Skip to main content
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 9
Capitolo 40, Problema 9

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?

Guida verificata passo dopo passo
1
Step 1: Understand the concept of penetration distance. In quantum mechanics, penetration distance refers to the distance a particle's wavefunction extends into a classically forbidden region (where the particle's energy is less than the potential energy). This is described by the exponential decay of the wavefunction in the forbidden region.
Step 2: Write the expression for the penetration distance. The penetration distance \( \lambda \) is given by \( \lambda = \frac{1}{\kappa} \), where \( \kappa \) is the decay constant. The decay constant \( \kappa \) is defined as \( \kappa = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}} \), where \( m \) is the mass of the particle, \( U_0 \) is the potential well depth, \( E \) is the energy of the particle, and \( \hbar \) is the reduced Planck's constant.
Step 3: Substitute the known values into the formula for \( \kappa \). For an electron, the mass \( m \) is approximately \( 9.11 \times 10^{-31} \; \text{kg} \), and \( \hbar \) is approximately \( 1.055 \times 10^{-34} \; \text{J·s} \). Convert the given energies \( U_0 \) and \( E \) from electron volts (eV) to joules (J) using the conversion factor \( 1 \; \text{eV} = 1.602 \times 10^{-19} \; \text{J} \).
Step 4: Calculate \( \kappa \) for each case: (a) \( E = 0.50 \; \text{eV} \), (b) \( E = 1.00 \; \text{eV} \), and (c) \( E = 1.50 \; \text{eV} \). Use the formula \( \kappa = \sqrt{\frac{2m(U_0 - E)}{\hbar^2}} \) and substitute the corresponding values of \( U_0 \) and \( E \) for each case.
Step 5: Once \( \kappa \) is determined for each case, calculate the penetration distance \( \lambda \) using \( \lambda = \frac{1}{\kappa} \). This will give the penetration distance for the electron in each scenario.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Finite Potential Well

A finite potential well is a region in quantum mechanics where a particle experiences a potential energy lower than its surroundings. The well has a finite depth (U₀) and width, allowing for bound states where particles can exist with energies less than U₀. Outside the well, the potential energy is higher, leading to a phenomenon known as quantum tunneling, where particles can penetrate into classically forbidden regions.
Video consigliato:
Percorso guidato
07:59
Magnetic Field due to Finite Wire

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 have a non-zero probability of being found in regions where their energy is less than the potential energy. The penetration distance into the barrier is influenced by the energy of the particle and the height of the barrier.

Penetration Distance

The penetration distance refers to how far a particle can travel into a potential barrier before its probability of being found drops significantly. It is determined by the energy of the particle relative to the potential barrier height. For an electron in a finite potential well, the penetration distance increases as the energy of the electron approaches the depth of the well, allowing for a greater likelihood of finding the electron within the barrier.
Video consigliato:
Percorso guidato
06:13
Displacement vs. Distance
Pratica correlata
Domanda del libro di testo

The graph in FIGURE EX40.15 shows the potential-energy function U(x) of a particle. Solution of the Schrödinger equation finds that the n = 3 level has E3 = 0.5 eV and that the n = 6 level has E6 = 2.0 eV. Redraw this figure and add to it the energy lines for the n = 3 and n = 6 states.

112
views
Domanda del libro di testo

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?

49
views
Domanda del libro di testo

Sketch the n = 8 wave function for the potential energy shown in FIGURE EX40.13.

118
views
Domanda del libro di testo

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?

39
views
Domanda del libro di testo

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?

83
views
Domanda del libro di testo

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?

60
views