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Quantum Tunneling

Quantum tunneling is a fascinating phenomenon within physics that defies classical expectations by allowing particles to pass through energy barriers they seemingly should not overcome. As part of modern physics, tunneling arises from the probabilistic nature of quantum mechanics, where particles are not confined to single trajectories. Instead, they are described by a wave function, which allows for non-zero probability of existing beyond a barrier. This non-classical behavior is deeply connected to concepts like wave-particle duality and the superposition of states.
To understand quantum tunneling, students must first explore the atomic physics foundations such as the structure of the atom and quantum numbers and electron configuration, which introduce quantized energy levels and the behavior of electrons in orbitals. In contexts like nuclear physics, tunneling explains how fission and fusion occur, as particles overcome potential barriers at the atomic nucleus level. Similarly, nuclear reactions and processes involving radioactivity and isotopes often hinge on tunneling probabilities.
In practical applications, tunneling is essential to understanding phenomena in condensed matter physics, including the operation of tunnel diodes, quantum dots, and scanning tunneling microscopes. The broader field of particle physics also relies on tunneling in understanding how fermions and bosons interact under the fundamental forces. These interactions are further modeled and refined in quantum field theory, which provides a comprehensive framework for understanding quantum-level dynamics, including tunneling effects.
Quantum tunneling is also entwined with deeper principles such as the Heisenberg’s uncertainty principle, which asserts inherent limits to how precisely one can know position and momentum simultaneously. These uncertainties enable particles to “borrow” energy momentarily to pass through barriers—a notion incompatible with classical determinism. Tunneling even plays a role in cosmological models and speculative theories, including those examined through the lens of relativity and explored statistically in statistical mechanics.
A strong grasp of tunneling encourages learners to bridge the micro- and macroscopic views of reality. From quantum entanglement and atomic behavior to subatomic dynamics within the Standard Model, tunneling offers insights not only into how nature functions but also into how modern technologies emerge from deep physical theories. Students exploring modern physics and its intersections with advanced systems will encounter tunneling as a striking example of how quantum rules defy classical boundaries.
Illustration of quantum tunneling. The image depicts a particle wave approaching an energy barrier, with part of the wave function penetrating and continuing on the other side, illustrating the tunneling effect.

Illustration of quantum tunneling. The image depicts a particle wave approaching an energy barrier, with part of the wave function penetrating and continuing on the other side, illustrating the tunneling effect.

Study Pathways for Quantum Tunneling

Quantum tunneling is best understood as part of a wider quantum mechanics pathway. The following pages help students connect tunneling with wavefunctions, wave-particle duality, superposition, uncertainty, and related quantum ideas.
Structure chart showing Quantum Tunneling within the Modern Physics and Quantum Mechanics pathway, connected to wave function, wave-particle duality, quantum superposition, uncertainty principle, and quantum entanglement topics.

Quantum Tunneling within the Quantum Mechanics pathway: this structure chart shows how tunneling connects with wavefunctions, wave-particle duality, superposition, uncertainty, and entanglement.

Modern Physics

Establishes the wider physics framework where particles, waves, probability, energy, and measurement are studied beyond classical mechanics.

Quantum Mechanics Overview

Places quantum tunneling within the broader framework of wavefunctions, probability, uncertainty, superposition, and measurement.

Quantum Tunneling

Explains how particles can pass through potential barriers because their wavefunctions can extend into regions that are classically forbidden.

Wave-Particle Duality

Explains why particles can show wave-like behaviour, allowing their probability waves to extend beyond classical boundaries.

Quantum Superposition

Shows how quantum systems can be represented as combinations of possible states before measurement.

Quantum Entanglement

Extends the study of quantum behaviour into correlated systems, where the state of one particle may be linked with another.

