General Formalism: 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 23:14, 31 August 2013

A central potential is a potential that depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant. We may use these properties to reduce this otherwise three-dimensional problem to an effective one-dimensional problem. The general form of the Hamiltonian for a particle immersed in such a potential is

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V(|{\hat {r}}|).$

Due to rotational symmetry, $[{\hat {H}},{\hat {L}}_{z}]=0\!$ and $[{\hat {H}},{\hat {L}}^{2}]=0.\!$ This allows us to find a complete set of states that are simultaneous eigenstates of ${\hat {H}},\!$ ${\hat {L}}_{z},\!$ and ${\hat {L}}^{2}.\!$ We will label these eigenstates as $|n,l,m\rangle ,\!$ where $l\!$ and $m\!$ are the orbital and magnetic quantum numbers, as defined in the previous chapter, and $n\!$ represents the quantum numbers that define the radial dependence of the wave function; this is the only part of the state that depends on the exact form of the potential, as we will see shortly.

Let us now write the Schrödinger equation for this system and solve for the angular dependence of the wave function, thus reducing the problem to an effective one-dimensional problem. The equation is

$-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V(r)\psi =E\psi .$

The Laplacian in spherical coordinates may be written as

$\nabla ^{2}={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r-{\frac {{\hat {\mathbf {L} }}^{2}}{\hbar ^{2}r^{2}}},$

so that the equation becomes

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}r+{\frac {{\hat {\mathbf {L} }}^{2}}{2mr^{2}}}+V(r)\right)\psi =E\psi .$

We already know the eigenfunctions of ${\frac {{\hat {\mathbf {L} }}^{2}}{2mr^{2}}}$ from the previous chapter, and thus the entire angular dependence of the wave function. We may therefore use separation of variables and write $\psi (r,\theta ,\phi )=f_{l}(r)Y_{l}^{m}(\theta ,\phi ),\!$ where $Y_{l}^{m}(\theta ,\phi )\!$ are the spherical harmonics. Substituting this into the Schrödinger equation, we obtain

$\left(-{\frac {\hbar ^{2}}{2m}}{\frac {1}{r}}{\frac {d^{2}}{dr^{2}}}r+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)f_{l}(r)=Ef_{l}(r).$

The term ${\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}$ is referred to as the centrifugal barrier, which is associated with the motion of the particle. The classical analogue is the term, ${\frac {l^{2}}{2mr^{2}}},$ that arises in treating central forces classically. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point.

Now, if we let $u_{l}(r)=rf_{l}(r)\!$ , we finally arrive at the effective one-dimensional Schrödinger equation for the radial dependence of the wave function, $\left(-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dr^{2}}}+{\frac {\hbar ^{2}l(l+1)}{2mr^{2}}}+V(r)\right)u_{l}(r)=Eu_{l}(r).$

Due to the boundary condition that $f_{l}(r)\!$ must be finite the origin, $u_{l}(r)\!$ must vanish.

In many cases, looking at the asymptotic behavior of $u_{l}(r)\!$ can be quite helpful, as we will see in later sections.

Nomenclature

Historically, the first four (previously five) values of $l\!$ have taken on names, and additional values of $l\!$ are referred to alphabetically:

{\begin{cases}l=0&{\mbox{s-wave (sharp)}}\\l=1&{\mbox{p-wave (principal)}}\\l=2&{\mbox{d-wave (diffuse)}}\\l=3&{\mbox{f-wave (fundamental)}}\\l=4&{\mbox{g-wave (previously called t-wave for thick)}}\\l=5&{\mbox{h-wave}}\\\end{cases}}

