Nonrelativistic Schrödinger Equation

The Nonrelativistic Schrödinger Equation is a fundamental equation in quantum mechanics, named after its creator, Erwin Schrödinger. Here's an explanation at a high school level: ### Concept: Imagine you're trying to predict where a tiny particle, like an electron, might be found around an atom. Unlike bigger objects, these tiny particles don't follow paths we can predict exactly; instead, they have a probability cloud where they might be. The Schrödinger Equation helps us calculate this probability. ### The Equation: The equation looks like this: \[ i\hbar \frac{\partial}{\partial t} \Psi(x,t) = \hat{H} \Psi(x,t) \] Here's what each part means: - **\( \Psi(x,t) \)** (called "psi"): This is the wave function. It's a mathematical function that gives us the probability of finding the particle at position \( x \) at time \( t \). Think of it like a wave that tells you how likely it is to find the particle somewhere. - **\( \hat{H} \)** (called "H-hat"): This is the Hamiltonian operator. It represents the total energy of the system, including kinetic (movement) and potential (position-related) energy. For a single particle in a simple case, it might look like: \[ \hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \] - **\( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} \)** represents the kinetic energy part, where \( m \) is the mass of the particle, and \( \hbar \) (h-bar) is a really small number related to Planck's constant. - **\( V(x) \)** is the potential energy, which can vary depending on where the particle is. - **\( i \)** : This is the imaginary unit, which is \( \sqrt{-1} \). It's used because the equation deals with waves, and in physics, we often use complex numbers to describe waves. - **\( \hbar \frac{\partial}{\partial t} \)** : This part represents how the wave function changes over time. ### What It Does: - **Time Evolution**: The equation tells us how the wave function, and thus the probabilities of where the particle could be, changes over time. - **Energy Levels**: By solving this equation for specific systems (like an electron in an atom), we find out the allowed energy levels or states the particle can be in. This is why atoms have distinct energy levels. - **Probability**: The square of the wave function \( |\Psi|^2 \) gives the probability density of finding the particle at \( x \) at time \( t \). ### Simple Analogy: Think of a soap bubble floating in the air. The Schrödinger Equation would describe how the bubble's surface (our wave function) moves and changes shape over time due to air currents (like forces on the particle). By knowing the shape of the bubble at any moment, we can predict where it might pop (where you might find the particle). ### Why It's Called "Nonrelativistic": This version of the Schrödinger Equation doesn't take into account effects that become significant when things move very fast, near the speed of light, where Einstein's relativity would be important. For everyday speeds of particles in atoms or molecules, this equation works fine. ### In Summary: The Schrödinger Equation is like a recipe to calculate the probability wave of a particle, telling us where it might be, how fast it might be moving, and how these probabilities change with time. It's essential in understanding the quantum world where particles behave very differently from what we observe in our everyday macroscopic world.

1) what?

there is just one Schrödinger equation and it's non relativistic, it's redundant to call it non relativistic, a relativistic qm wave equation would be the Dirac equation

Yes, waves of a medium, not point particles in empty space.

Where did you find this? I have some concerns.

i disagree

yeah welcome to atmospheric range not time relishing

pretty sure thats how you start up warp drive once you can figure out how to create gravity.

Is this non-euclidean