Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Quantum mechanics → Schrödinger Equation ↓


Quantum Tunneling


Quantum tunneling is a fundamental concept in quantum mechanics. It allows particles to pass through potential barriers, even when classically they do not have enough energy to do so. This phenomenon is well described by the Schrödinger equation, which is a key element of quantum mechanics.

Schrödinger equation

The Schrödinger equation is a differential equation that describes how the quantum state of a physical system changes over time. In its time-independent form, it is expressed as:

Hψ = Eψ

Where:

  • H is the Hamiltonian operator, which represents the total energy of the system.
  • ψ is the wave function of the system, which provides information about the probability amplitude.
  • E is the energy eigenvalue associated with the wave function.

Tunneling across the barrier

To understand quantum tunneling, consider a particle that encounters a one-dimensional potential barrier. Classically, if the energy of the particle is less than the potential height of the barrier, the particle cannot pass through it. However, in quantum mechanics, due to its wave nature, there is a possibility for the particle to pass through the barrier even in this case.

Let's visualize this concept:

Particle approach Potential barrier through particle tunnels

Mathematical description of quantum tunneling

For a deeper understanding, let us consider the potential as a rectangular barrier of height V and width a For a particle with total energy E < V , the probabilities of reflection and transmission through the barrier can be calculated using the Schrödinger equation.

Zone 1: Before the obstacle

Here, the potential energy U(x) = 0 , and the Schrödinger equation becomes:

-ħ²/2m * d²ψ/dx² = Eψ

The solutions of this equation are plane waves, which are given as:

ψ₁(x) = A e^(ikx) + B e^(-ikx)

Where:

  • k = sqrt(2mE/ħ²)
  • A and B are coefficients representing the amplitudes of the wave traveling in the positive and negative directions, respectively.

Area 2: Inside the barrier

For the constraint 0 < x < a , the potential energy U(x) = V , and the Schrödinger equation becomes:

-ħ²/2m * d²ψ/dx² = (EV)ψ

Here the solution involves exponential decay because E < V :

ψ₂(x) = C e^(κx) + D e^(-κx)

Where:

  • κ = sqrt(2m(VE)/ħ²)
  • C and D are constants determined by the marginal conditions.

Area 3: Beyond the barrier

As in region 1, the solution again takes the form of free travelling waves:

ψ₃(x) = F e^(ikx)

where F is a constant. In normal tunneling problems, conditions are set such that no wave escapes this region.

Continuity and boundary conditions

Wave functions and their derivatives must be continuous across the boundaries:

  • At x = 0 : ψ₁(0) = ψ₂(0) and (dψ₁/dx) at 0 = (dψ₂/dx) at 0
  • At x = a : ψ₂(a) = ψ₃(a) and (dψ₂/dx) at a = (dψ₃/dx) at a

Solving these conditions allows us to find a relation between the coefficients A , F describing the incident and transmitted wave amplitudes. The transmission coefficient T , which indicates the probability of the particle passing through, is given by:

T = |F/A|²

This coefficient provides a quantitative measure of the tunnel effect.

Examples of quantum tunneling in nature

Alpha decay

In nuclear physics, alpha decay is an example of quantum tunneling. An alpha particle trapped inside the nucleus crosses the nuclear potential barrier to escape. This process cannot be explained through classical mechanics, because the energy of the alpha particle is less than the height of the barrier.

Fusion in stars

In the stellar core, protons undergo nuclear fusion by overcoming electrostatic repulsion barriers. Quantum tunneling allows this process to occur even at temperatures lower than classically required.

Visualization of quantum tunneling

Imagine lighting a flashlight with a low battery. Traditionally, if we think of light as particles, they cannot pass through a thick, opaque barrier. However, with quantum tunneling, there is a small chance that some particles pass through, and a faint light appears on the other side.

Potential barrier Emitted light (tunnel example)

Intuitive understanding of quantum tunneling

To understand quantum tunneling, it is helpful to consider wave-particle duality in quantum mechanics. Particles behave as waves with specific acceptor probabilities, meaning that there is a non-zero probability that they can appear on the other side of the barrier, even if this is theoretically impossible according to classical standards.

Consider the mathematical metaphor of a narrow footbridge crossing a river. Conventionally, without sufficient kinetic energy or thrust, one cannot cross the bridge. But if you were a quantum particle with a wave-like nature, you could find yourself surprisingly quickly on the other side due to your "spread out" probability distribution.

Conclusion

Quantum tunneling demonstrates the strangeness and beauty of the quantum world. Through the Schrödinger equation, quantum mechanics not only predicts this amazing behaviour, but also provides the mathematical framework for understanding and calculating tunnelling probabilities. It affects a variety of natural processes and has important implications in technologies such as semiconductors and tunnelling microscopes.

Understanding and embracing the story of quantum tunneling encourages a shift from classical intuition, and reveals a universe that is more subtle and interconnected than conventional physics allows.


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Quantum Tunneling

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