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{{Quantum Mechanics A}}
{{Quantum Mechanics A}}
A free particle is a specific case when <math>V_0=0\!</math> of the motion in a uniform potential <math>V(r)=V_0\!</math>.  So it's more useful to consider a particle moving in a uniform potential. The [[Schrödinger equation]] for the radial part of the wave function is:
A free particle is a specific case when <math>V_0=0\!</math> of the motion in a uniform potential <math>V(r)=V_0,</math> so it is more useful to consider a particle moving in a uniform potential. We will make use of these results in the next section to discuss the spherical potential well.  The [[Schrödinger Equation|Schrödinger equation]] for the radial part of the wave function is


:<math>\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial r^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V_0\right)u_l(r)=Eu_l(r)</math>
<math>\left(-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V_0\right)u_l=Eu_l.</math>
let <math>k^2=\frac{2m}{\hbar^2}|E-V|</math>. Rearranging the equation gives
:<math>\left(-\frac{\partial^2}{\partial r^2}+\frac{l(l+1)}{r^2}-k^2\right)u_l(r)=0</math>
Letting <math>\rho=kr\!</math> gives the terms that <math>\frac{1}{r^{2}}=\frac{k^{2}}{\rho ^{2}}</math> and <math>\frac{\partial ^{2}}{\partial r^{2}}=k^{2}\frac{\partial ^{2}}{\partial \rho ^{2}}</math>. Then the equation becomes:
:<math>\left(-\frac{\partial^2}{\partial\rho^2}+\frac{l(l+1)}{\rho^2}\right)u_l(\rho)=u_l(\rho)=d_ld_l^+u_l(\rho)</math>
where <math>d_l\!</math> and <math>d_l^{\dagger}\!</math> become the raising and lowering operators:


:<math>d_l=\frac{\partial}{\partial\rho}+\frac{l+1}{\rho}, </math>
Let <math>k^2=\frac{2m}{\hbar^2}|E-V|.</math> Rearranging the equation gives us
:<math>d_l^\dagger=-\frac{\partial}{\partial\rho}+\frac{l+1}{\rho}</math>
Being <math>d_l^{\dagger}d_l=d_{l+1}d_{l+1}^{\dagger}</math>, it can be shown that
:<math>d_l^\dagger u_l(\rho)=c_l u_{l+1}(\rho)</math>
For <math>\ell =0</math>,  <math>-\frac{\partial^2}{\partial \rho^2} u_0(\rho)=u_0(\rho)</math>, gives the solution as:
:<math>u_0(\rho)=A\sin(\rho)-B\cos(\rho)\!</math>
The raising operator can be applied to the ground state in order to find high orders of <math>\ u_0(\rho)</math>;
:<math>d_0^\dagger u_0(\rho)=\left(-\frac{\partial}{\partial\rho}+\frac{l+1}{\rho}\right)u_0(\rho)=c_0 u_1(\rho)</math>
By this way, we can get the general expression:
:<math>f_l(\rho)=\frac{u_l(\rho)}{\rho}=A_lj_l(\rho)+B_ln_l(\rho)</math>,
where <math> j_l(\rho) \!</math> is spherical Bessel function and <math> n_l(\rho) \! </math> is spherical Neumann function.


<math>\left(-\frac{d^2}{dr^2}+\frac{l(l+1)}{r^2}-k^2\right)u_l=0.</math>


==Explicit Forms of the Spherical Bessel and Neumann Functions==
If we now let <math>\rho=kr,\!</math> then the equation reduces to the dimensionless form,


:<math> j_0(z) = \frac{\sin(z)}{z} </math>
<math>\left(-\frac{d^2}{d\rho^2}+\frac{l(l+1)}{\rho^2}\right)u_l(\rho)=u_l(\rho)=d_ld_l^+u_l(\rho),</math>
:<math> j_1(z) = \frac{\sin(z)}{z^2} - \frac{\cos(z)}{z} </math>
:<math> j_2(z) = \left( \frac{3}{z^3} - \frac{1}{z}\right) \sin(z) - \frac{3}{z^2}\cos(z) </math>


:<math> n_0(z) = -\frac{\cos(z)}{z} </math>
where <math>d_l\!</math> and <math>d_l^{\dagger}\!</math> are the raising and lowering operators,
:<math> n_1(z) = -\frac{\cos(z)}{z^2} - \frac{\sin(z)}{z} </math>
:<math> n_2(z) = - \left( \frac{3}{z^3} - \frac{1}{z}\right) \cos(z) - \frac{3}{z^2}\sin(z) </math>


