Quantum Tunneling

One-Dimensional Infinitely Rigid Box: Energy Eigenvalues and Eigenfunctions, Normalization

The one-dimensional infinitely rigid box, also known as the particle in a box model, is a fundamental concept in quantum mechanics that illustrates the quantization of energy levels. This system provides a foundation for understanding more complex quantum systems.

Introduction to the One-Dimensional Infinitely Rigid Box

The one-dimensional infinitely rigid box consists of a particle of mass mmm confined to a region 0≤x≤L , where L is the length of the box. The potential energy function for this system is defined as:

This means the particle is completely free inside the box but cannot exist outside due to infinitely high potential walls.

Time-Independent Schrödinger Equation for the System

The quantum state of the particle is described by the time-independent Schrödinger equation:

Since V(x)=0 inside the box (0≤x≤L), the equation simplifies to:

Rearranging:

Where

This is a standard second-order differential equation with the general solution:

where A and B are constants determined by boundary conditions.

Applying Boundary Conditions

Since the potential at x=0 and x=L is infinite, the wave function must vanish at these points:

1. At x=0: ψ(0)=0
   
   So, the wave function simplifies to:
   
2. At x=L: ψ(L)=0

Since A≠0(otherwise, the wave function would be zero everywhere), we require:
sin⁡(kL)=0
This is satisfied when:
kL=nπ,n=1,2,3,…
Solving for k:
k=nπ/L

Energy Eigenvalues

The energy levels can be found using:

​

Substituting k=nπ/L​:

This result shows that energy levels are quantized, meaning the particle cannot have arbitrary energy values but only discrete levels.

Energy Eigenfunctions

Using k=nπ/L​, the normalized wave function (eigenfunction) is:

Normalization Condition

To find A, we use the normalization condition:

Using the integral result:

​

we get:

Solving for A:

Thus, the normalized eigenfunction is:

Key Observations

1. Quantized Energy Levels: The allowed energies En depend on n^2, meaning higher energy levels are farther apart.
   
2. Wave Function Properties: The wave function has n−1 nodes (zero crossings) for each state.
   
3. No Ground State at Zero Energy: The lowest energy state (n=1) has energy ​, meaning the particle always has some energy (due to the Heisenberg Uncertainty Principle).
   
4. Probability Density: The probability of finding the particle varies with xxx and is highest at antinodes of ψn​(x).
   

Conclusion

The one-dimensional infinitely rigid box is a simple yet powerful model that illustrates the fundamental principles of quantum mechanics, including wave function quantization, discrete energy levels, and normalization. These principles play a crucial role in understanding atomic and molecular behavior in quantum physics.

Quantum Dots

Quantum dots (QDs) are nanoscale semiconductor particles that exhibit unique quantum mechanical properties. They are sometimes called artificial atoms because their energy levels are discrete, much like those of individual atoms. Quantum dots have found extensive applications in optoelectronics, biomedical imaging, quantum computing, and solar energy.

What are Quantum Dots?

Quantum dots are tiny semiconductor particles, typically ranging from 2 to 10 nanometers in diameter. Due to their small size, they exhibit size-dependent optical and electronic properties, a direct consequence of quantum confinement.

• Material Composition:
  
  • Most quantum dots are made from semiconductors such as CdSe (Cadmium Selenide), CdTe (Cadmium Telluride), InP (Indium Phosphide), ZnS (Zinc Sulfide), and PbS (Lead Sulfide).
    
  • They are often coated with another material to enhance stability and reduce toxicity.
    

Quantum Confinement Effect

In a bulk semiconductor, electrons move freely, but in a quantum dot, the movement of charge carriers (electrons and holes) is restricted in all three spatial dimensions. This is known as quantum confinement, leading to discrete energy levels rather than continuous bands.

• When the dot size decreases, the bandgap increases, meaning smaller dots emit higher-energy (bluer) light, while larger dots emit lower-energy (redder) light.
  
• The energy levels in a quantum dot are analogous to those in a particle in a box, where the energy is inversely proportional to the square of the size of the dot.
  

Mathematically, the energy of an electron confined in a quantum dot is given by:

where:

• n is the quantum number,
  
• h is Planck’s constant,
  
• m is the effective mass of the electron,
  
• L is the quantum dot size.
  

Energy Band Gap and Emission Spectrum

• Tunable Emission: The color (wavelength) of emitted light depends on the dot size.
  
  • Smaller QDs → Higher energy (Blue light)
    
  • Larger QDs → Lower energy (Red light)
    
• This tunability makes quantum dots extremely useful in display technologies and imaging.
  