Quantum Tunneling: Concept and Definition

Quantum tunneling, also spelled quantum tunnelling in British English, describes how a quantum particle can pass through an energy barrier that classical physics would not allow. In classical mechanics, if a particle encounters a potential energy barrier higher than its total energy, it cannot cross the barrier. However, in the quantum world, particles behave both as particles and as waves. This duality allows particles to “tunnel” through barriers, even if they lack the energy to go over them.
This behavior arises from the wavefunction (ψ) that describes the quantum state of a particle. Unlike classical particles, which are localized, quantum particles have a probability of being found in different regions of space. When a particle approaches a potential barrier, its wavefunction doesn’t abruptly drop to zero but instead exponentially decays within the barrier. If the barrier is thin or low enough, the wavefunction can extend to the other side, giving the particle a non-zero probability of appearing beyond the barrier.

Quantum Tunneling: Mathematical Description

The mathematical explanation of quantum tunneling comes from the time-independent Schrödinger equation. For a particle moving in one dimension, the equation may be written as:
−(ℏ2 / 2m) (d2ψ(x) / dx2) + V(x)ψ(x) = Eψ(x)
In this equation:
  • ℏ is the reduced Planck constant.
  • m is the mass of the particle.
  • ψ(x) is the particle’s wavefunction.
  • V(x) is the potential energy as a function of position.
  • E is the total energy of the particle.
In classical physics, a particle with energy E cannot pass through a barrier if the barrier height V0 is greater than the particle’s energy. In other words, if:
V0 > E
then the particle should be reflected. Quantum mechanics gives a different result because the particle is described by a wavefunction, not by a perfectly localized classical point.

Wavefunction Inside the Barrier

Inside a simple rectangular barrier where V(x) = V0 and V0 > E, the wavefunction does not suddenly become zero. Instead, it takes an exponential form:
ψ(x) = A e−κx + B eκx
where:
κ = √[ 2m(V0 − E) / ℏ2 ]
The term e−κx shows that the wavefunction decays exponentially inside the barrier. This means the probability of finding the particle inside the barrier becomes smaller as the particle moves deeper into the barrier, but it does not immediately vanish.
If the barrier is thin enough, or if the particle’s energy E is close to the barrier height V0, part of the wavefunction can still extend beyond the barrier. This creates a non-zero probability that the particle will be detected on the other side.

Tunneling Probability

For a simple rectangular barrier of width L, the tunneling probability T is approximately proportional to:
T ∝ e−2κL
This expression shows why tunneling depends strongly on the barrier width and barrier height. A wider barrier makes L larger, which reduces T. A higher barrier makes V0 − E larger, which increases κ and also reduces T.
Therefore, tunneling is more likely when the barrier is thin, when the barrier is not much higher than the particle’s energy, or when the particle has a smaller mass.

Key Idea

Quantum tunneling occurs because the wavefunction does not immediately become zero inside a classically forbidden barrier. Instead, it decays exponentially, leaving a small but real probability that the particle can appear on the other side.

Quantum Tunneling: Physical Interpretation

  • Quantum tunneling is a direct result of the wave-like nature of particles in quantum mechanics. It demonstrates that particles are not confined to well-defined paths but are described by probabilities. Even when a barrier seems impenetrable by classical standards, there is a non-zero chance that the particle will appear on the other side due to its probabilistic nature.

Real-World Applications of Quantum Tunneling

Quantum tunneling is not just a theoretical curiosity; it has profound real-world applications that are integral to modern technology and natural phenomena.

Semiconductors and Electronic Devices

Quantum tunneling is vital in the design and function of modern electronic components, especially at the nanoscale.
  • Tunnel Diodes: Tunnel diodes exploit quantum tunneling for high-speed switching. These devices operate based on the tunneling of electrons across a very thin depletion layer, allowing them to function at extremely high frequencies with low power consumption.
Tunnel Diode, highlighting its heavily doped p-n junction and quantum tunneling effect

Tunnel Diode, highlighting its heavily doped p-n junction and quantum tunneling effect

  • Transistors (MOSFETs): In modern transistors, especially in MOSFETs with nanometer-scale gate oxides, electrons can tunnel through thin insulating layers, leading to leakage currents. This effect must be carefully managed in the design of smaller, faster electronic devices.
  • Flash Memory: Quantum tunneling is used in flash memory devices. Data is stored by moving electrons onto or off a floating gate through a thin insulating layer via tunneling, enabling non-volatile data storage.
  • Quantum Dot Devices: Quantum dots exploit tunneling to confine and control electron movement at the quantum scale, promising advanced applications in quantum computing and display technologies.

