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 ↓


Particle in a box


The concept of a "particle in a box" is a fundamental idea in quantum mechanics that many undergraduate physics students encounter. It is a simplified model that helps us understand the basic principles of quantum mechanics and describe a particle that is confined within an infinitely deep potential well. This model allows us to explore the quantization of energy levels, wave functions, and probability distributions in quantum systems.

Understanding the basics

In classical physics, if we have a particle that is trapped in a box, it can move freely within the box as long as it does not make contact with the walls. However, the situation is quite different in the quantum realm. Here, the position of the particle is uncertain, and its behavior is best described by a probability wave function derived from the Schrödinger equation.

Schrödinger equation

The time-independent Schrödinger equation for a particle inside a one-dimensional box is given as:

    Hψ = Eψ

Where:

  • H is the Hamiltonian operator
  • ψ (psi) is the wave function of the particle
  • E is the energy of the particle

The Hamiltonian operator for a free particle is given as:

    h = - (ħ² / 2m) (d²/dx²)

Where:

  • ħ is the reduced Planck constant
  • m is the mass of the particle
  • d²/dx² is the second derivative with respect to x

Infinite potential well

For the "particle in a box" model, we consider a potential V(x) that is zero inside the box and infinite outside it. Mathematically, it is expressed as:

    V(x) = 0, for 0 < x < L
    V(x) = ∞, otherwise

This means that the particle is completely confined within the box, which is defined between x = 0 and x = L

Solving the Schrödinger equation

Inside the box (0 < x < L), the potential is zero, so the Schrödinger equation simplifies to:

    - (ħ² / 2m) (d²ψ/dx²) = Eψ

On rearranging the terms, we get:

    d²ψ/dx² + (2mE/ħ²)ψ = 0

Defining the constant k² = 2mE/ħ², we have the differential equation:

    d²ψ/dx² + k²ψ = 0

This is a second-order linear differential equation. The solutions of this equation are sinusoids:

    ψ(x) = A sin(kx) + B cos(kx)

Applying boundary conditions

Since the walls of the box are infinite, the wave function must vanish at both boundaries, i.e. at x = 0 and x = L Applying these boundary conditions:

  • At x = 0, ψ(0) = A sin(0) + B cos(0) = B = 0 Thus, B = 0.
  • At x = L, ψ(L) = A sin(kL) = 0.

The condition A sin(kL) = 0 implies sin(kL) = 0, which is true when:

    kL = nπ

where n is an integer (1, 2, 3, ...). So, k = nπ/L.

Quantization of energy

Now, substituting k = nπ/L again in the expression k² = 2mE/ħ², we get:

    (nπ/L)² = 2mE/ħ²

Solving for E, the energy levels are quantized as:

    E_n = n²h²/(8mL²)

where h is Planck's constant. This quantization shows that the particle can only exist at discrete energy levels within the box.

Wave functions and visualization

The generalized wave functions satisfying the boundary conditions are given as follows:

    ψ_n(x) = sqrt(2/L) sin(nπx/L)

The wave functions represent stationary waves inside the box. Let's imagine some wave functions:

n = 1

Wave function for n = 1

n = 2

Wave function for n = 2

Probability distribution

The probability density for finding the particle in a particular region is given by the square of the absolute value of the wave function:

    |ψ_n(x)|² = (2/L) (sin(nπx/L))²

This tells us where the particle is most likely to be found. Visualizing these probabilities helps to understand the concept of particle confinement and spatial probabilities:

n = 1

Probability distribution for n = 1

Insights and applications

The "particle in a box" model is more than a theoretical exercise; it forms the basis for understanding more complex quantum systems. Some of the insights and applications of this model are as follows:

  • Quantization of energy levels: The discrete energy levels predicted by the particle in the box model are similar to the quantized energy levels in atoms and molecules.
  • Wave–particle duality: Wave functions reflect the wave-like nature of particles, which is an integral part of quantum mechanics.
  • Limiting conditions: The importance of limiting conditions in determining physical phenomena is highlighted by the model.
  • Nanotechnology: This idea is used in the design of quantum wells used in semiconductors and nanostructures.

Conclusion

"Particle in a Box" provides a glimpse into the unique realm of quantum mechanics, where particles behave in unexpected ways that defy classical intuition. By exploring this simple model, students develop a nuanced understanding of fundamental quantum principles. As they progress in their studies, they will encounter more sophisticated quantum systems, but the fundamental concepts of quantization, wave functions, and probability distributions learned here serve them well on their journey to understanding the universe at its most basic level.


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Time-independent Schrödinger equation
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Particle in a box

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