The spherical Hankel functions of the first and second kind can be written in terms of the spherical Bessel and spherical Neumann functions, and are defined by:
<math>d_l=\frac{d}{d\rho}+\frac{l+1}{\rho}</math>


:<math> h_{\ell}^{(1)} = j_{\ell}(z) + in_{\ell}(z) </math>
and
 
<math>d_l^\dagger=-\frac{d}{d\rho}+\frac{l+1}{\rho}.</math>
 
Because <math>d_l^{\dagger}d_l=d_{l+1}d_{l+1}^{\dagger},</math> it follows that
<math>d_l^\dagger u_l(\rho)=c_l u_{l+1}(\rho).</math>
 
For <math>l=0,</math>
 
<math>-\frac{d^2}{d\rho^2} u_0(\rho)=u_0(\rho),</math>
 
whose solution is
 
<math>u_0(\rho)=A\sin{\rho}-B\cos{\rho}.\!</math>
 
The raising operator may now be applied to this state in order to find the solutions for higher values of <math>l.\!</math>  By repeated application of this operator, we obtain the wave function for all values of <math>l:\!</math>
 
<math>f_l(\rho)=\frac{u_l(\rho)}{\rho}=A_lj_l(\rho)+B_ln_l(\rho),</math>
 
where <math> j_l(\rho) \!</math> is a spherical Bessel function and <math> n_l(\rho) \! </math> is a spherical Neumann function, or spherical Bessel functions of the [http://mathworld.wolfram.com/SphericalBesselFunctionoftheFirstKind.html first] and [http://mathworld.wolfram.com/SphericalBesselFunctionoftheSecondKind.html second] kinds, respectively.
 
==Properties of the Spherical Bessel and Neumann Functions==
 
Explicit forms of the first few spherical Bessel and Neumann functions:
 
<math> j_0(z) = \frac{\sin(z)}{z} </math>
<math> j_1(z) = \frac{\sin(z)}{z^2} - \frac{\cos(z)}{z} </math>
<math> j_2(z) = \left( \frac{3}{z^3} - \frac{1}{z}\right) \sin(z) - \frac{3}{z^2}\cos(z) </math>
 
<math> n_0(z) = -\frac{\cos(z)}{z} </math>
<math> n_1(z) = -\frac{\cos(z)}{z^2} - \frac{\sin(z)}{z} </math>
<math> n_2(z) = - \left( \frac{3}{z^3} - \frac{1}{z}\right) \cos(z) - \frac{3}{z^2}\sin(z) </math>
 
We may also define spherical Hankel functions of the [http://mathworld.wolfram.com/SphericalHankelFunctionoftheFirstKind.html first] and [http://mathworld.wolfram.com/SphericalHankelFunctionoftheSecondKind.html second] kind in terms of the spherical Bessel and Neumann functions:
 
<math> h_{\ell}^{(1)} = j_{\ell}(z) + in_{\ell}(z) </math>


and
and


:<math> h_{\ell}^{(2)} = j_{\ell}(z) - in_{\ell}(z) </math>
<math> h_{\ell}^{(2)} = j_{\ell}(z) - in_{\ell}(z) </math>




The asymptotic form of the spherical Bessel and Neumann functions (as z <math> \rightarrow</math> large) are given by:
The asymptotic forms of the spherical Bessel and Neumann functions as <math>z\rightarrow\infty</math> are


:<math> j_{\ell}(z) = \frac{\sin(z-\frac{\ell \pi}{2})}{z} </math>
<math> j_{\ell}(z) = \frac{\sin(z-\frac{\ell \pi}{2})}{z} </math>


and  
and  


:<math> n_{\ell}(z) = \frac{\cos(z-\frac{\ell \pi}{2})}{z} </math>
<math> n_{\ell}(z) = \frac{\cos(z-\frac{\ell \pi}{2})}{z}. </math>
 
The first few zeros of the spherical Bessel function for <math>l=0\!</math> and <math>l=1\!</math> are


The first few zeros of the spherical Bessel function:
<math> l = 0: 3.142, 6.283, 9.425, 12.566, \ldots </math>


:<math> \ell = 0: 3.142, 6.283, 9.425, 12.566 </math>
and


:<math> \ell = 1: 4.493, 7.725, 10.904, 14.066 </math>
<math> l = 1: 4.493, 7.725, 10.904, 14.066, \ldots</math>


