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Chapter 40: Problem 27

A proton is bound in a square well of width \(4.0 \mathrm{fm}=\) \(4.0 \times 10^{-15} \mathrm{~m} .\) The depth of the well is six times the ground-level energy \(E_{1-\text { IDW }}\) of the corresponding infinite well. If the proton makes a transition from the level with energy \(E_{1}\) to the level with energy \(E_{3}\) by absorbing a photon, find the wavelength of the photon.

Short Answer

Expert verified
After putting the values through calculations, you will get the wavelength of the photon, which is the final answer. This is an advanced quantum mechanics problem that uses fundamental concepts like energy levels and states, Planck's equation, and the principle of transitioning between energy states by photon absorption in a quantum system.

Step by step solution

01

Ground Level Energy for Infinite Potential Well

The ground level energy of a proton in an infinite potential well can be calculated using the formula: \[E_{1-IDW}=\frac{h^2}{8mL^2}\], where \(h\) is the Planck's constant (\(h = 6.63 \times 10^{-34}\) Js), \(m\) is the mass of the proton (\(m=1.67 \times 10^{-27}\) kg), and \(L\), the width of the well, is \(4.0 \times 10^{-15}\) m.
02

Energy Levels in the Square Well

The depths of square wells are often given in terms of the ground level energy of the infinite well, so we use this information to find the energy of the well. The depth of the well is six times the ground level energy of the infinite well: therefore, if \(E_{1-IDW}=E\), then \(E_{1}=6E\) and \(E_{3}=9E\). In other words, \(E_{1}=6 \times E_{1-IDW}\), and \(E_{3}=9 \times E_{1-IDW}\).
03

Energy Difference

The energy difference between the first and the third level is given by: \(螖E = E_{3} - E_{1}\). Substitute the energy values into this equation to find the difference.
04

Wavelength of the Photon

Use the energy difference and Planck鈥檚 equation (\(E = \frac{hc}{位}\)) to find the wavelength of the photon. Here, \(c\) is the speed of light (\(c = 3.0 \times 10^8\) m/s). Rearrange the equation to solve for the wavelength of the photon: \(位 = \frac{hc}{螖E}.\)

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Key Concepts

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

Infinite Potential Well
An infinite potential well is a fundamental concept in quantum mechanics that describes a particle that is "trapped" in a box, experiencing potential energy that is zero inside the box and infinitely high outside. This signifies that the particle cannot escape and is confined to the region inside the well. Such a model allows us to explore quantum behavior in a simplified manner.

The dimensions of the well, such as its width, dictate the range within which the particle is confined. Usually, the width is denoted by \(L\), and this determines the possible solutions to the Schr枚dinger equation, which are the energy levels that the particle can occupy.
  • The particle's wave function, which tells us the probability of finding the particle at a given position, must vanish at the walls of the infinite potential well.
  • This results in quantized energy levels, meaning the particle can only have certain discrete energies.
  • As the width of the well changes, so do the energy levels.
Energy Levels
In the context of quantum mechanics, energy levels refer to the specific energies that a particle can occupy when trapped in a potential well. For an infinite potential well, these levels are quantized, meaning they are distinct and non-continuous.

The ground state energy, which is the lowest energy level, can be determined using the formula:\[E_{n}=\frac{h^2n^2}{8mL^2}\]where \(n\) is a positive integer called the principal quantum number, \(h\) is Planck鈥檚 constant, \(m\) is the mass of the particle, and \(L\) is the width of the well.
  • Energy levels increase as the quantum number \(n\) increases, leading to higher energy states.
  • The deeper the well, the more energy levels there are.
  • Transitions between these levels involve gaining or losing energy in the form of photons.
Photon Wavelength
Photon wavelength relates to the color of light in visible terms and is determined by the energy of the photon itself. This energy is connected to transitions between energy levels in quantum systems.

When an electron transitions from one energy level to another in an atom or particle in a potential well, it either absorbs or emits energy in the form of a photon. The relationship between energy and wavelength is given by:\[E = \frac{hc}{\lambda}\]where \(E\) is the energy of the photon, \(h\) is Planck鈥檚 constant, \(c\) is the speed of light in a vacuum, and \(\lambda\) is the photon wavelength.
  • This equation implies that energy and wavelength are inversely proportional; higher energy photons have shorter wavelengths.
  • Visible light ranges from about 400 nm (for violet) to 700 nm (for red), with each color corresponding to a different energy and wavelength.
  • In quantum mechanics, observing the emitted photon wavelength can help deduce the difference in energy levels.
Quantum Transitions
Quantum transitions refer to the movement of particles between different energy levels in a quantum system. These transitions are crucial as they explain phenomena such as spectral lines in atomic spectra and light emission.

In an infinite potential well, when a particle like a proton moves from a lower energy level to a higher one, it absorbs energy. This energy is supplied by a photon whose wavelength can be calculated using known equations for energy and light.
  • To transition upwards (absorb energy), a particle requires an external energy input, typically from a photon.
  • Conversely, when falling to a lower energy level, a particle emits energy, often visible as light.
  • The precise transition energies depend on the initial and final states' energy levels.
  • These quantum transitions form the basis of laser technology, various spectroscopy techniques, and more.

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Most popular questions from this chapter

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.)

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 particle moving in one dimension (the \(x\) -axis) is described by the wave function $$ \psi(x)=\left\\{\begin{array}{ll} A e^{-b x}, & \text { for } x \geq 0 \\ A e^{b x}, & \text { for } x<0 \end{array}\right. $$ where \(b=2.00 \mathrm{~m}^{-1}, A>0,\) and the \(+x\) -axis points toward the right. (a) Determine \(A\) so that the wave function is normalized. (b) Sketch the graph of the wave function. (c) Find the probability of finding this particle in each of the following regions: (i) within \(50.0 \mathrm{~cm}\) of the origin, (ii) on the left side of the origin (can you first guess the answer by looking at the graph of the wave function?), (iii) between \(x=0.500 \mathrm{~m}\) and \(x=1.00 \mathrm{~m}\).

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?

An electron is bound in a square well that has a depth equal to six times the ground-level energy \(E_{1-\mathrm{IDW}}\) of an infinite well of the same width. The longest-wavelength photon that is absorbed by this electron has a wavelength of \(582 \mathrm{nm}\). Determine the width of the well.
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