) , determine \(A\) (including its units) so that the wave function is normalized.

Consider a potential well defined as \(U(x)=\infty\) for \(x < 0, U(x)=0\) for \(0 < x < L,\) and \(U(x)=U_{0} > 0\) for \(x > L\) (Fig. \(\mathrm{P} 40.70 ) .\) Consider a particle with mass \(m\) and kinetic energy \(E < U_{0}\) that is trapped in the well. (a) The boundary condition at the infinite wall \((x=0)\) is \(\psi(0)=0 .\) What must the form of the function \(\psi(x)\) for \(0 < x < L\) be in order to satisfy both the Schrodinger equation and this boundary condition (b) The wave function must remain finite as \(x \rightarrow \infty .\) What must the form of the function \(\psi(x)\) for \(x>L\) be in order to satisfy both the Schrodinger equation and this boundary condition at infinity? (c) Impose the boundary conditions that \(\psi\) and \(d \psi / d x\) are continuous at \(x=L .\) Show that the energies of the allowed levels are obtained from solutions of the equation \(k \cot k L=-\kappa,\) where \(k=\sqrt{2 m E} / \hbar\) and \(\kappa=\sqrt{2 m\left(U_{0}-E\right) / \hbar}\)

(a) The wave nature of particles results in the quantum-mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls. Use this to show that an electron confined in a one-dimensional box of length \(L\) will have energy levels given by $$E_{n}=\frac{n^{2} h^{2}}{8 m L^{2}}$$ (Hint: Recall that the relationship between the de Broglie wave-length and the speed of a nonrelativistic particle is \(m v=h / \lambda\) . The energy of the particle is \(\frac{1}{2} m v^{2} . )\) (b) If a hydrogen atom is modeled as a one- dimensional box with length equal to the Bohr radius, what is the energy (in electron volts) of the lowest energy level of the electron?

Consider a particle in a box with rigid walls at \(x=0\) and \(x=L .\) Let the particle be in the ground level. Calculate the probability \(|\psi|^{2} d x\) that the particle will be found in the interval \(x\) to \(x+d x\) for \((\) a \() x=L / 4 ;(\) b) \(x=L / 2 ;(\) c) \(x=3 L / 4\)

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.
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