Schrödinger Equation

 Introduction

The Schrödinger Equation is the backbone of quantum mechanics, revealing the secrets of how particles behave in the bizarre and fascinating quantum world. Developed by Erwin Schrödinger in 1926, this equation describes the evolution of the wave function, which contains all the probabilistic information about a quantum system's state. By bridging the classical and quantum realms, the Schrödinger Equation provides a powerful tool to understand phenomena like electron orbitals, quantum tunneling, and energy levels in atoms. It stands as a cornerstone of modern physics, revolutionizing our understanding of the microscopic universe.

Non-relativistic quantum mechanics and  Einstein's theory of relativity

Non-relativistic quantum mechanics is the study of quantum systems where the effects of relativity are negligible, typically focusing on particles moving at speeds much slower than the speed of light. It forms the foundation of quantum theory and is governed by principles such as wave-particle duality, superposition, and uncertainty. 

When the effects of relativity are negligible, it means that the velocities of particles under study are much slower than the speed of light, and their energy scales are not high enough to involve relativistic corrections. In such cases, the theory of non-relativistic quantum mechanics is sufficient to describe the system.

This approach simplifies calculations and focuses on the quantum behavior of particles without incorporating Einstein's theory of relativity. Non-relativistic quantum mechanics is widely applicable in areas like atomic physics, molecular chemistry, and materials science, where relativistic effects, such as time dilation or length contraction, are insignificant. Understanding general relativity is not strictly required to grasp quantum mechanics, as the two theories describe different domains of physics: general relativity deals with gravity and the large-scale structure of the universe, while quantum mechanics focuses on the behavior of particles at microscopic scales. However, their connection becomes crucial when scientists attempt to merge these frameworks into a single theory, such as in the pursuit of quantum gravity.

Einstein's theory of relativity: Einstein's theory of relativity revolutionized our understanding of the universe by uniting space, time, and gravity into a cohesive framework. It comprises two key components: Special relativity and General Relativity. Special Relativity, introduced in 1905, focuses on objects moving at constant speeds, especially those approaching the speed of light. It asserts that the laws of physics remain consistent for all observers and that the speed of light is constant regardless of the observer's motion. This theory brought forward groundbreaking concepts like time dilation, where time slows for objects traveling near light speed; length contraction, where objects appear shorter along their direction of motion; and mass-energy equivalence, famously expressed as (E = mc^2), which illustrates the interchangeability of mass and energy.

General Relativity, presented in 1915, expands upon these ideas by incorporating gravity as the curvature of spacetime caused by massive objects. This framework explains phenomena like spacetime distortion, where gravity results from the bending of spacetime by massive bodies; gravitational time dilation, where time moves slower in stronger gravitational fields; and gravitational waves, ripples in spacetime produced by accelerating massive objects, confirmed observationally in 2015. Together, these theories provide profound insights into cosmic phenomena like black holes and the universe's expansion, while also enabling practical applications such as the accuracy of GPS systems. Einstein's work reshaped classical mechanics and opened the door to a new era of understanding both the cosmos and the fabric of reality.

Non-relativistic quantum mechanics serves as the fundamental study of quantum systems where relativistic effects, like those due to high-speed motion or intense gravity, can be ignored. It applies to particles moving much slower than the speed of light and is based on principles such as wave-particle duality, superposition, and uncertainty.

The Schrödinger equation is central to this field, describing how the wave function (( \Psi )) evolves and encapsulates the probabilities of different states a particle might occupy. Key concepts include Heisenberg's Uncertainty Principle, which highlights the limits of simultaneously measuring properties like position and momentum; superposition, where particles exist in multiple states until observed; and quantum tunneling, which enables particles to pass through barriers, defying classical expectations.

 Pic illustrates an electron passing through an obstacle: quantum tunneling Gif Reference:https://medium.com/quantaphy/quantum-tunneling-explained-299c2b417112

Pic illustrates the Superposition of observer effect Gif reference and credit: https://www.nist.gov/image/superpositiongif

In classical physics, a skateboarder can only be in one position at a time, such as either the left side of the ramp (representing 0) or the right side (representing 1). However, if the skateboarder behaved like a quantum object, such as an atom, they could exist in a state known as "superposition." This means they wouldn't just be on one side or the other; instead, they could simultaneously be in both places—effectively existing as a combination of 0 and 1 at the same time.

Non-relativistic quantum mechanics finds applications in fields like atomic physics, where it explains electron orbitals and energy levels; condensed matter physics, which explores phenomena in materials like semiconductors and superconductors; and chemical bonding, providing insights into molecular interactions. Its methodologies underpin advancements in cutting-edge areas such as quantum computing, material science, and nanotechnology.

Erwin Schrödinger 

Erwin Schrödinger was an Austrian physicist and one of the pioneers of quantum mechanics. Born on August 12, 1887, in Vienna, Austria, he is best known for formulating the Schrödinger Equation, a fundamental equation that describes the behavior of quantum systems. His work laid the foundation for wave mechanics, a key aspect of quantum theory.

