Motion in One Dimension: Difference between revisions

	Quantum Mechanics A
	Schrödinger Equation; The most fundamental equation of quantum mechanics; given a Hamiltonian , it describes how a state 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 13:27, 18 January 2014

In this chapter, we will study the quantum mechaincs of one-dimensional systems. We study such problems for two reasons. Not only is it interesting to study the simplest cases to demonstrate some of the general properties of quantum mechanical systems, but it also turns out that there are two- and three-dimensional systems that can be reduced to effective one-dimensional problems, such as a particle in a central potential (also see the problem below).

We will discuss both bound and scattering states of one-dimensional potentials, thus illustrating the basic features of each, such as the discrete energy spectrum of the bound states, as opposed to the continuous scattering state spectrum. We also give an introduction to scattering from one-dimensional potentials, and show how to calculate the probabilities of transmission and reflection from such a potential. Finally, we treat two special cases, namely the Dirac delta function potential and periodic potentials, the latter of which will introduce the concept of energy bands, a topic of great importance in, for example, the study of the electronic properties of crystalline solids.

Chapter Contents

Problem

(Based on Problem 3.19 in Schaum's Theory and Problems of Quantum Mechanics)

Consider a particle of mass $m\!$ in a three dimensional potential of the form, $V(x,y,z)=X(x)+Y(y)+Z(z).\!$ Show that we can treat the problem as three independent one-dimensional problems. Relate the energy of the three-dimensional state to the effective energies of the one-dimensional problems.

Solution

@@ Line 1: / Line 1: @@
 {{Quantum Mechanics A}}
-We study the one dimensional problems in quantum theory, not only because the interest of study the simplest cases to learn about the general properties. Actually there are many cases in two and three dimensions that can be reduced to one-dimensional problem like the cases in central potentials.
+In this chapter, we will study the quantum mechaincs of one-dimensional systems.  We study such problems for two reasons.  Not only is it interesting to study the simplest cases to demonstrate some of the general properties of quantum mechanical systems, but it also turns out that there are two- and three-dimensional systems that can be reduced to effective one-dimensional problems, such as a particle in a central potential (also see the problem below).
-Let's consider the motion in 1 direction of a particle in the potential V(x). Supposing that V(x) has finite values when x goes to infinity:
+We will discuss both bound and scattering states of one-dimensional potentials, thus illustrating the basic features of each, such as the discrete energy spectrum of the bound states, as opposed to the continuous scattering state spectrum.  We also give an introduction to scattering from one-dimensional potentials, and show how to calculate the probabilities of transmission and reflection from such a potential.  Finally, we treat two special cases, namely the Dirac delta function potential and periodic potentials, the latter of which will introduce the concept of energy bands, a topic of great importance in, for example, the study of the electronic properties of crystalline solids.
-:<math>\lim_{x \to -\infty}V(x)=V_-,  \lim_{x \to +\infty}V(x)=V_+</math>
+==Chapter Contents==
-and assuming that:     <math>V_-<V_+ \!</math>
+* [[One-Dimensional Bound States]]
+* [[The Dirac Delta Function Potential]]
+* [[Oscillation Theorem]]
+* [[Scattering States, Transmission and Reflection]]
+* [[Motion in a Periodic Potential]]
+* [[Summary of One-Dimensional Systems]]
-The Schrodinger equation becomes:
+==Problem==
-:<math>\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x)</math>
+(Based on Problem 3.19 in Schaum's ''Theory and Problems of Quantum Mechanics'')
-<math>\rightarrow \frac{d^2}{dx^2}\psi(x)+\frac{2m}{\hbar^2}(E-V(x))\psi(x)=0</math>
-From this equation we can discuss some general properties of 1-D motion as follows:
+Consider a particle of mass <math>m\!</math> in a three dimensional potential of the form, <math>V(x,y,z) = X(x)+Y(y)+Z(z).\!</math>  Show that we can treat the problem as three independent one-dimensional problems.  Relate the energy of the three-dimensional state to the effective energies of the one-dimensional problems.
+[[Phy5645/Problem 1D sample|Solution]]
-If <math>E>V_+\!</math>:
-<math>E-V(x)>0\!</math> at both <math>-\infty</math> and <math>+\infty</math>. Therefore, the solutions of Schrodinger equation can be express as linear combinations of trigonometric functions (sine or cosine). The wave function is oscillating at both <math>-\infty</math> and <math>+\infty</math>. The particle is in an unbound state. The energy spectrum is continous. Both oscillating solutions are allowed, and the energy levels are two-fold degenerate.
-If <math>V_-\le E \le V_+</math>:
-<math>E-V(x)>0\!</math> at <math>-\infty</math> but <math>E-V(x)<0</math> at <math>+\infty</math>. Therefore, the wave function is oscillating at  <math>-\infty</math> but decaying exponentially at <math>+\infty</math>. The energy spectrum is still continous but no longer degenerate.
-If <math>E<V_-\!</math>:
-<math>E-V(x)\!<0</math> at both <math>-\infty</math> and <math>+\infty</math>. Therefore, the wave function decays exponentially at both <math>-\infty</math> and <math>+\infty</math>. The particle is in a bound state. The energy spectrum is discrete and non-degenerate.
-[http://wiki.physics.fsu.edu/wiki/index.php/Phy5645/Problem_1D_sample Example]

Motion in One Dimension: Difference between revisions

Latest revision as of 13:27, 18 January 2014

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