Brief Derivation of Schrödinger Equation

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Quantum Mechanics A
Schrödinger Equation
The most fundamental equation of quantum mechanics; given a Hamiltonian \mathcal{H}, it describes how a state |\Psi\rangle evolves in time.
Basic Concepts and Theory of Motion
UV Catastrophe (Black-Body Radiation)
Photoelectric Effect
Stability of Matter
Double Slit Experiment
Stern-Gerlach Experiment
The Principle of Complementarity
The Correspondence Principle
The Philosophy of Quantum Theory
Brief Derivation of Schrödinger Equation
Relation Between the Wave Function and Probability Density
Stationary States
Heisenberg Uncertainty Principle
Some Consequences of the Uncertainty Principle
Linear Vector Spaces and Operators
Commutation Relations and Simultaneous Eigenvalues
The Schrödinger Equation in Dirac Notation
Transformations of Operators and Symmetry
Time Evolution of Expectation Values and Ehrenfest's Theorem
One-Dimensional Bound States
Oscillation Theorem
The Dirac Delta Function Potential
Scattering States, Transmission and Reflection
Motion in a Periodic Potential
Summary of One-Dimensional Systems
Harmonic Oscillator Spectrum and Eigenstates
Analytical Method for Solving the Simple Harmonic Oscillator
Coherent States
Charged Particles in an Electromagnetic Field
WKB Approximation
The Heisenberg Picture: Equations of Motion for Operators
The Interaction Picture
The Virial Theorem
Commutation Relations
Angular Momentum as a Generator of Rotations in 3D
Spherical Coordinates
Eigenvalue Quantization
Orbital Angular Momentum Eigenfunctions
General Formalism
Free Particle in Spherical Coordinates
Spherical Well
Isotropic Harmonic Oscillator
Hydrogen Atom
WKB in Spherical Coordinates
Feynman Path Integrals
The Free-Particle Propagator
Propagator for the Harmonic Oscillator
Differential Cross Section and the Green's Function Formulation of Scattering
Central Potential Scattering and Phase Shifts
Coulomb Potential Scattering

Imagine a particle constrained to move along the x-axis, subject to a potential energy V(x,t)\!. Classically, we could model this system by writing down its Hamiltonian H,\! given by

H = \frac{p^2}{2m} + V(x,t).

We then employ Hamilton's equations of motion,

\dot{p}=-\frac{\partial H}{\partial x},\,\dot{x}=\frac{\partial H}{\partial p},

where a dot denotes a time derivative, to determine the motion of the particle. Applying these equations to the above Hamiltonian, we can recover Newton's second law,

m\ddot{a}=-\frac{\partial V}{\partial x}=F.

Now by applying the appropriate initial conditions for our particle, we obtain a solution for the trajectory of the particle. As we will see, the above relation is only an approximation to actual physical reality. As we attempt to describe increasingly smaller objects, we enter the quantum mechanical regime, where we can no longer neglect the particles' wave properties. Allowing \displaystyle{p \rightarrow \frac{\hbar}{i}\frac{\partial}{\partial x}} and \displaystyle{H \rightarrow i\hbar \frac{\partial}{\partial t}}, we can use the Hamiltonian for a classical particle above to find an equation that describes this wave nature. We find that the wave function \Psi(x,t)\! satisfies the Schrödinger equation for a scalar potential V(x,t)\! in one dimension:

 i\hbar\frac{\partial}{\partial t}\Psi(x,t)=\left[-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+V(x,t)\right]\Psi(x,t)

A similar equation may be derived in multiple dimensions:

 i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r},t)=\left[-\frac{\hbar^2}{2m}\nabla^2+V(\textbf{r},t)\right]\Psi(\textbf{r},t)

Given a solution which satisfies the above Schrödinger equation, quantum mechanics provides a mathematical description of the laws obeyed by the probability amplitudes associated with quantum motion.

We can also generalize the Schrödinger equation to a system which contains  N \! particles. We assume that the wave function is  \Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t) and that the Hamiltonian of the system can be expressed as

  H= \sum_{k=1}^N \frac{\textbf{p}^2_k}{2m_k}+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N).

Making a similar replacement as before, we may obtain the Schrödinger equation for a many-particle system:

  i\hbar\frac{\partial}{\partial t}\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)=\left[\sum_{k=1}^N \frac{\hbar^2}{2m_k}\nabla_{k}^2+V(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N)\right]\Psi(\textbf{r}_1,\textbf{r}_2, \ldots, \textbf{r}_N, t)

Here, \nabla_{k}^2 is the Lapalacian operator that acts on the coordinates of particle k.\!

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