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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: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblema 11
Capitolo 40, Problema 11

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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Step 1: Understand the concept of quantum tunneling. In quantum mechanics, particles such as atoms can penetrate into regions where their energy is less than the potential energy barrier (U₀). This is known as the classically forbidden region.
Step 2: Recall the formula for the penetration distance (decay length) in the classically forbidden region: \( \lambda = \frac{\hbar}{\sqrt{2m(U_0 - E)}} \), where \( \hbar \) is the reduced Planck's constant, \( m \) is the mass of the particle, \( U_0 \) is the potential energy barrier, and \( E \) is the energy of the particle.
Step 3: Identify the given values from the problem: \( U_0 - E = 1.0 \, \text{eV} \). Convert this energy difference into joules using the conversion factor \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \).
Step 4: Determine the mass of the helium atom. The mass of a helium atom is approximately \( 4.0026 \, \text{u} \), where \( 1 \, \text{u} = 1.6605 \times 10^{-27} \, \text{kg} \). Multiply these values to find the mass in kilograms.
Step 5: Substitute the values for \( \hbar \) (\( 1.054 \times 10^{-34} \, \text{J·s} \)), \( m \), and \( U_0 - E \) into the formula for \( \lambda \). Perform the necessary calculations to find the penetration distance.

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Finite Potential Well

A finite potential well is a region where a particle experiences a lower potential energy compared to its surroundings. In quantum mechanics, this well can confine particles like electrons or atoms, allowing them to exist in discrete energy states. The depth and width of the well determine the energy levels and the behavior of particles within it, including their ability to penetrate into classically forbidden regions.
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Classically Forbidden Region

The classically forbidden region refers to areas where a particle's energy is less than the potential energy, making it classically impossible for the particle to exist there. However, due to quantum tunneling, particles can penetrate these regions with a certain probability. The extent of this penetration is influenced by the energy difference between the particle and the potential barrier, as well as the characteristics of the potential well.
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Quantum Tunneling

Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential barrier that it classically should not be able to surmount. This occurs due to the wave-like nature of particles, allowing them to exist in a superposition of states. The probability of tunneling is related to the energy of the particle and the width and height of the barrier, which can be calculated using the Schrödinger equation.
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