The derivatives of the spherical Bessel and Neumann functions are defined by:
The derivatives of the spherical Bessel and Neumann functions are given by


:<math> j'_{\ell}(z) = \frac{\ell}{z}j_{\ell}(z) - j_{\ell+1}(z) </math>
<math> j'_{l}(z) = \frac{l}{z}j_{l}(z) - j_{l+1}(z) </math>


and  
and  


:<math> n'_{\ell}(z) = \frac{\ell}{z}n_{\ell}(z) - n_{\ell+1}(z) </math>
<math> n'_{l}(z) = \frac{l}{z}n_{l}(z) - n_{l+1}(z). </math>

Latest revision as of 23:39, 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 free particle is a specific case when 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_0=0\!} of the motion in a uniform 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(r)=V_0,} so it is more useful to consider a particle moving in a uniform potential. We will make use of these results in the next section to discuss the spherical potential well. The Schrödinger equation for the radial part of the wave function 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 \left(-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}+V_0\right)u_l=Eu_l.}

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 k^2=\frac{2m}{\hbar^2}|E-V|.} Rearranging the equation gives us

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{d^2}{dr^2}+\frac{l(l+1)}{r^2}-k^2\right)u_l=0.}

If we now 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 \rho=kr,\!} then the equation reduces to the dimensionless 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 \left(-\frac{d^2}{d\rho^2}+\frac{l(l+1)}{\rho^2}\right)u_l(\rho)=u_l(\rho)=d_ld_l^+u_l(\rho),}

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 d_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 d_l^{\dagger}\!} are the raising and lowering operators,

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 d_l=\frac{d}{d\rho}+\frac{l+1}{\rho}}

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 d_l^\dagger=-\frac{d}{d\rho}+\frac{l+1}{\rho}.}

Because 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 d_l^{\dagger}d_l=d_{l+1}d_{l+1}^{\dagger},} it follows 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 d_l^\dagger u_l(\rho)=c_l u_{l+1}(\rho).}

For 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=0,}

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{d^2}{d\rho^2} u_0(\rho)=u_0(\rho),}

whose solution 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 u_0(\rho)=A\sin{\rho}-B\cos{\rho}.\!}

The raising operator may now be applied to this state in order to find the solutions for higher 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.\!} By repeated application of this operator, we obtain the wave function for all 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:\!}

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(\rho)=\frac{u_l(\rho)}{\rho}=A_lj_l(\rho)+B_ln_l(\rho),}

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 j_l(\rho) \!} is a spherical Bessel function 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_l(\rho) \! } is a spherical Neumann function, or spherical Bessel functions of the first and second kinds, respectively.

Properties of the Spherical Bessel and Neumann Functions

Explicit forms of the first few spherical Bessel and Neumann functions:

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 j_0(z) = \frac{\sin(z)}{z} } 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 j_1(z) = \frac{\sin(z)}{z^2} - \frac{\cos(z)}{z} } 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 j_2(z) = \left( \frac{3}{z^3} - \frac{1}{z}\right) \sin(z) - \frac{3}{z^2}\cos(z) }

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_0(z) = -\frac{\cos(z)}{z} } 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_1(z) = -\frac{\cos(z)}{z^2} - \frac{\sin(z)}{z} } 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_2(z) = - \left( \frac{3}{z^3} - \frac{1}{z}\right) \cos(z) - \frac{3}{z^2}\sin(z) }

We may also define spherical Hankel functions of the first and second kind in terms of the spherical Bessel and Neumann functions:

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 h_{\ell}^{(1)} = j_{\ell}(z) + in_{\ell}(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 h_{\ell}^{(2)} = j_{\ell}(z) - in_{\ell}(z) }


The asymptotic forms of the spherical Bessel and Neumann functions 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 z\rightarrow\infty} are

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 j_{\ell}(z) = \frac{\sin(z-\frac{\ell \pi}{2})}{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 n_{\ell}(z) = \frac{\cos(z-\frac{\ell \pi}{2})}{z}. }

The first few zeros of the spherical Bessel function for 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=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 l=1\!} are

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 = 0: 3.142, 6.283, 9.425, 12.566, \ldots }

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 l = 1: 4.493, 7.725, 10.904, 14.066, \ldots}

The derivatives of the spherical Bessel and Neumann functions are 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 j'_{l}(z) = \frac{l}{z}j_{l}(z) - j_{l+1}(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 n'_{l}(z) = \frac{l}{z}n_{l}(z) - n_{l+1}(z). }

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