@@ Line 1: / Line 1: @@
 {{Quantum Mechanics A}}
-A central potential does not depend on time, but rather depends only on the absolute value of the distance away from the potential's center. A central potential is rotationally invariant, not depending on the orientation. These properties can effectively reduce a three dimensional problem into a one dimensional problem.
+A central potential is a potential that depends only on the absolute value of the distance away from the potential's center.  A central potential is rotationally invariant.  We may use these properties to reduce this otherwise three-dimensional problem to an effective one-dimensional problem.  The general form of the Hamiltonian for a particle immersed in such a potential is
-:<math>H=\frac{p^2}{2m}+V(|r|)</math>
+<math>\hat{H}=\frac{\hat{p}^2}{2m}+V(|\hat{r}|).</math>
-Due to the rotational symmetry, <math>[H,L_z]=0\!</math> and <math>[H,L^2]=0\!</math>, and the eigenstates of <math>L^2\!</math>, <math>L_z\!</math> are non-degerate. This allows us to find a complete set of states that are simultaneous eigenfunctions of <math>H\!</math>, <math>L_z\!</math>, and <math>L^2\!</math>. We can label these states by their eigenvalues of <math>|Elm\rangle\!</math>.
+Due to rotational symmetry, <math>[\hat{H},\hat{L}_z]=0\!</math> and <math>[\hat{H},\hat{L}^2]=0.\!</math>  This allows us to find a complete set of states that are simultaneous eigenstates of <math>\hat{H},\!</math> <math>\hat{L}_z,\!</math> and <math>\hat{L}^2.\!</math>  We will label these eigenstates as <math>|n,l,m\rangle,\!</math> where <math>l\!</math> and <math>m\!</math> are the orbital and magnetic quantum numbers, as defined in the [[Eigenvalue Quantization|previous chapter]], and <math>n\!</math> represents the quantum numbers that define the radial dependence of the wave function; this is the only part of the state that depends on the exact form of the potential, as we will see shortly.
-From this we can get a state of the same energy for a given <math>l\!</math> with a degeneracy of <math>2l+1\!</math>.
+Let us now write the [[Schrödinger Equation|Schrödinger equation]] for this system and solve for the angular dependence of the wave function, thus reducing the problem to an effective one-dimensional problem.  The equation is
-We can rewrite the Laplacian as
-:<math>\nabla^2=\frac{1}{r}\frac{\partial^2}{\partial r^2}r-\frac{L^2}{\hbar^2 r^2}</math>
-This makes the [[Schrödinger equation]]
+<math>-\frac{\hbar^2}{2m}\nabla^2\psi+V(r)\psi=E\psi.</math>
-:<math>\left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{L^2}{2mr^2}+V(r)\right)\psi(r,\theta,\phi)=E\psi(r,\theta,\phi)</math>
-Using separation of variables, <math>\psi(r,\theta,\phi)=f_l(r)Y_{lm}(\theta,\phi)\!</math>, we get:
+The Laplacian in spherical coordinates may be written as
-:<math>\left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)f_l(r)Y_{lm}(\theta,\phi)=Ef_l(r)Y_{lm}(\theta,\phi)</math>
-The term <math> \frac{\hbar^2 l(l+1)}{2mr^2} </math> is referred to as the centrifugal barrier, which is associated with the motion of the particle. The classical analogue is <math> \frac{l^2}{2mr^2} </math>. The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point. Multiplying both sides by <math>Y_{l^\prime m'}\!</math> and integrating over the angular dependence reduces the equation to merely a function of <math>r\!</math>.
+<math>\nabla^2=\frac{1}{r}\frac{\partial^2}{\partial r^2}r-\frac{\hat{\mathbf{L}}^2}{\hbar^2 r^2},</math>
-Now if we let <math>u_l(r)=rf_l(r)\!</math>, this gives the radial Schrödinger equation:
+so that the equation becomes
-:<math>\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)u_l(r)=Eu_l(r)</math>
-Due to the boundary condition that <math>f_l(r)\!</math> must be finite the origin, <math>u_l(r)\!</math> must vanish.
+<math>\left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{\partial^2}{\partial r^2}r+\frac{\hat{\mathbf{L}}^2}{2mr^2}+V(r)\right)\psi=E\psi.</math>
-Often looking at the asymptotic behavior of <math>u_l(r)\!</math> can be quite helpful.
+We already know the eigenfunctions of <math>\frac{\hat{\mathbf{L}}^2}{2mr^2}</math> from the [[Orbital Angular Momentum Eigenfunctions|previous chapter]], and thus the entire angular dependence of the wave function.  We may therefore use separation of variables and write <math>\psi(r,\theta,\phi)=f_l(r)Y_l^m(\theta,\phi),\!</math> where <math>Y_l^m(\theta,\phi)\!</math> are the spherical harmonics.  Substituting this into the [[Schrödinger Equation|Schrödinger equation]], we obtain
-As <math>r\rightarrow 0\!</math> and <math>V(r)\ll\frac{1}{r^2}\!</math>, the dominating term becomes the centrifugal barrier giving the approximate Hamiltonian:
+<math>\left(-\frac{\hbar^2}{2m}\frac{1}{r}\frac{d^2}{dr^2}r+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)f_l(r)=Ef_l(r).</math>
-:<math>-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2 l(l+1)}{2mr^2}</math>
-which has the solutions <math>u_l(r)\sim r^{l+1},r^{-l}\!</math> where only the first term is physically possible because the second blows up at the origin.
-As <math>r\rightarrow\infty\!</math> and <math>rV(r)\rightarrow 0</math>(which does not include the monopole <math>\frac{1}{r}</math> coulomb potential), the Hamiltonian approximately becomes
+The term <math> \frac{\hbar^2 l(l+1)}{2mr^2} </math> is referred to as the centrifugal barrier, which is associated with the motion of the particle.  The classical analogue is the term, <math> \frac{l^2}{2mr^2},</math> that arises in treating central forces classically.  The centrifugal barrier prevents the particle from reaching the center of force, causing the wave function to vanish at this point.
-:<math>-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}u_l(r)=Eu_l(r)</math>.
-Letting <math>k=-i\sqrt{\frac{2mE}{\hbar^2}}</math>
+Now, if we let <math>u_l(r)=rf_l(r)\!</math>, we finally arrive at the effective one-dimensional [[Schrödinger Equation|Schrödinger equation]] for the radial dependence of the wave function,
-gives a solution of <math>u_l(r)=Ae^{kr}+Be^{-kr}\!</math>,  where when <math>k\!</math> is real, <math>B=0\!</math>, but both terms are needed when <math>k\!</math> is imaginary.
+<math>\left(-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2 l(l+1)}{2mr^2}+V(r)\right)u_l(r)=Eu_l(r).</math>
+Due to the boundary condition that <math>f_l(r)\!</math> must be finite the origin, <math>u_l(r)\!</math> must vanish.
+In many cases, looking at the asymptotic behavior of <math>u_l(r)\!</math> can be quite helpful, as we will see in later sections.
 ==Nomenclature==
@@ Line 47: / Line 43: @@
 l = 5  & \mbox{h-wave}\\
 \end{cases} </math>
-[[Phy5645/HydrogenAtomProblem2|Worked Problem]] involving the energy levels in a central potential.

General Formalism: Difference between revisions

Latest revision as of 23:14, 31 August 2013

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