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Chapter 40: Problem 30

An electron is bound in a square well with a depth equal to six times the ground-level energy \(E_{1-\mathrm{LDW}}\) of an infinite well of the same width. The longest-wavelength photon that is absorbed by the electron has a wavelength of 400.0 \(\mathrm{nm} .\) Determine the width of the well.

Short Answer

Expert verified
Use the relation \(E_{\text{photon}} = \frac{3h^2}{8mL^2}\) to solve for \(L\) using given \(\lambda = 400 nm\) and constants.

Step by step solution

01

Identify Given Variables

We have a square well of depth equal to 6 times the ground-level energy of an infinite well of the same width. Additionally, the wavelength of the absorbed photon is 400.0 nm. Let the width of the well be \(L\) and its depth be \(V_0\). The energy of the infinite well is \(E_{1- ext{LDW}}\). The relationship given is \(V_0 = 6E_{1- ext{LDW}}\).
02

Energy Level Relation

For an infinite square well, the ground-level energy is given by \(E_{1- ext{LDW}} = \frac{h^2}{8mL^2}\), where \(h\) is Planck's constant and \(m\) is the electron mass.
03

Photon Energy Calculation

The energy of a photon with a wavelength \(\lambda\) is given by \(E_{\text{photon}} = \frac{hc}{\lambda}\), where \(c\) is the speed of light. Substitute \(\lambda = 400 nm = 400 \times 10^{-9} m\) to find \(E_{\text{photon}}\).
04

Compare Photon Energy with Transition

The photon is absorbed when the electron transitions from a lower energy level to a higher energy level in the well. Therefore, the energy of the photon \(E_{\text{photon}}\) corresponds to the difference in energy levels of the electron in the well. Since this is the longest-wavelength photon absorbed, this transition is likely to occur from the ground state to the first excited state.
05

Express the Difference in Energy Levels

The energy difference \(\Delta E\) between the first and the ground state levels in a finite well can be approximately analyzed as \(\Delta E = E_2 - E_1 \approx E_{\text{photon}}\).
06

Express Energy Levels in Terms of L

Express the energies \(E_1\) and \(E_2\) using the terms of energies in the infinite square well as approximations, where \(E_n = \frac{n^2h^2}{8mL^2}\). In a real finite well, these energies will be slightly adjusted due to the potential \(V_0\), but can be an initial reference for approximations, thus \(E_2 - E_1 = \frac{3h^2}{8mL^2}\).
07

Equate Photon Energy to Energy Level Difference

Set \(E_{\text{photon}} = \frac{3h^2}{8mL^2}\) and solve for \(L\). Use \(E_{\text{photon}} = \frac{hc}{400 \times 10^{-9}}\).
08

Calculate Well Width

Calculate \(L\) by substituting known values (Planck's constant \(h\), electron mass \(m\), and speed of light \(c\)) into the equation for \(L\).

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Key Concepts

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

Infinite Square Well
In quantum mechanics, the infinite square well is a fundamental concept used to study particle confinement. Imagine a particle, often an electron, trapped in a narrow box or well where the walls hold infinite height and the bottom is infinitely deep. This idealization makes calculations more straightforward, guiding students through foundational quantum theories without too much complexity.

  • The width of this well, denoted as \( L \), is significant because it dictates the spatial boundaries within which our particle must exist.
  • Since there's no probability of the particle existing outside the well, it can't "escape", balancing on quantum principles.
  • Understanding this concept helps in visualizing how particles, such as electrons, behave at a quantum level.
The infinite square well also offers insight into how energy levels stack up for particles: they are "quantized", meaning particles can only occupy specific levels.
Energy Levels
In any given well, when dealing with electrons, the concept of energy levels becomes crucial. These levels are quantized, which means that only certain energy values are allowed for an electron in the well.

  • For an infinite square well, the ground-level energy \( E_{1} \) is defined mathematically as \( E_{1} = \frac{h^2}{8mL^2} \), where \( h \) is Planck's constant and \( m \) is the mass of the electron.
  • Electrons can transition between these levels but require a specific amount of energy to do so.
  • The energy difference between levels, such as between \( E_1 \) (ground state) and \( E_2 \) (first excited state), can often be triggered by external influences like photon absorption.
Identifying the energy difference between levels helps in determining the characteristics of electron movement and behavior in a quantum mechanical context.
Photon Absorption
Photon absorption is an event where an electron within a well absorbs a photon, or a particle of light, which carries energy. This additional energy allows the electron to move to a higher energy level.