Quantum Dot Size	Emission Color
2 nm	Blue (High energy)
4 nm	Green
6 nm	Red (Low energy)

Quantum Dots as an Example of a One-Dimensional Infinite Potential Well

Quantum dots can be modeled using the one-dimensional infinite potential well (rigid box) concept in quantum mechanics.

• The quantum dot traps an electron in a confined space.
  
• The energy levels are quantized due to the small size.
  
• The wave function of the electron inside the quantum dot resembles that of a particle in a box.
  

By solving Schrödinger’s equation for an electron trapped in a quantum dot, we get:

where:

• n represents different energy levels,
  
• L is the dot size.
  

This equation shows that as the size L decreases, the energy E increases, explaining why smaller quantum dots emit higher-energy light (blue shift).

Fabrication of Quantum Dots

Quantum dots are synthesized using several methods:

1. Colloidal Synthesis
   
   • Most common method.
     
   • Chemical reaction in a solution to produce nanoparticles.
     
   • High control over dot size and properties.
     
2. Epitaxial Growth (Molecular Beam Epitaxy – MBE, Metal-Organic Chemical Vapor Deposition – MOCVD)
   
   • Used for semiconductor quantum dots.
     
   • Involves layer-by-layer deposition of materials.
     
3. Electrochemical Methods
   
   • Used for self-assembly of quantum dots on a substrate.
     
4. Lithographic Techniques
   
   • Uses nano-patterning to create controlled QDs.
     

Applications of Quantum Dots

Due to their size-dependent optical and electronic properties, quantum dots are used in various fields:

Display Technologies (Quantum Dot Displays – QLEDs)

• Used in TVs, monitors, and smartphone displays.
  
• Provide high color accuracy and energy efficiency.
  
• Samsung, LG, and Sony use quantum dots in QLED TVs.
  

Biomedical Imaging & Drug Delivery

• Fluorescent quantum dots are used as biomarkers for imaging inside the body.
  
• Can track cancer cells, drug delivery, and gene expression.
  
• More stable and brighter than traditional organic dyes.
  

Quantum Computing

• QDs can act as qubits, the fundamental unit of a quantum computer.
  
• Quantum dots in silicon are being explored for scalable quantum computers.
  

Solar Cells & Renewable Energy

• Quantum dot solar cells (QDSCs) can capture wider solar spectrum than traditional silicon-based cells.
  
• High potential for next-generation photovoltaic devices.
  

Photodetectors and Sensors

• Used in low-light imaging, infrared sensors, and security applications.
  

Medical Therapy

• Near-Infrared (NIR) emitting quantum dots are used in deep-tissue imaging.
  
• Potential use in targeted drug therapy.
  

Advantages of Quantum Dots

 Size-Dependent Properties → Tunable light emission
  High Brightness & Stability → Brighter than organic dyes
  Wide Color Gamut → Useful in displays and imaging
  Quantum Efficiency → High efficiency in energy applications
  Scalability → Can be mass-produced using chemical synthesis

Challenges and Limitations

 Toxicity → Heavy-metal-based quantum dots (e.g., CdSe) can be harmful. Research is ongoing for eco-friendly alternatives (e.g., Perovskite QDs).
  Stability Issues → Some quantum dots degrade over time.
  Manufacturing Complexity → Precise size control is challenging in large-scale production.

Conclusion

Quantum dots are a revolutionary advancement in nanotechnology with vast applications in electronics, medicine, and renewable energy. Their unique quantum confinement effects, tunable optical properties, and energy efficiency make them an essential component of future technologies. The particle-in-a-box model provides a fundamental theoretical understanding of their behavior, illustrating how quantum mechanics governs nanoscale materials.

Quantum Mechanical Tunneling in a One-Dimensional Rectangular Potential Barrier

Quantum mechanical tunneling is a fascinating phenomenon where particles penetrate through a potential barrier even when they classically do not have enough energy to overcome it. This effect is a direct consequence of the wave-like nature of particles in quantum mechanics, as described by Schrödinger’s equation.

What is Quantum Mechanical Tunneling?

In classical physics, if a particle does not have sufficient energy to overcome a barrier, it is completely reflected. However, quantum mechanics allows the particle to have a finite probability of passing through the barrier, even if its energy is less than the barrier height. This phenomenon is called quantum tunneling.

• The probability of tunneling depends on barrier width and height.
  
• Smaller barriers (narrow and lower barriers) allow for higher tunneling probability.
  

