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Chapter 40: Problem 59

In your research on new solid-state devices, you are studying a solid-state structure that can be modeled accurately as an electron in a one-dimensional infinite potential well (box) of width \(L\). In one of your experiments, electromagnetic radiation is absorbed in transitions in which the initial state is the \(n=1\) ground state. You measure that light of frequency \(f=9.0 \times 10^{14} \mathrm{~Hz}\) is absorbed and that the next higher absorbed frequency is \(16.9 \times 10^{14} \mathrm{~Hz}\). (a) What is quantum number \(n\) for the final state in each of the transitions that leads to the absorption of photons of these frequencies? (b) What is the width \(L\) of the potential well? (c) What is the longest wavelength in air of light that can be absorbed by an electron if it is initially in the \(n=1\) state?

Short Answer

Expert verified
The final quantum numbers for the transitions are approximately \(n = 2\) and \(n = 3\). The width of the potential well is \(1.78 \times 10^{-9}\) m. The longest wavelength that can be absorbed by an electron initially in the \(n = 1\) state is \(8.29 \times 10^{-7}\) m.

Step by step solution

01

Calculate Final Quantum Numbers

We can calculate the final quantum numbers using the relationship between the energy difference \(\Delta E\) and the frequency of the light, \(f\), absorbed by the electron. \(\Delta E = h \cdot f\), where \(h\) is Planck's constant. The energy levels in the infinite potential well are given by \(E_n = n^2 \cdot E_1\), where \(E_1\) is the energy of the ground state (n=1) and \(n\) is the quantum number of the excited state. Therefore, \(\Delta E = E_n - E_1 = (n^2 - 1) \cdot E_1\). Solving these two expressions for equal \(\Delta E\), we obtain \(n^2 - 1 = f \cdot h / E_1\). We need to compute this for \(f = 9.0 \times 10^{14} Hz\) and \(f = 16.9 \times 10^{14} Hz\). Obtaining \(n\) from the square root of 1 plus the above expression will yield the final quantum numbers.
02

Compute Width of Potential Well

We should start by calculating the energy of the ground state \(E_1\) from the first transition (from \(n=1\) to the \(n\) found in Step 1, with frequency \(f=9.0 \times 10^{14} Hz\)). We find this from the formula \(\Delta E = E_n - E_1 = h \cdot f\) we used in Step 1, so \(E_1 = E_n - h \cdot f\). With \(E_1 = (h^2 \cdot n^2) / (8ml^2)\), where \(m\) is the mass of an electron and \(l\) is the width of the well, we can calculate \(l\) (or L) from the equation by substituting known values (planck's constant \(h\), the electron mass \(m\) and the energy level \(E_1\) found above).
03

Find Longest Wavelength of Absorbed Light

The longest wavelength of light that can be absorbed by the electron corresponds to the smallest difference in energy level. This difference is between the ground state at \(n=1\) and the first level at \(n=2\).We already calculated \(E_1\) in Step 2, so we can use \(E_2 = 2^2 \cdot E_1\) to find \(E_2\). The energy difference is then \(\Delta E = E_2 - E_1 = 3 \cdot E_1\). We can find the frequency \(f\) of this transition from \(\Delta E = h \cdot f\), and then find the wavelength from \(f = c / \lambda\), where \(c\) is the speed of light. Thus the required wavelength is \(\lambda = c / f\).

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Key Concepts

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

Quantum Number
The concept of the quantum number is foundational to understanding the behavior of electrons within an atom or, as in our example problem, within a solid-state structure envisioned as an infinite potential well. In this scenario, the quantum number, denoted by the symbol 'n', represents an integer value that quantifies the energy level of an electron. The energy levels available to an electron are not continuous; they are discrete, like rungs on a ladder that the electron can 'step' on.

The quantum number is crucial because it determines the energy state of the electron: The higher the quantum number, the higher the energy level. In an infinite potential well, energy of an electron at a certain level 'n' is proportional to the square of the quantum number, mathematically expressed as \( E_n = n^2 \times E_1 \), where \( E_1 \) is the energy of the electron at the base level when \( n=1 \). This relationship is key in predicting the behavior of electrons when energy, such as that from absorbed photons, is added or subtracted from the system.
Electron Energy Levels
Diving deeper into electron energy levels, especially in the context of an infinite potential well, allows us to predict how an electron will behave when it absorbs or emits energy. In quantum mechanics, these energy levels are represented by the allowed states an electron can have within the system. Just like the quantum number, the energy levels are not arbitrary; they are defined by the physical constraints of the system, which in our case is the width of the potential well, symbolized by 'L'.

The energy levels are spaced out in such a way that each level is increasingly higher in energy than the previous one. This is important because when an electron transitions between these levels, it must either absorb or release a precise amount of energy equal to the difference in energy between the two levels. In mathematical terms, the difference in energy (\( \Delta E \)) when an electron transitions from one level with quantum number 'n' to another level is given by the formula \( E_n - E_1 \), with \( E_1 \) being the first energy state. This detailed understanding of energy levels is essential when calculating transitions, as seen in our original exercise.
Absorption of Photons
The absorption of photons is essentially how electrons gain energy to move to a higher energy level. A photon is a particle representing a quantum of light, and when it collides with an electron, it can transfer its energy to that electron. For a photon to be absorbed, the energy it carries must match exactly the energy gap between the electron鈥檚 current energy level and the next available higher level.

