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

문제 32a

40장, 문제 32a

A particle confined in a rigid one-dimensional box of length 10 fm has an energy level E_n = 32.9 MeV and an adjacent energy level E_n+1 = 51.4 MeV. Determine the values of n and n+1.

검증된 단계별 안내

Step 1: Recall the formula for the energy levels of a particle in a one-dimensional box:

Eₙ = (n²h²)/(8mL²)

, where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

Step 2: Write the formula for the difference between adjacent energy levels:

ΔE = Eₙ+₁ - Eₙ

. Substitute the given values for

Eₙ = 32.9 \, \(\text{MeV}\)

and

Eₙ+₁ = 51.4 \, \(\text{MeV}\)

to find

ΔE = 18.5 \, \(\text{MeV}\)

Step 3: Express the difference in energy levels using the formula:

ΔE = \(\frac{h²}{8mL²}\)((n+1)² - n²)

. Expand the term

(n+1)²

to get

n² + 2n + 1

, and simplify the expression to

ΔE = \(\frac{h²}{8mL²}\)(2n + 1)

Step 4: Rearrange the formula to solve for n:

n = \(\frac{ΔE \cdot 8mL²}{2h²}\) - \(\frac{1}{2}\)

. Substitute the known values for

ΔE = 18.5 \, \(\text{MeV}\)

L = 10 \, \(\text{fm}\)

, and the constants h and m (Planck's constant and the mass of the particle, typically the mass of an electron or proton depending on the context).

Step 5: Once n is determined, calculate

n+1

by simply adding 1 to the value of n. Verify that the calculated energy levels match the given values of

Eₙ

and

Eₙ+₁

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주요 개념

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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, where particles can only occupy specific energy states. This framework is essential for understanding phenomena like the energy levels of a particle in a potential well, such as a one-dimensional box.

Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics problem that describes a particle confined to a one-dimensional region with infinitely high potential walls. The energy levels of the particle are quantized and can be calculated using the formula Eₙ = n²h²/(8mL²), where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box. This model helps in determining the allowed energy states of the particle.

Energy Level Calculation

In quantum mechanics, energy levels are calculated based on the quantum number associated with the state of the particle. For a particle in a box, the energy levels are proportional to the square of the quantum number (n²). By knowing the energy of adjacent levels, one can derive the quantum numbers n and n+1, which correspond to the specific energy states of the particle in the box.