Stationary States: Difference between revisions

	Quantum Mechanics A
	Schrödinger Equation; The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|\Psi\rangle} evolves in time.
	Physical Basis of Quantum Mechanics
	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
	Schrödinger Equation
	Brief Derivation of Schrödinger Equation; Relation Between the Wave Function and Probability Density; Stationary States; Heisenberg Uncertainty Principle; Some Consequences of the Uncertainty Principle
	Operators, Eigenfunctions, and Symmetry
	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
	Motion in One Dimension
	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
	Discrete Eigenvalues and Bound States
	Harmonic Oscillator Spectrum and Eigenstates; Analytical Method for Solving the Simple Harmonic Oscillator; Coherent States; Charged Particles in an Electromagnetic Field; WKB Approximation
	Time Evolution and the Pictures of Quantum Mechanics
	The Heisenberg Picture: Equations of Motion for Operators; The Interaction Picture; The Virial Theorem
	Angular Momentum
	Commutation Relations; Angular Momentum as a Generator of Rotations in 3D; Spherical Coordinates; Eigenvalue Quantization; Orbital Angular Momentum Eigenfunctions
	Central Forces
	General Formalism; Free Particle in Spherical Coordinates; Spherical Well; Isotropic Harmonic Oscillator; Hydrogen Atom; WKB in Spherical Coordinates
	The Path Integral Formulation of Quantum Mechanics
	Feynman Path Integrals; The Free-Particle Propagator; Propagator for the Harmonic Oscillator
	Continuous Eigenvalues and Collision Theory
	Differential Cross Section and the Green's Function Formulation of Scattering; Central Potential Scattering and Phase Shifts; Coulomb Potential Scattering

Latest revision as of 16:49, 12 August 2013

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(\textbf{r})} , the Schrödinger equation takes the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(\textbf{r},t)=e^{-iEt/\hbar}\psi(\textbf{r})} .

Obviously, for such state the probability density is

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(\textbf{r},t)|^2=|\psi(\textbf{r})|^2}

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

The Schrödinger equation now becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(\textbf{r})} and eigenvalue Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x,p),\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle Q(x,p)\rangle = \int \psi^{\ast}(x) Q\left(x,\frac{\hbar}{i} \frac{d}{dx}\right) \psi(x)\,dx }

Problem

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m} \nabla^2 \psi \left(\mathbf{r} \right) = E \psi\left(\mathbf{r} \right) }

Typically, one lets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = \frac{\hbar^2 k^2}{2m},} obtaining

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left( \nabla^2 + k^2 \right) \psi \left( \mathbf{r} \right) = 0. }

Show that

(a) a plane wave Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\left(\mathbf{r} \right) = e^{ikz}, \!} and

(b) a spherical wave Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi\left(\mathbf{r} \right) = \frac{e^{ikr}}{r}, \! } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r = \sqrt{x^2 + y^2 + z^2}, \! }

satisfy the equation. In either case, the wave length of the solution is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda = \frac{2\pi}{k} \!} and the momentum by de Broglie's relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \hbar k. \! }

Solution

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+{{Quantum Mechanics A}}
 Stationary states are the energy eigenstates of the [[Commutation relations and simultaneous eigenvalues#Hamiltonian|Hamiltonian operator]]<nowiki />. These states are called "stationary" because their probability distributions are independent of time.
 For a conservative system with a time independent potential, <math>V(\textbf{r})</math>, the [[Schrödinger equation]] takes the form:
-:<math> 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)</math>
+:<math> 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)</math>
 Since the potential and the Hamiltonian do not depend on time, solutions to this equation can be written as
-:<math>\psi(\textbf{r},t)=e^{-iEt/\hbar}\psi(\textbf{r})</math>.
+:<math>\Psi(\textbf{r},t)=e^{-iEt/\hbar}\psi(\textbf{r})</math>.
 Obviously, for such state the probability density is
-:<math>|\psi(\textbf{r},t)|^2=|\psi(\textbf{r})|^2</math>
+:<math>|\Psi(\textbf{r},t)|^2=|\psi(\textbf{r})|^2</math>
-which is independent of time. Hence, the name is "stationary state".
+which is independent of time, hence the term, "stationary state".
-The same thing happens in calculating the expectation value of any dynamical variable.
+The Schrödinger equation now becomes
+:<math>\left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\textbf{r})\right]\psi(\textbf{r})=E\psi(\textbf{r})</math>
+which is an eigenvalue equation with eigenfunction <math>\psi(\textbf{r})</math> and eigenvalue <math>E\!</math>. This equation is known as the time-independent Schrödinger equation.
-For some operator <math> \langle Q(x,p,t) \rangle \!</math>
+Something similar happens when calculating the expectation value of any dynamical variable.
-:<math> \langle Q(x,p)\rangle = \int \psi^*(x) Q\left(x,\frac{\hbar}{i} \frac{d}{dx}\right) \psi(x) dx </math>
-The Schrödinger equation now becomes
+For any time-independent operator <math>Q(x,p),\!</math>
+:<math> \langle Q(x,p)\rangle = \int \psi^{\ast}(x) Q\left(x,\frac{\hbar}{i} \frac{d}{dx}\right) \psi(x)\,dx </math>
+==Problem==
+The time-independent Schrodinger equation for a free particle is given by
+:<math>
+-\frac{\hbar^2}{2m} \nabla^2 \psi \left(\mathbf{r} \right) = E \psi\left(\mathbf{r} \right)
+</math>
+Typically, one lets <math> E = \frac{\hbar^2 k^2}{2m},</math> obtaining
+:<math>
+ \left( \nabla^2 + k^2 \right) \psi \left( \mathbf{r} \right) = 0.
+</math>
+Show that
-:<math>\left[ -\frac{\hbar^2}{2m}\nabla^2 + V(\textbf{r})\right]\psi(\textbf{r},t)=E\psi(\textbf{r})</math>
+'''(a)''' a plane wave <math> \psi\left(\mathbf{r} \right) = e^{ikz}, \!</math> and
-which is an eigenvalue equation with eigenfunction <math>\psi(\textbf{r})</math> and eigenvalue <math>E\!</math>. This equation is known as the time-independent Schrödinger equation.
+'''(b)''' a spherical wave <math> \psi\left(\mathbf{r} \right) = \frac{e^{ikr}}{r}, \! </math> where <math> r = \sqrt{x^2 + y^2 + z^2}, \! </math>
+satisfy the equation.  In either case, the wave length of the solution is given by <math> \lambda = \frac{2\pi}{k} \!</math> and the momentum by de Broglie's relation <math> p = \hbar k. \! </math>
-A sample problem:  [http://wiki.physics.fsu.edu/wiki/index.php/Phy5645/Free_particle_SE_problem A free particle Schrödinger equation].
+[[Phy5645/Free particle SE problem|Solution]]

Stationary States: Difference between revisions

Latest revision as of 16:49, 12 August 2013

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