In the context of our problem, when light of a certain frequency is absorbed, the electron in the infinite potential well transitions between energy levels 鈥� the difference in energy before and after the absorption corresponds to the electron moving from a lower quantum number to a higher quantum number state. The relationship between the frequency of the absorbing light (represented by 'f') and the energy levels can be described using Planck's equation: \( E = h \times f \), where 'E' is the energy of the photon, 'h' is Planck's constant, and 'f' is the frequency of the light. This equation says that the energy contained in a photon is directly proportional to its frequency, which allows scientists to predict and understand which photons will be absorbed by an electron based on its energy requirements to reach a higher state.

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Most popular questions from this chapter

An electron is in a box of width \(3.0 \times 10^{-10} \mathrm{~m} .\) What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in (a) the \(n=1\) level; (b) the \(n=2\) level; (c) the \(n=3\) level? In each case how does the wavelength compare to the width of the box?

A particle with mass \(m\) is in a one-dimensional box with width \(L\). If the energy of the particle is \(9 \pi^{2} \hbar^{2} / 2 m L^{2},\) (a) what is the linear momentum of the particle and (b) what is the ratio of the width of the box to the de Broglie wavelength \(\lambda\) of the particle?

The penetration distance \(\eta\) in a finite potential well is the distance at which the wave function has decreased to \(1 / e\) of the (b) wave function at the classical turning point: $$ \psi(x=L+\eta)=\frac{1}{e} \psi(L) $$ The penetration distance can be shown to be $$\eta=\frac{\hbar}{\sqrt{2 m\left(U_{0}-E\right)}}$$ The probability of finding the particle beyond the penetration distance is nearly zero. (a) Find \(\eta\) for an electron having a kinetic energy of \(13 \mathrm{eV}\) in a potential well with \(U_{0}=20 \mathrm{eV} .\) (b) Find \(\eta\) for a \(20.0 \mathrm{MeV}\) proton trapped in a 30.0 -MeV-deep potential well.

A particle is confined to move on a circle with radius but is otherwise free. We can parameterize points on this circle by using the distance \(x\) from a reference point or by using the angle \(\theta=x / R .\) since \(x=0\) and \(x=2 \pi R\) describe the same point, the wave function must satisfy \(\psi(x)=\psi(x+2 \pi R)\) and \(\psi^{\prime}(x)=\psi^{\prime}(x+2 \pi R)\) (a) Solve the free-particle time-independent Schr枚dinger equation subject to these boundary conditions. You should find solutions \(\psi_{n}^{\pm}\), where \(n\) is a positive integer and where lower \(n\) corresponds to lower energy. Express your solutions in terms of \(\theta\) using unspecified normalization constants \(A_{n}^{+}\) and \(A_{n}^{-}\) corresponding, respectively, to modes that move "counterclockwise" toward higher \(x\) and "clockwise" toward lower \(x\). (b) Normalize these functions to determine \(A_{n}^{\pm}\). (c) What are the energy levels \(E_{n} ?\) (d) Write the time-dependent wave functions \(\Psi_{n}^{\pm}(x, t)\) corresponding to \(\Psi_{n}^{\pm}(x) .\) Use the symbol \(\omega\) for \(E_{1} / \hbar .\) (e) Consider the nonstationary state defined at \(t=0\) by \(\Psi(x, 0)=\frac{1}{\sqrt{2}}\left[\Psi_{1}^{+}(x)+\Psi_{2}^{+}(x)\right] .\) Determine the probability density \(|\Psi(x, t)|^{2}\) in terms of \(R, \omega,\) and \(t .\) Simplify your result using the identity \(1+\cos \alpha=2 \cos ^{2}(\alpha / 2) .\) (f) With what angular speed does the density peak move around the circle? (g) If the particle is an electron and the radius is the Bohr radius, then with what speed does its probability peak move?

40.42 - Hydrogen emits radiation with four prominent visible wavelengths - one red, one cyan, one blue, and one violet. The respective frequencies are \(656 \mathrm{nm}, 486 \mathrm{nm}, 434 \mathrm{nm},\) and \(410 \mathrm{nm} .\) We can model the hydrogen atom as an electron in a one-dimensional box, and attempt to match four adjacent emission lines in the predicted spectrum to the visible part of the hydrogen spectrum. (a) Determine the photon energies associated with the visible part of the hydrogen spectrum. (b) The electron-in-a-box emission spectrum is \(E_{n_{i} \rightarrow n_{\mathrm{f}}}=\left(n_{\mathrm{i}}^{2}-n_{\mathrm{f}}^{2}\right) \epsilon\) where \(n_{\mathrm{i}}\) and \(n_{\mathrm{f}}\) are the initial and final quantum numbers of the electron when it drops to a lower energy level and \(\epsilon\) is the energy determined by the Schr枚dinger equation. What is the smallest possible value of \(n_{\mathrm{i}}\) that can accommodate four emission lines? (c) Using the value from part (b) for \(n_{i}\), estimate the order of magnitude of \(\epsilon\) by dividing the four photon energies by the relevant differences \(n_{\mathrm{i}}^{2}-n_{\mathrm{f}}^{2}\) for transitions in the possible sets of \(\left(n_{i}, n_{f}\right)\) pairings. (d) Using Eq. (40.31) to identify \(\epsilon\), and using the mass of the electron, use your result from part (c) to estimate the length \(L\) of the box. (e) What is the ratio of your estimate of \(L\) to twice the Bohr radius? (Note: The hydrogen atom is better modeled using the Coulomb potential rather than as a particle in a box.)
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