Nuclear Fusion in Stars

Quantum tunneling is fundamental to the process of nuclear fusion, which powers stars, including our Sun.
  • Electrostatic Repulsion: In the core of stars, positively charged hydrogen nuclei (protons) repel each other due to Coulombic (electrostatic) forces. Classically, these protons would require extremely high kinetic energy to overcome this repulsion.
  • Tunneling Effect: Quantum tunneling allows protons to penetrate the Coulomb barrier and fuse, despite not having sufficient classical energy. This fusion releases massive amounts of energy in the form of light and heat, sustaining the star’s luminosity and stability.
Proton-Proton Chain Reaction: p + p →tunneling d + e+ + νe
Without quantum tunneling, the core temperatures of stars would not be high enough to initiate fusion, and stars like the Sun would not exist.

Radioactive Decay

Quantum tunneling explains alpha decay, a type of radioactive decay.
  • Alpha Particles: In heavy nuclei, alpha particles (helium nuclei) are bound within the nucleus by the strong nuclear force. Classically, they do not have enough energy to escape the nucleus due to the potential barrier created by the nuclear and electrostatic forces.
  • Tunneling Escape: Quantum tunneling allows alpha particles to “escape” through the nuclear potential barrier, resulting in radioactive decay.
238U →tunneling 234Th + α
This phenomenon is responsible for the decay of many heavy radioactive elements.

Quantum Computing

In quantum computing, tunneling is used in certain types of qubits and quantum logic operations.
  • Quantum Annealing: Quantum tunneling is exploited in quantum annealing to find low-energy states in complex optimization problems. Devices like D-Wave systems use tunneling to overcome energy barriers and escape local minima.
  • Josephson Junctions: These are superconducting circuits where electrons tunnel through thin insulators between superconductors. They are key components in superconducting qubits, used by companies like IBM and Google in their quantum computers.
A highly advanced supercomputer data center, featuring towering server racks and quantum processing units with glowing cooling systems.

A highly advanced supercomputer data center, featuring towering server racks and quantum processing units with glowing cooling systems.

Scanning Tunneling Microscopy (STM)

STM is a powerful imaging technique that uses quantum tunneling to map surfaces at the atomic scale. A sharp conductive tip scans a material, and the tunneling current between the tip and the surface provides detailed atomic-level images.
Scanning Tunneling Microscopy (STM), depicting a sharp conductive tip scanning a surface at the atomic level using quantum tunneling

Scanning Tunneling Microscopy (STM), depicting a sharp conductive tip scanning a surface at the atomic level using quantum tunneling

Josephson Junctions and Superconductors

Superconducting quantum devices, such as Josephson junctions, rely on Cooper pairs of electrons tunneling through an insulating layer between superconductors. This principle underlies SQUIDs (Superconducting Quantum Interference Devices), which are used in medical imaging and quantum computing.
Superconducting Quantum Interference Device (SQUID) - Used for ultra-sensitive magnetic field detection

Superconducting Quantum Interference Device (SQUID) – Used for ultra-sensitive magnetic field detection

Experimental Evidence of Quantum Tunneling

Quantum tunneling has been observed and confirmed in numerous experiments:
  • Scanning Tunneling Microscope (STM): The STM uses tunneling to image surfaces at the atomic level. A sharp tip is brought extremely close to a surface, and electrons tunnel between the tip and the sample, producing a measurable current that reveals atomic structures.
  • Cold Emission of Electrons: Electrons can tunnel out of metal surfaces in the presence of strong electric fields, a phenomenon used in field electron emission devices.

Quantum Tunneling: Limitations and Challenges

While quantum tunneling enables incredible technologies, it also presents challenges:
  • Leakage Current in Microelectronics: As electronic components shrink, tunneling leads to unwanted leakage currents, increasing power consumption and heat generation.
  • Controlling Tunneling Probability: In devices where precise control is needed, managing the barrier width and height becomes technologically challenging.

