An electron is in a box of width 3.0*10^-10 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?

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Understand the concept of quantization in a box: An electron in a box is a model where the electron is confined to a one-dimensional region with fixed boundaries. The energy levels are quantized, meaning the electron can only occupy certain discrete energy states.

Use the formula for the de Broglie wavelength: The de Broglie wavelength $ \lambda $ of a particle is given by $ \lambda = \frac{h}{p} $, where $ h $ is Planck's constant and $ p $ is the momentum of the particle.

Determine the momentum of the electron: The momentum $ p $ of an electron in a box is quantized and can be expressed as $ p = \frac{n h}{2L} $, where $ n $ is the quantum number, $ h $ is Planck's constant, and $ L $ is the width of the box.

Calculate the de Broglie wavelength for each energy level: Substitute the expression for momentum into the de Broglie wavelength formula to find $ \lambda = \frac{2L}{n} $. Calculate $ \lambda $ for $ n = 1 $, $ n = 2 $, and $ n = 3 $.

Compare the wavelength to the width of the box: For each energy level, compare the calculated de Broglie wavelength $ \lambda $ to the width of the box $ L = 3.0 \times 10^{-10} $ m to understand how the wavelength changes with different quantum levels.

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Key Concepts

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

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics that describes the wave-like nature of particles. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant, and p is the momentum of the particle. This concept is crucial for understanding how particles like electrons exhibit both particle and wave characteristics, especially in confined systems like a box.

Particle in a Box Model

The particle in a box model is a quantum mechanical system where a particle is confined to move within a perfectly rigid and impenetrable box. The energy levels of the particle are quantized, meaning the particle can only occupy certain discrete energy states. The energy levels are determined by the quantum number n, and the wave function solutions are sinusoidal, reflecting the standing wave nature of the particle within the box.

Quantum Numbers and Energy Levels

Quantum numbers are integers that describe the quantized energy levels of a system. In the context of a particle in a box, the principal quantum number n determines the energy level and the corresponding wave function of the particle. Higher quantum numbers correspond to higher energy levels and shorter wavelengths, as the particle's wave function must fit more nodes within the box's fixed width.

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A free particle moving in one dimension has wave function ψ(x,t) = A[e^i(kx-ωt) -e^i(2kx-4ωt)] where k and v are positive real constants. (c) Calculate v_av as the distance the maxima have moved divided by the elapsed time.

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Textbook Question

An electron with initial kinetic energy $5.0$ eV encounters a barrier with height $U sub 0$ and width $0.60$ nm. What is the transmission coefficient if (a) $U sub 0 equals 7.0$ eV; (b) $U sub 0 equals 9.0$ eV; (c) $U sub 0 equals 13.0$ eV?

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Textbook Question

While undergoing a transition from the $n equals 1$ to the $n equals 2$ energy level, a harmonic oscillator absorbs a photon of wavelength $6.50$ $μ$ m. What is the wavelength of the absorbed photon when this oscillator undergoes a transition (a) from the $n equals 2$ to the $n equals 3$ energy level and (b) from the $n equals 1$ to the $n equals 3$ energy level?

(c) What is the value of $\sqrt{(k^{'} / m)}$ , the angular oscillation frequency of the corresponding Newtonian oscillator?

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