Stationary States

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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

Stationary states are the energy eigenstates of the Hamiltonian operator. These states are called "stationary" because their probability distributions are independent of time.

For a conservative system with a time independent potential, V(\textbf{r}), the Schrödinger equation takes the form:

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

Since the potential and the Hamiltonian do not depend on time, solutions to this equation can be written as


Obviously, for such state the probability density is


which is independent of time, hence the term, "stationary state".

The Schrödinger equation now becomes

\left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\textbf{r})\right]\psi(\textbf{r})=E\psi(\textbf{r})

which is an eigenvalue equation with eigenfunction \psi(\textbf{r}) and eigenvalue E\!. This equation is known as the time-independent Schrödinger equation.

Something similar happens when calculating the expectation value of any dynamical variable.

For any time-independent operator Q(x,p),\!

 \langle Q(x,p)\rangle = \int \psi^{\ast}(x) Q\left(x,\frac{\hbar}{i} \frac{d}{dx}\right) \psi(x)\,dx


The time-independent Schrodinger equation for a free particle is given by

-\frac{\hbar^2}{2m} \nabla^2 \psi \left(\mathbf{r} \right) = E \psi\left(\mathbf{r} \right)

Typically, one lets  E = \frac{\hbar^2 k^2}{2m}, obtaining

 \left( \nabla^2 + k^2 \right) \psi \left( \mathbf{r} \right) = 0.

Show that

(a) a plane wave  \psi\left(\mathbf{r} \right) = e^{ikz}, \! and

(b) a spherical wave  \psi\left(\mathbf{r} \right) = \frac{e^{ikr}}{r}, \! where  r = \sqrt{x^2 + y^2 + z^2}, \!

satisfy the equation. In either case, the wave length of the solution is given by  \lambda = \frac{2\pi}{k} \! and the momentum by de Broglie's relation  p = \hbar k. \!


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