Why Study Quantum Tunneling

Crossing Energy Barriers with Quantum Probability

Quantum tunneling allows particles to pass through potential barriers that would be insurmountable classically. Students learn how this behavior arises from the probabilistic nature of wave functions. It challenges intuitive ideas about energy and confinement. It demonstrates how quantum systems behave fundamentally differently from classical ones.

Mathematical Description and Barrier Analysis

Students analyze tunneling through potential wells and barriers using the time-independent Schrödinger equation. They calculate transmission probabilities and penetration depths. These exercises develop fluency in solving boundary value problems. They illustrate how abstract mathematics models real quantum behavior.

Applications in Electronics and Astrophysics

Tunneling is used in devices such as tunnel diodes, scanning tunneling microscopes, and flash memory. It also explains nuclear fusion in stars, where particles overcome Coulomb barriers via tunneling. Students see how theoretical physics applies to technology and astrophysics. It underscores the interdisciplinary impact of quantum phenomena.

Role in Alpha Decay and Quantum Processes

Alpha decay of nuclei is a classic example of quantum tunneling in nuclear physics. Students study how quantum probability governs the escape of alpha particles from potential wells. This connects tunneling with radioactive decay and half-life predictions. It reinforces understanding of nuclear structure and decay processes.

Foundational Implications for Quantum Theory

Quantum tunneling illustrates the uncertainty and non-locality intrinsic to quantum systems. Students reflect on how tunneling defies deterministic boundaries in space and energy. It encourages exploration of quantum paradoxes and advanced theoretical concepts. It reveals the richness and complexity of quantum reality.

Quantum Tunneling: Conclusion

Quantum tunneling is a striking manifestation of the non-intuitive principles of quantum mechanics, allowing particles to cross barriers they shouldn’t be able to in the classical sense. It plays a critical role in natural processes like nuclear fusion in stars and technological advancements in semiconductors, quantum computing, and nanotechnology. By bridging the gap between theoretical physics and practical applications, quantum tunneling continues to shape our understanding of the universe and drives innovation in cutting-edge technologies.

Frequently Asked Questions: Quantum Tunnelling

1. What is quantum tunnelling?

Quantum tunnelling is a phenomenon where a particle has a non-zero probability of passing through an energy barrier that it would not be able to cross according to classical physics. Because the particle is described by a spread-out wavefunction, there is a chance of finding it on the other side of the barrier, as if it had “tunnelled” through.

2. Why is tunnelling impossible in classical physics but allowed in quantum mechanics?

In classical physics, if a particle does not have enough energy to climb over a barrier, it is completely reflected. In quantum mechanics, particles behave as waves, and their wavefunctions can extend into and beyond the barrier. This penetration gives a finite probability of transmission even when the particle’s energy is less than the barrier height.

3. How is quantum tunnelling described mathematically?

Mathematically, tunnelling is described by solving the Schrödinger equation for a potential barrier. The solution shows an exponentially decaying wavefunction inside the barrier and a smaller, but non-zero, transmitted wave on the far side. The square of the transmitted wave’s amplitude gives the tunnelling probability.

4. Which factors affect the probability of tunnelling through a barrier?

The tunnelling probability depends primarily on the barrier’s height, width, and shape, as well as the particle’s mass and energy. Higher or wider barriers and heavier particles generally reduce the probability exponentially, while particles with energy closer to the barrier height tunnel more easily.

5. What are some natural examples of quantum tunnelling?

Important natural examples include alpha decay, where an alpha particle tunnels out of an atomic nucleus, and nuclear fusion in stars, where protons tunnel through their mutual electrostatic repulsion to get close enough for the strong nuclear force to bind them. Without tunnelling, many nuclear processes in the universe would be extremely improbable.

6. How is quantum tunnelling used in modern technology?

Quantum tunnelling is exploited in devices such as tunnel diodes, Josephson junctions, and scanning tunnelling microscopes (STMs). In these devices, electrons tunnel across thin barriers, enabling ultra-fast switching, highly sensitive detectors, and imaging of surfaces with atomic-scale resolution.

7. Does a particle lose energy when it tunnels through a barrier?

In an ideal, time-independent potential, a particle that tunnels through a barrier emerges with the same energy it had before encountering the barrier. The process changes the probability of where the particle can be found, but it does not require the particle to “spend” energy to pass through.