Quantum tunneling has crucial applications in semiconductor physics, nuclear fusion, and scanning tunneling microscopy.

One-Dimensional Rectangular Potential Barrier

The most common model used to understand quantum tunneling is the one-dimensional rectangular potential barrier. It is defined as:

where:

• V0​ is the potential barrier height.
  
• L is the barrier width.
  
• A particle with energy E approaches the barrier from Region I.
  

If E>V0​ ​, the particle can pass over the barrier (classical motion).
If E<V0​ ​, classically, the particle would be reflected, but quantum mechanically, it has a probability of tunneling through.

Schrödinger Equation for Tunneling

The Schrödinger equation governs the motion of the particle in each region:

(A) Region I: Before the Barrier (x < 0)

Solution:

where

• The first term represents a wave moving towards the barrier.
  
• The second term represents a reflected wave.
  

(B) Region II: Inside the Barrier (0 ≤ x ≤ L)

Solution:

where

• The wave decays exponentially inside the barrier, indicating the evanescent wave.
  
• The particle has a nonzero probability of being inside the barrier.
  

(C) Region III: After the Barrier (x > L)

Solution:

• There is only a transmitted wave, meaning a fraction of the wave has tunneled through.
  

Transmission and Reflection Coefficients

The probability of tunneling is given by the Transmission Coefficient T, which is derived from continuity conditions at the boundaries.

• When L (barrier width) increases, T decreases exponentially.
  
• When V0​ (barrier height) increases, T decreases.
  
• If E is close to V0​, tunneling is more likely.
  

Similarly, the Reflection Coefficient RRR is given by:

R=1−T

which represents the probability of the particle being reflected.

Factors Affecting Tunneling Probability

Several factors influence how much tunneling occurs:

1. Barrier Height (V0​)
   
   • Higher V0​ → Lower tunneling probability.
     
   • If V0​≫E, tunneling is almost impossible.
     
2. Barrier Width (L)
   
   • Wider barrier → Lower tunneling probability.
     
   • If L→0, tunneling becomes very likely.
     
3. Particle Energy (E)
   
   • Higher E → Higher tunneling probability.
     
   • If E≈V0​​, tunneling is significant.
     

Applications of Quantum Tunneling

Quantum tunneling plays a vital role in many technological and natural processes.

(A) Scanning Tunneling Microscope (STM)

• Uses tunneling of electrons to image surfaces at the atomic level.
  
• A sharp metallic tip is brought close to a surface → Electrons tunnel between tip and surface.
  
• Extremely high-resolution imaging of nanomaterials.
  

(B) Semiconductor Devices (Tunnel Diodes)

• Tunnel diodes use quantum tunneling for high-speed switching.
  
• Found in high-frequency oscillators, amplifiers, and memory cells.
  

(C) Nuclear Fusion in Stars

• Fusion reactions in the Sun rely on tunneling.
  
• Protons classically do not have enough energy to overcome the Coulomb barrier, but they tunnel through, allowing fusion.
  

(D) Quantum Computing

• Quantum bits (qubits) use tunneling for superposition and entanglement.
  
• Josephson junctions in superconductors rely on tunneling effects.
  

(E) DNA Mutations

• Tunneling of protons in DNA molecules can lead to spontaneous mutations.
  
• Important in genetics and evolution.
  

Classical vs. Quantum View

Classical Mechanics	Quantum Mechanics
Particles cannot cross barriers if E<V0	Particles have a finite probability of tunneling.
Deterministic outcomes.	Probabilistic outcomes.
No wave properties.	Wave-like behavior of particles.

Conclusion

Quantum mechanical tunneling is a fundamental concept in modern physics, illustrating how wave-particle duality allows particles to bypass classical restrictions. By solving the Schrödinger equation for a one-dimensional potential barrier, we derive the transmission probability, which depends on barrier width, height, and particle energy.

Key Takeaways:

 Tunneling occurs due to wave nature of particles.
The probability decreases exponentially with barrier width and height.
  STM, tunnel diodes, nuclear fusion, and quantum computing rely on tunneling.
  Quantum mechanics defies classical intuition, enabling new-age technology.

One-Dimensional Linear Harmonic Oscillator

The one-dimensional quantum harmonic oscillator is one of the most important models in quantum mechanics. It describes a particle moving in a quadratic potential well, which is widely used in atomic physics, molecular vibrations, quantum field theory, and nanotechnology.

