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Ch 40: One-Dimensional Quantum Mechanics
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblem 1b
Chapter 40, Problem 1b

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. How long is the box in which the electrons are confined?

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Step 1: Understand the problem. The electrons are confined in a rigid box, which is a quantum mechanical system described by the particle-in-a-box model. The energy levels of the electrons are quantized, and the transition between these levels emits photons. The wavelength of the emitted photon is given as 1484 nm, and we need to determine the length of the box.
Step 2: Recall the energy difference formula for the particle-in-a-box model. The energy levels are given by \( E_n = \frac{n^2 h^2}{8mL^2} \), where \( n \) is the quantum number, \( h \) is Planck's constant, \( m \) is the mass of the electron, and \( L \) is the length of the box. The energy difference between levels \( n_3 \) and \( n_2 \) is \( \Delta E = E_3 - E_2 \).
Step 3: Relate the energy difference to the wavelength of the emitted photon. The energy of the photon is given by \( E_{photon} = \frac{hc}{\lambda} \), where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength of the photon. Set \( \Delta E = E_{photon} \) to find the relationship between the box length \( L \) and the wavelength.
Step 4: Substitute the expressions for \( E_3 \) and \( E_2 \) into \( \Delta E \). This gives \( \Delta E = \frac{9h^2}{8mL^2} - \frac{4h^2}{8mL^2} = \frac{5h^2}{8mL^2} \). Equate this to \( \frac{hc}{\lambda} \) and solve for \( L \): \( L = \sqrt{\frac{5h\lambda}{8mc}} \).
Step 5: Plug in the known values for \( h \) (Planck's constant), \( \lambda \) (1484 nm converted to meters), \( m \) (mass of the electron), and \( c \) (speed of light) into the formula \( L = \sqrt{\frac{5h\lambda}{8mc}} \) to calculate the length of the box.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Quantum Mechanics

Quantum mechanics is the branch of physics that deals with the behavior of particles at the atomic and subatomic levels. It introduces concepts such as quantization of energy levels, wave-particle duality, and the uncertainty principle, which are essential for understanding how electrons transition between energy states and emit photons.
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Energy Levels and Transitions

In quantum mechanics, electrons occupy discrete energy levels within an atom or a confined space, such as a box. When an electron transitions from a higher energy level to a lower one, it emits energy in the form of a photon. The wavelength of the emitted photon is inversely related to the energy difference between these levels, which can be calculated using the formula E = hc/λ.
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Particle in a Box Model

The particle in a box model is a fundamental concept in quantum mechanics that describes a particle confined to a rigid, impenetrable box. This model helps determine the quantized energy levels of the particle based on the box's length. The relationship between the box length and the wavelength of emitted photons is crucial for calculating the dimensions of the box when given the wavelength of the emitted light.
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Related Practice
Textbook Question

A finite potential well has depth U₀ = 2.00 eV. What is the penetration distance for an electron with energy (a) 0.50 eV, (b) 1.00 eV, and (c) 1.50 eV?

Textbook Question

A 16-nm-long box has a thin partition that divides the box into a 4-nm-long section and a 12-nm-long section. An electron confined in the shorter section is in the n = 2 state. The partition is briefly withdrawn, then reinserted, leaving the electron in the longer section of the box. What is the electron’s quantum state after the partition is back in place?

Textbook Question

A helium atom is in a finite potential well. The atom’s energy is 1.0 eV below U₀. What is the atom’s penetration distance into the classically forbidden region?

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Textbook Question

The electrons in a rigid box emit photons of wavelength 1484 nm during the 3→2 transition. What kind of photons are they—infrared, visible, or ultraviolet?