8. How is quantum tunnelling related to the uncertainty principle?

Quantum tunnelling is consistent with the uncertainty principle because a particle’s position and momentum cannot both be known with arbitrary precision. The spread in momentum and energy implied by the uncertainty principle allows a non-zero chance that the particle is found on the far side of a barrier, even when its average energy is below the barrier height.

9. Can large or macroscopic objects tunnel through barriers?

In principle, quantum mechanics applies to all objects, but for macroscopic objects the tunnelling probability becomes unimaginably small because of their large mass and size. As a result, tunnelling of everyday objects is effectively impossible, and classical physics remains an excellent approximation at human scales.

10. How do physicists observe and study quantum tunnelling experimentally?

Physicists study tunnelling by measuring currents through thin insulating barriers, observing radioactive decays, or using techniques like scanning tunnelling microscopy. By varying barrier thickness, voltage, or other parameters and comparing data with theoretical predictions, they confirm the characteristic exponential dependence of tunnelling probability.

11. What is a tunnelling time, and is it meaningful to ask how long tunnelling takes?

The question of how long tunnelling takes is subtle in quantum mechanics because time is not an operator like position or momentum. Various definitions of tunnelling time exist, and experiments suggest that tunnelling does not involve travelling through the barrier in a classical sense. Instead, the theory focuses on probabilities rather than a simple transit time.

12. Why is learning about quantum tunnelling useful for students?

Learning about quantum tunnelling helps students see how quantum rules lead to phenomena with no classical counterpart, and how abstract ideas like wavefunctions and probability amplitudes have concrete consequences in nature and technology. It also provides a bridge between conceptual quantum ideas and practical applications in electronics, nuclear physics, and astrophysics.

Quantum Tunneling: Review Questions and Answers

1. What is quantum tunneling?
Answer: Quantum tunneling is a phenomenon in which a particle has a non-zero probability of passing through an energy barrier that it would not be able to cross according to classical physics.

2. How does the wavefunction contribute to quantum tunneling?
Answer: The wavefunction describes the probability amplitude of a particle’s position. Even in a region where the particle’s energy is lower than the barrier height, the wavefunction may still have a non-zero value, allowing a finite probability of tunneling.

3. What factors affect the probability of tunneling?
Answer: Tunneling probability depends on the barrier’s width, the barrier’s height, the particle’s mass, and the particle’s energy. Tunneling becomes more likely when the barrier is thinner, lower, or when the particle is lighter and has energy closer to the barrier height.

4. How is quantum tunneling used in modern electronics?
Answer: Quantum tunneling is used in devices such as tunnel diodes, flash memory, and nanoscale electronic components. In these systems, electrons tunnel through very thin barriers, allowing current flow or charge storage to be controlled.

5. What role does quantum tunneling play in nuclear fusion?
Answer: In nuclear fusion, tunneling allows atomic nuclei to get through the electrostatic repulsion barrier more often than classical physics would predict. This helps explain how fusion can occur in stars at temperatures lower than a purely classical calculation would require.

6. How can the tunneling effect be mathematically described?
Answer: Tunneling is described by solving the Schrödinger equation for a potential barrier. Inside the classically forbidden region, the wavefunction decays exponentially rather than immediately becoming zero.

7. What is the significance of the transmission coefficient in quantum tunneling?
Answer: The transmission coefficient represents the probability that a particle will pass through a barrier. It is used to quantify how likely tunneling is for a given particle and barrier.

8. How does quantum tunneling challenge classical mechanics?
Answer: Classical mechanics predicts that a particle without enough energy cannot cross a barrier. Quantum tunneling shows that microscopic particles can still have a probability of appearing on the other side because they are described by wavefunctions.

9. In what experimental contexts has quantum tunneling been observed?
Answer: Quantum tunneling appears in scanning tunneling microscopy, radioactive decay, semiconductor devices, tunnel diodes, Josephson junctions, and other quantum systems.

10. What are some potential future applications of quantum tunneling?
Answer: Future applications may include improved quantum devices, advanced sensors, lower-power electronics, nanoscale components, and quantum technologies that depend on controlled tunneling behaviour.