Potential Energy and Hamiltonian

The harmonic oscillator potential is given by:

where:

• m = Mass of the particle
  
• ω = Angular frequency of the oscillator
  
• x = Displacement from equilibrium
  

The total Hamiltonian operator (total energy operator) is:

where:

• ​ is the momentum operator
  
• X̂  is the position operator
  

---

Eigenvalues of Energy (Quantization of Energy Levels)

Solving the Schrödinger equation for this system leads to discrete energy levels, meaning that the system can only have certain quantized energy values. The energy eigenvalues are given by:

where:

• n is the quantum number (integer values: 0,1,2,…0, 1, 2, …0,1,2,…)
  
• ℏ is the reduced Planck’s constant (h/2π)
  
• ωis the angular frequency of the oscillator
  

Key Features of the Energy Spectrum:
  Energy levels are equally spaced: The difference between successive energy levels is always ℏω
  Lowest possible energy is not zero: The minimum energy (E0​) is nonzero, unlike in classical mechanics.

Zero-Point Energy (Ground State Energy, E0)

The lowest possible energy state (when n=0) is called the ground state energy or zero-point energy:

Physical Meaning of Zero-Point Energy:

• Even at absolute zero temperature (0 K), a quantum harmonic oscillator retains some energy.
  
• This zero-point energy is a fundamental consequence of the Heisenberg uncertainty principle.
  
• No quantum particle can be completely at rest, unlike in classical physics.
  

Eigenfunctions of the Harmonic Oscillator

The eigenfunctions ψn(x) (wavefunctions) of the harmonic oscillator are given by:

where:

• Hn(ξ) are Hermite polynomials (special functions in mathematics).
  
• ξ=x/α​ is a dimensionless variable, with characteristic length:
  ​​
• Nn​ is a normalization constant ensuring that:
  

Properties of the Eigenfunctions:

 Ground state wavefunction (n=0):

• It is a Gaussian function (bell-shaped curve).
  
• The probability is highest at x=0, meaning the particle is most likely near the equilibrium position.
  

 Higher order wavefunctions (n>0):

• Involve Hermite polynomials, making them more complex.
  
• Each successive wavefunction has more nodes (zero crossings).
  
• The wavefunctions extend further from the center as n increases.
  

Key Differences: Classical vs. Quantum Oscillator

Feature	Classical Oscillator	Quantum Oscillato
Energy Levels	Continuous (any value)	Discrete (quantized)
Ground State Energy	Can be zero	E0​=1/2​ℏω (nonzero)
Particle Position	Moves between fixed limits	Has probability distribution
Behavior at High n	Same motion repeats	Approaches classical behavior (correspondence principle)

Feature	Classical Oscillator	Quantum Oscillator
Energy Levels	Continuous (any value)	Discrete (quantized)
Ground State Energy	Can be zero	E0=12ℏωE_0 = \frac{1}{2} \hbar \omegaE0​=21​ℏω (nonzero)
Particle Position	Moves between fixed limits	Has probability distribution
Behavior at High nnn	Same motion repeats	Approaches classical behavior (correspondence principle)

Applications of the Quantum Harmonic Oscillator

The quantum harmonic oscillator is one of the most important models in physics because many real-world systems behave similarly.

(A) Vibrational Energy Levels in Molecules

• The vibrational modes of diatomic molecules (e.g., H₂, CO) follow the harmonic oscillator model.
  
• Infrared spectroscopy analyzes molecular vibrations using this concept.
  

(B) Quantum Dots and Nanotechnology

• Quantum dots (nanoscale semiconductor structures) have discrete energy levels similar to a quantum harmonic oscillator.
  
• Used in optoelectronics, quantum computing, and medical imaging.
  

(C) Phonons in Solids

• Lattice vibrations in crystals and semiconductors behave like quantum oscillators.
  
• Phonons (quantized sound waves) follow the same principles.
  

(D) Quantum Field Theory and Particle Physics

• The quantum harmonic oscillator forms the basis for quantum field theory.
  
• Each mode of a field behaves as an independent oscillator.
  

(E) Bose-Einstein Condensates (BECs)

• Ultracold gases confined in magnetic traps behave as harmonic oscillators.
  
• Used in atomic physics and quantum simulations.
  

Conclusion

The one-dimensional quantum harmonic oscillator is a fundamental model that appears in multiple branches of physics.

 The energy levels are quantized, and the zero-point energy is nonzero.
The wavefunctions involve Hermite polynomials and describe probabilistic distributions of particle positions.
  It serves as a foundation for molecular vibrations, nanotechnology, solid-state physics, and quantum field theory.