Schrödinger shared the 1933 Nobel Prize in Physics with Paul Dirac for their contributions to quantum mechanics. Beyond his groundbreaking equation, he is also famous for the thought experiment known as Schrödinger's Cat, which illustrates the concept of superposition and wave function collapse in quantum mechanics.

Throughout his career, Schrödinger explored various fields, including statistical mechanics, thermodynamics, and even the philosophy of science. He authored the influential book What Is Life?, which examined the connection between physics and biology, inspiring future research in molecular biology.

Schrödinger passed away on January 4, 1961, in Vienna, leaving behind a legacy that continues to shape modern physics.

Equation of motion in non-relativistic quantum mechanics

The Schrödinger Equation, the fundamental equation of motion in non-relativistic quantum mechanics, dictates how the quantum state of a physical system evolves over time. It exists in two primary forms:

1. The Time-Dependent Schrödinger Equation:

iħ ∂Ψ(r, t)/∂t = Ĥ Ψ(r, t)

This equation describes the temporal evolution of the wave function, denoted by Ψ(r, t), which encapsulates all the information about a quantum system as a function of both position (r) and time (t). The left side of the equation features the imaginary unit (i) and the reduced Planck constant (ħ), multiplied by the first partial derivative of the wave function with respect to time (∂Ψ(r, t)/∂t), representing how the wave function changes instantaneously. The right side involves the Hamiltonian operator (Ĥ) acting on the wave function. The Hamiltonian operator is a mathematical representation of the total energy of the system, typically comprising the kinetic energy operator and the potential energy operator. Solving this equation yields how the probability distribution of a quantum particle or system changes over time under the influence of a given potential. 

Watch a YouTube video demonstrating: Time-Dependent Schrödinger Equation

2. The Time-Independent Schrödinger Equation:

Ĥ ψ(r) = E ψ(r)

or more explicitly:

(-ħ²/2m ∇² + V(r)) ψ(r) = E ψ(r)

This form is applicable when the potential energy (V(r)) of the system does not change with time. It allows us to find the stationary states of the system, which are quantum states with definite energy that do not exhibit any temporal changes in their probability distributions. Here, ψ(r) represents the time-independent part of the wave function, depending only on position. The equation states that when the Hamiltonian operator (Ĥ) acts on this time-independent wave function, it results in the wave function being multiplied by a constant, E, which represents the energy eigenvalue or the specific allowed energy level of that stationary state. The expanded form explicitly shows the kinetic energy operator (-ħ²/2m ∇², involving the mass 'm' and the Laplacian operator ∇²) and the potential energy operator (V(r)) contributing to the total energy. Solving this equation for a given time-independent potential provides the quantized energy levels that the system can occupy and the corresponding spatial wave functions for each energy level.

Watch a YouTube video demonstrating and explaining: The Time-Independent Schrödinger Equation

What We Need To Know About Wave Function

Imagine a particle, like an electron. In the quantum world, we can't know its exact position and momentum at the same time with perfect certainty. Instead, we describe the state of that particle using a mathematical object called the wave function, often represented by the Greek letter ψ (psi).

Key Things to Understand:

• Describes the Quantum State: The wave function contains all the information we can possibly know about a quantum system. This includes things like the probability of finding a particle at a specific location, its momentum, and its energy.
• Probability Interpretation: The square of the absolute value of the wave function (|ψ|²) at a particular point in space tells us the probability density of finding the particle at that location. A higher value of |ψ|² means a higher probability of finding the particle there.
• Wave-like Behavior: The wave function itself behaves like a wave. It can undergo phenomena like superposition (being in multiple states at once) and interference (waves adding or canceling each other out), which explains the wave-like behavior of particles.
• Evolves in Time: The wave function is not static; it changes over time according to the Schrödinger equation, which is a fundamental law of quantum mechanics. This equation describes how the quantum state of a system evolves.
• Not Directly Measurable: The wave function itself is not a directly measurable quantity. What we measure are physical observables (like position or momentum), and the wave function helps us predict the probabilities of obtaining different measurement outcomes.

Conclusion

In conclusion, the Schrödinger Equation and non-relativistic quantum mechanics together form the bedrock of our understanding of the quantum realm. They enable us to model and predict the behavior of particles on an atomic and subatomic scale, unlocking the mysteries of wave-particle duality, superposition, and quantum tunneling. Meanwhile, Einstein's theory of relativity, with its revolutionary concepts of spacetime curvature and energy-mass equivalence, broadens our understanding of the universe's large-scale structure.

While non-relativistic quantum mechanics and relativity describe distinct domains of physics, their eventual unification through a quantum gravity framework represents one of the greatest challenges—and opportunities—in modern science. By combining these insights, we bridge the microscopic and macroscopic worlds, deepening our grasp of the intricate laws that govern existence. This synthesis not only advances technology and scientific exploration but also inspires us to reflect on the profound interconnectedness of the universe.