Quantum Tunneling: Thought-Provoking Questions and Answers

1. How might a deeper understanding of quantum tunneling reshape our approach to energy generation?
Answer: A better understanding of tunneling could improve models of nuclear fusion, materials design, and energy conversion. However, tunneling alone does not automatically make practical energy generation easy. It must be combined with engineering, plasma control, materials science, and efficient device design.

2. What implications does quantum tunneling have for our understanding of the limits of classical physics?
Answer: Quantum tunneling shows that classical ideas about barriers and motion are incomplete at microscopic scales. A particle is not simply a tiny classical object with a fixed path; it is described by a wavefunction that can extend into classically forbidden regions.

3. How could improvements in controlling tunneling rates impact semiconductor technology?
Answer: Better control of tunneling could improve transistors, memory devices, nanoscale circuits, and low-power electronic components. At very small scales, tunneling can be either useful or problematic, depending on whether engineers want to exploit it or suppress unwanted leakage.

4. In what ways might quantum tunneling influence the design of future quantum computers?
Answer: Some quantum computing platforms involve controlled tunneling between quantum states or across barriers. Understanding tunneling can help improve qubit control, stability, coherence, and device architecture.

5. Could quantum tunneling be exploited to create new types of sensors?
Answer: Yes. Tunneling-based sensors can detect extremely small changes in distance, current, field strength, or surface structure. Scanning tunneling microscopy is already a major example of tunneling used for high-resolution measurement.

6. How does quantum tunneling affect our theoretical understanding of chemical reactions?
Answer: Tunneling can allow particles such as electrons or protons to pass through energy barriers during reactions. This can affect reaction rates, especially at low temperatures, and is important in chemistry, catalysis, and molecular physics.

7. What are the potential consequences of quantum tunneling for matter in extreme environments?
Answer: In stars and other extreme environments, tunneling can influence nuclear reactions and the behaviour of dense matter. For example, tunneling helps fusion occur in stellar cores even when nuclei do not classically have enough energy to overcome repulsion directly.

8. How might quantum tunneling be applied to improve solar cells?
Answer: Tunneling can influence charge transport across thin layers and junctions in advanced solar-cell designs. If controlled well, it may help reduce losses or improve carrier movement, although practical improvements depend on materials and device structure.

9. In what ways does quantum tunneling contribute to radioactive decay?
Answer: In alpha decay, an alpha particle can escape the nucleus by tunneling through the nuclear potential barrier. This explains why decay can occur even when the alpha particle does not have enough classical energy to climb over the barrier.

10. How do researchers use tunneling phenomena to test quantum mechanics?
Answer: Researchers compare measured tunneling currents, decay rates, and transmission probabilities with predictions from quantum theory. Agreement between experiment and theory supports the quantum description of particles as wave-like systems.

11. What challenges do scientists face when trying to observe quantum tunneling in macroscopic systems?
Answer: Macroscopic systems interact strongly with their environment, causing decoherence. This makes it difficult to preserve the quantum behaviour needed to observe tunneling-like effects at larger scales.

12. How could simulation and computational modelling improve our understanding of tunneling?
Answer: Simulations can help estimate tunneling rates, model quantum barriers, design nanoscale devices, and study systems that are difficult to measure directly. Better modelling can support both theoretical understanding and practical device development.

Numerical Problems and Solutions

1. Calculate the energy of a photon with wavelength 450 nm.

Solution:
E = (hc) / λ
Using h = 4.1357 × 10−15 eV·s, c = 3.0 × 108 m/s, and λ = 450 × 10−9 m:
E = (4.1357 × 10−15 × 3.0 × 108) / (450 × 10−9)
E ≈ 2.76 eV
Answer: The photon energy is approximately 2.76 eV.

2. Estimate the tunneling probability for an electron through a barrier of width 0.5 nm, with U − E = 2.0 eV.

Solution:
κ = √[ 2m(U − E) ] / ℏ
For an electron and U − E = 2.0 eV:
κ ≈ 7.24 × 109 m−1
T ≈ e−2κL
T ≈ e−2(7.24 × 109)(0.5 × 10−9) = e−7.24 ≈ 7.15 × 10−4
Answer: The estimated tunneling probability is approximately 7.15 × 10−4, or about 0.0715%.