  • The photon's energy is represented by the equation \( E_{\text{photon}} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon.
  • The energy from the photon must match the energy difference required for the electron to transition from one quantized state to another within the well.
  • In this context, the longest wavelength photon absorbed prompts the smallest possible transition, usually from the ground state to the first excited state.
Photon absorption not only illustrates the electromagnetic interaction but also helps calculate essential characteristics of the system, such as the well width.
Electron Transitions
Electron transitions within a quantum well entail an electron moving from one energy level to another. This can occur naturally or through external catalysts like photon interactions.

  • These transitions are essential as they link photon absorption processes to electron movement within and between energy levels.
  • In our exercise, the absorbed photon facilitates a transition from the lowest energy level \( E_1 \) to a higher one \( E_2 \).
  • The energy required for this transition is equivalent to the photon's energy, reinforced by the energy equation \( \Delta E = E_2 - E_1 \).
Electron transitions not only explain how and why electrons move within a quantum system but also help in calculating system parameters like the potential well's width.

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Most popular questions from this chapter

When a hydrogen atom undergoes a transition from the \(n=2\) to the \(n=1\) level, a photon with \(\lambda=122 \mathrm{nm}\) is emitted. (a) If the atom is modeled as an electron in a one-dimensional box, what is the width of the box in order for the \(n=2\) to \(n=1\) transition to correspond to emission of a photon of this energy? (b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain. (Hint: Compare the spacing between adjacent energy levels as a function of \(n . )\)

A proton is in a box of width \(L .\) What must the width of the box be for the ground-level energy to be \(5.0 \mathrm{MeV},\) a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus- that is, on the order of \(10^{-14} \mathrm{m} .\)

Chemists use infrared absorption spectra to identify chemicals in a sample. In one sample, a chemist finds that light of wavelength 5.8\(\mu \mathrm{m}\) is absorbed when a molecule makes a transition from its ground harmonic oscillator level to its first excited level. (a) Find the energy of this transition. (b) If the molecule can be treated as a harmonic oscillator with mass \(5.6 \times 10^{-26} \mathrm{kg},\) find the force constant.

A particle with mass \(m\) and total energy \(E\) tunnels through a square barrier of height \(U_{0}\) and width \(L .\) When the trans- mission coefficient is not much less than unity, it is given by $$T=\left[1+\frac{\left(U_{0} \sinh \kappa L\right)^{2}}{4 E\left(U_{0}-E\right)}\right]^{-1}$$ where \(\sinh \kappa L=\left(e^{\kappa L}-e^{-\kappa L}\right) / 2\) is the hyperbolic sine of \(\kappa L .\) (a) Show that if \(\kappa L \gg 1\) , this expression for \(T\) approaches Eq. \((40.42) .\) (b) Explain why the restriction \(\kappa L>1\) in part (a) implies either that the barrier is relatively wide or the energy \(E\) is relatively low compared to \(U_{0 .}\) (c) Show that as the particle's incident kinetic energy \(E\) approaches the barrier height \(U_{0}\) . \(T\) approaches \(\left[1+(k L / 2)^{2}\right]^{-1},\) where \(k=\sqrt{2 m E / \hbar}\) is the wave number of the incident particle. (Hint: If \(|z|<1,\) then \(\sinh z \approx z . )\)

Wave functions like the one in Problem 40.50 can represent free particles moving with velocity \(v=p / m\) in the \(x\) -direction. Consider a beam of such particles incident on a potential-energy step \(U(x)=0,\) for \(x < 0,\) and \(U(x)=U_{0} < E,\) for \(x > 0 .\) The wave function for \(x < 0\) is \(\psi(x)=A e^{i k_{1} x}+B e^{-i k_{1} x}\) representing incident and reflected particles, and for \(x > 0\) is \(\psi(x)=C e^{i k_{2} x},\) representing transmitted particles. Use the conditions that both \(\psi\) and its first derivative must be continuous at \(x=0\) to find the constants \(B\) and \(C\) in terms of \(k_{1}, k_{2},\) and \(A .\)
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