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 ↓


Time-independent Schrödinger equation


The time-independent Schrödinger equation is one of the cornerstone concepts in quantum mechanics. It describes how the quantum state of a physical system changes with space but not with time. In simple terms, it helps us understand where a particular particle, such as an electron, can be found in a specific region of space, without considering how it changes with time. To understand this concept in depth, let's look at its description, history, and application in clear and simple terms.

Historical background

This equation is named after Austrian physicist Erwin Schrödinger. He developed the Schrödinger equation in 1925, which provides the basis for wave mechanics. Schrödinger's work was revolutionary. It expanded the ideas of Max Planck and Albert Einstein on quantum theory and thus provided a powerful tool for explaining the behavior of atomic and subatomic particles.

What is the time-independent Schrödinger equation?

Before we dive deeper into the specific form of the time-independent version, we should mention the general Schrödinger equation, which is time-dependent:

iħ (∂ψ/∂t) = Ĥψ

Here:

  • i is the imaginary unit.
  • ħ is the decreasing Planck constant.
  • ψ is the wave function of the quantum system.
  • Ĥ is the Hamiltonian operator, corresponding to the total energy of the system.

The time-independent Schrödinger equation can be obtained from this. When the Hamiltonian operator Ĥ does not depend on time, the wave function can be split into a product of spatial and temporal parts:

ψ(x, t) = ψ(x)ϕ(t)

If we substitute this product into the time-dependent equation, separate the variables, and solve for the time-independent part, we arrive at the following equation:

Ĥψ(x) = Eψ(x)

In this equation:

  • Ĥ is the Hamiltonian operator.
  • ψ(x) is the spatial part of the wave function.
  • E is the energy eigenvalue corresponding to the wave function.

Understanding the equation

At its core, the time-independent Schrödinger equation tells us how particles behave in a quantum state without the effects of time shift. Solutions to this equation give us the energy levels (eigenvalues) of the system as well as the shape of the wave functions (eigenstates).

Visual examples and intuitive understanding

Let's analyze and visualize how potential energy, wave function and probability density are related:

Distance (x) potential energy Wave function

In this graph, the blue wave represents a wave function within the potential energy landscape. The behavior of the particle can be visualized by the oscillation curve indicating the probability amplitude - a larger amplitude means there is a greater probability of finding the particle at that point, if measured.

Key terms defined

Knowing a few key terms can make this concept much easier to understand:

  • Wave function (ψ): This is an important concept in quantum mechanics, representing the quantum state of a particle or system. The square of its absolute value (|ψ| 2) gives the probability density of finding a particle at a particular location.
  • Hamiltonian operator (Ĥ): This operator represents the total energy of the system, which includes both kinetic and potential energy.
  • Eigenvalues (E): These represent the possible energy levels of a quantum system.

Solving the time-independent Schrödinger equation

Solving the Schrödinger equation is necessary to determine the possible states of energy and the corresponding wave functions. Here is a simplified method:

  1. Identify the potential energy function: This could be a harmonic oscillator, a particle in a box, or some other form, depending on the physical system under examination.
  2. Apply the Hamiltonian of the individual system: Use the potential energy function and the kinetic energy operator to construct the Hamiltonian.
  3. Solve the differential equation: Solve the resulting equation, which is usually a second-order differential equation, to find solutions for ψ(x) and E.
  4. Normalize the wave functions: Make sure the wave functions are normalized so that the overall probability of finding the particle is 1.

Examples of specific systems

Different systems exhibit unique potential energy functions, so each requires a customized approach:

Particle in a one-dimensional box

In this simple model, a particle is confined in a box with impenetrable walls. The potential energy is zero inside the box and infinite outside, which confines the particle within a fixed region:

Ĥψ(x) = - (ħ^2 / 2m) (d^2ψ(x)/dx^2) = Eψ(x) for 0 < x < L

Here, L is the length of the box, and m is the mass of the particle. Solving this gives the discrete energy levels and the sine wave function:

ψ_n(x) = √(2/L) sin(nπx/L) E_n = n^2π^2ħ^2 / (2mL^2)

Harmonic oscillator

Another important example is the quantum harmonic oscillator. Here the potential energy is given by:

V(x) = 1/2 mω^2x^2

Solving for this potential gives the Hermite polynomial solution in terms of wave functions and energy levels:

E_n = (n + 1/2)ħω

in which n is a non-negative integer.

Applications in physics and chemistry

The time-independent Schrödinger equation is important in a variety of areas:

  • Quantum mechanics: Lays the foundation for understanding the quantum behavior of particles.
  • Chemistry and molecular physics: Helps to understand molecular bonds and energy states. This is important for quantum chemistry and predicting molecular behavior.
  • Solid State Physics: Helps in explaining the properties of semiconductors and designing new materials based on electronic band structure calculations.

Challenges and intuition

Understanding the time-independent Schrödinger equation can be mathematically challenging, but gaining intuitive insight is equally important. Recognize that solutions often represent not just mathematical results, but also profound physical realities and limitations in our ability to accurately determine the dynamics of a quantum system.

Conclusion

The time-independent Schrödinger equation is fundamental in understanding the quantum world without the direct effects of time. It enables physicists and chemists to predict energy levels, analyse quantum states, and explore the interactions that govern atomic and subatomic particles. By understanding this equation and its solutions, we open up the realm of understanding for quantum theory, which is important not only for theoretical physics but also for practical applications in various scientific fields.


Undergraduate → 5.2.1


U
username
0%
completed in Undergraduate

← Prev (5.2)
Schrödinger Equation
Next (5.2.2) →
Particle in a box

Comments 



Time-independent Schrödinger equation

https://www.physicsflow.com/ug/5.2.1?pfal=en