3. An electron with energy 1.5 eV approaches a barrier of 3.0 eV. Calculate κ.

Solution:
U − E = 3.0 eV − 1.5 eV = 1.5 eV = 2.403 × 10−19 J
κ = √[ 2m(U − E) ] / ℏ
κ ≈ 6.27 × 109 m−1
Answer: κ ≈ 6.27 × 109 m−1.

4. Find the barrier width if the tunneling probability is 0.1 and κ = 7.0 × 108 m−1.

Solution:
T = e−2κL  ⇒  L = ln(1 / T) / (2κ)
L = ln(10) / [ 2(7.0 × 108) ] = 2.3026 / (1.4 × 109)
L ≈ 1.65 × 10−9 m
Answer: The barrier width is approximately 1.65 nm.

5. In an STM, estimate the reduction factor in current when distance increases from 0.4 nm to 0.5 nm, if κ = 1.0 × 1010 m−1.

Solution:
ΔL = 0.5 nm − 0.4 nm = 0.1 nm = 0.1 × 10−9 m
Reduction factor = e−2κ ΔL = e−2(1.0 × 1010)(0.1 × 10−9) = e−2 ≈ 0.135
Answer: The current becomes about 0.135 of its original value, or about 13.5%.

6. Estimate the electron energy required for tunneling probability approximately 0.5 through a 4.0 eV barrier.

Solution:
This problem cannot be solved uniquely from the barrier height alone. The tunneling probability also depends on the barrier width, the particle mass, and the detailed shape of the barrier.
For a simple barrier, the estimate depends on:
T ≈ e−2κL    where    κ = √[ 2m(U − E) ] / ℏ
Answer: More information is needed. A tunneling probability near 0.5 would generally require a thin barrier and an energy E close to the barrier height U = 4.0 eV.

7. A particle has momentum uncertainty 1.0 × 10−25 kg·m/s. Estimate the position uncertainty.

Solution:
Using the uncertainty relation:
Δx ≥ ℏ / (2 Δp)
Δx ≥ (1.055 × 10−34) / [ 2(1.0 × 10−25) ] = 5.28 × 10−10 m
Answer: The position uncertainty is at least 5.28 × 10−10 m.

8. Calculate the tunneling probability for a barrier width of 0.8 nm and κ = 8.0 × 108 m−1.

Solution:
T = e−2κL = e−2(8.0 × 108)(0.8 × 10−9) = e−1.28 ≈ 0.278
Answer: The tunneling probability is approximately 0.278, or 27.8%.

9. In an STM, the current drops by a factor of 20 when the distance increases by 0.2 nm. Estimate κ.

Solution:
A drop by a factor of 20 means e−2κ ΔL = 1/20.
Taking natural logarithms:
2κ ΔL = ln(20)  ⇒  κ = ln(20) / (2 ΔL)
κ = ln(20) / [ 2(0.2 × 10−9) ] ≈ 7.49 × 109 m−1
Answer: κ ≈ 7.5 × 109 m−1.

10. If the tunneling probability is 0.01, determine the value of 2κL.

Solution:
T = e−2κL = 0.01  ⇒  2κL = ln(100) ≈ 4.61
Answer: 2κL ≈ 4.61.

11. A semiconductor has effective electron mass 0.1m0 and barrier height 0.8 eV. Estimate κ.

Solution:
κ = √[ 2 m* U ] / ℏ
Using m* = 0.1 m0 and U = 0.8 eV:
κ ≈ 1.45 × 109 m−1
Answer: κ ≈ 1.45 × 109 m−1.

12. If increasing the barrier width by 0.1 nm reduces current by 30%, estimate κ.

Solution:
A 30% reduction means the current becomes 70% of its original value:
0.70 = e−2κ ΔL
Taking natural logarithms:
κ = ln(1 / 0.70) / (2 ΔL) = ln(1.4286) / [ 2(0.1 × 10−9) ] ≈ 1.78 × 109 m−1
Answer: κ ≈ 1.78 × 109 m−1.
Last updated: 27 Jul 2026