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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: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 40: One-Dimensional Quantum MechanicsProblem 48a
Chapter 40, Problem 48a

What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a STM probe if the work function is 4.0 eV?

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Step 1: Understand the concept of quantum tunneling. Quantum tunneling is a phenomenon where particles like electrons can pass through a potential barrier even if their energy is less than the height of the barrier. The probability of tunneling depends on the width of the barrier, the energy of the particle, and the height of the potential barrier (work function in this case).
Step 2: Use the formula for tunneling probability. The probability of tunneling (P) can be approximated using the equation: P = e - k , where k is the decay constant. The decay constant is given by: k = 2 m ħ ² ( U - E ) , where m is the mass of the electron, ħ is the reduced Planck's constant, U is the potential barrier (work function), and E is the energy of the electron.
Step 3: Substitute the given values into the decay constant formula. The width of the barrier is 0.50 nm (convert to meters: 0.50 × 10 -9 m), the work function U is 4.0 eV (convert to joules: 4.0 × 1.6 -19 J), and the mass of the electron is 9.11 × 10 -31 kg. Use these values to calculate k .
Step 4: Calculate the tunneling probability using the formula P = e - k × d , where d is the width of the barrier (0.50 nm). Substitute the calculated value of k and d into the equation.
Step 5: Interpret the result. The tunneling probability will be a very small number, indicating that the likelihood of the electron tunneling through the air gap is extremely low. This aligns with the quantum mechanical nature of tunneling, which becomes significant only for very small barriers or specific conditions.

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Key Concepts

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

Quantum Tunneling

Quantum tunneling is a quantum mechanical phenomenon where a particle can pass through a potential energy barrier, even if it does not have enough energy to overcome that barrier classically. This occurs due to the wave-like nature of particles at the quantum level, allowing for a non-zero probability of finding the particle on the other side of the barrier.

Work Function

The work function is the minimum energy required to remove an electron from the surface of a material. It is a critical parameter in understanding electron emission processes, such as photoemission and thermionic emission, and plays a significant role in determining the likelihood of tunneling events in quantum mechanics.
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Potential Barrier

A potential barrier is a region in space where the potential energy of a particle is higher than its total energy, effectively preventing the particle from passing through it under classical physics. In quantum mechanics, however, particles can tunnel through these barriers, and the height and width of the barrier, along with the particle's energy, influence the tunneling probability.
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Related Practice
Textbook Question

In a nuclear physics experiment, a proton is fired toward a Z = 13 nucleus with the diameter and neutron energy levels shown in Figure 40.17. The nucleus, which was initially in its ground state, subsequently emits a gamma ray with wavelength 1.73×10−4 nm. What was the minimum initial speed of the proton? Hint: Don't neglect the proton-nucleus collision.

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Textbook Question

Figure 40.17 showed that a typical nuclear radius is 4.0 fm. As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is En = −20 MeV. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

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Textbook Question

If a 10% current change is reliably detectable, what is the smallest height change the STM can detect?

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Textbook Question

The probe passes over an atom that is 0.050 nm “tall.” By what factor does the tunneling current increase?

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Textbook Question

A proton’s energy is 1.0 MeV below the top of a 10-fm-wide energy barrier. What is the probability that the proton will tunnel through the barrier?

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