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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  
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>\hat{H}=\frac{\hat{p}^2}{2m}+V(|\hat{r}|).</math>
<math>\hat{H}=\frac{\hat{p}^2}{2m}+V(|\hat{r}|).</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 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.
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>
 
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
 
<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>


Often looking at the asymptotic behavior of <math>u_l(r)\!</math> can be quite helpful.
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.


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:
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,
:<math>-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2 l(l+1)}{2mr^2}</math>
<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>
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
Due to the boundary condition that <math>f_l(r)\!</math> must be finite the origin, <math>u_l(r)\!</math> must vanish.
:<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>
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.
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.


==Nomenclature==
==Nomenclature==

Revision as of 23:12, 31 August 2013

Quantum Mechanics A
SchrodEq.png
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.
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

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

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

Due to rotational symmetry, 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 [\hat{H},\hat{L}_z]=0\!} and 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 [\hat{H},\hat{L}^2]=0.\!} This allows us to find a complete set of states that are simultaneous eigenstates of 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 \hat{H},\!} 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 \hat{L}_z,\!} and 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 \hat{L}^2.\!} We will label these eigenstates 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 |n,l,m\rangle,\!} 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 l\!} and 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 m\!} are the orbital and magnetic quantum numbers, as defined in the previous chapter, and 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 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

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+V(r)\psi=E\psi.}

The Laplacian in spherical coordinates may 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 \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

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}\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 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{\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 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(r,\theta,\phi)=f_l(r)Y_l^m(\theta,\phi),\!} 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 Y_l^m(\theta,\phi)\!} are the spherical harmonics. Substituting this into the Schrödinger equation, we obtain

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}\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 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 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, 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{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 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 u_l(r)=rf_l(r)\!} , we finally arrive at the effective one-dimensional Schrödinger equation for the radial dependence of the wave function, 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}\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 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 f_l(r)\!} must be finite the origin, 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 u_l(r)\!} must vanish.

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

Nomenclature

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

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 \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} }

Worked Problem involving the energy levels in a central potential.

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