Ch 40: One-Dimensional Quantum Mechanics

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 40: One-Dimensional Quantum Mechanics

Problem 17b

Chapter 40, Problem 17b

INT An electron is confined in a harmonic potential well that has a spring constant of 2.0 N/m. What wavelength photon is emitted if the electron undergoes a 3→1 quantum jump?

Verified step by step guidance

Step 1: Understand the problem. The electron is confined in a harmonic potential well, which means its energy levels are quantized. The energy levels for a quantum harmonic oscillator are given by the formula:

E

_n = (n + 1/2)ℏω, where

n

is the quantum number,

ℏ

is the reduced Planck's constant, and

ω

is the angular frequency of the oscillator.

Step 2: Calculate the angular frequency

ω

. The angular frequency is related to the spring constant

k

and the mass

m

of the electron by the formula:

ω = √(k/m)

. Use the given spring constant

k = 2.0 \, \(\text{N/m}\)

and the mass of the electron

m = 9.11 \(\times\) 10^{-31} \, \(\text{kg}\)

to find

ω

Step 3: Determine the energy difference between the quantum levels

n = 3

and

n = 1

. Using the formula for energy levels, calculate

E

₃ and

E

₁, then find the energy difference:

ΔE = E

₃ - E₁.

Step 4: Relate the energy difference to the wavelength of the emitted photon. The energy of a photon is given by

E = h \(\cdot\) c / λ

, where

h

is Planck's constant,

c

is the speed of light, and

λ

is the wavelength. Rearrange this formula to solve for

λ

λ = h \(\cdot\) c / ΔE

Step 5: Substitute the values for Planck's constant

h = 6.63 \(\times\) 10^{-34} \, \(\text{J·s}\)

, the speed of light

c = 3.00 \(\times\) 10^{8} \, \(\text{m/s}\)

, and the calculated energy difference

ΔE

into the formula to find the wavelength

λ

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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. In this context, the electron's energy levels in the harmonic potential well are quantized, meaning it can only occupy specific energy states.

Harmonic Potential Well

A harmonic potential well is a model used to describe a particle subject to a restoring force proportional to its displacement from an equilibrium position, akin to a mass on a spring. The energy levels of a particle in this potential are quantized and can be calculated using the formula E_n = (n + 1/2)ħω, where n is the quantum number, ħ is the reduced Planck's constant, and ω is the angular frequency related to the spring constant.

Photon Emission and Wavelength

When an electron transitions between energy levels, it can emit or absorb a photon, with the energy of the photon corresponding to the difference in energy between the two levels. The wavelength of the emitted photon can be calculated using the equation E = hc/λ, where E is the energy difference, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship allows us to determine the wavelength of the photon emitted during the electron's quantum jump.

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Related Practice

Textbook Question

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Textbook Question

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