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Updated with archived mats.
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BIN
23_NLTE_ionization.pdf
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23_NLTE_ionization.pdf
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hw7.motes
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hw7.motes
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@ -0,0 +1,78 @@
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Problem 1
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━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━
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a) The wave function ψ(ξ) = ξᵖ exp(-ξ²/2) L(ξ²), should satisfy the boundary condition for either even or odd fuctions of ξ with p=0 or p=1, respectively, for any function L(u).
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Recall ξ = √(mω/ħ) x.
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dξ = √(mω/ħ) dx = α dx.
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ψ(ξ) = ξᵖ exp(-ξ²/2) L(ξ²).
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ψ'(ξ) = d/dξ (ξᵖ exp(-ξ²/2) L(ξ²))
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= (d/dξ ξᵖ) exp(-ξ²/2) L(ξ²)
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+ ξᵖ (d/dξ exp(-ξ²/2)) L(ξ²)
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+ ξᵖ exp(-ξ²/2) (d/dξ L(ξ²)).
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For an even state, ψ'(0) = 0 and p=0, so
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ψ'(ξ) = (d/dξ 1) exp(-ξ²/2) L(ξ²)
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+ 1 d/dξ exp(-ξ²/2) L(ξ²)
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+ 1 exp(-ξ²/2) d/dξ L(ξ²).
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ψ'(ξ) = - ξ exp(-ξ²/2) L(ξ²)
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+ exp(-ξ²/2) L'(ξ²).
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ψ'(0) = L'(0).
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This must be wrong, somehow... in this case the boundary condition only applies if L'(0) = 0.
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∎
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For an odd state, ψ(0) = 0 and p=1, so
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ψ(0) = 0ᵖ * ... = 0. This is trivial.
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∎
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b) Equation 2.72 from Griffiths:
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dHₙ/dξ = 2nHₙ₋₁(ξ).
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Substituting the state ψ(ξ), I can obtain the associated Laguerre differential equation
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uL'' + (p - ½ + 1 - u)L' + kL = 0, with u = ξ².
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A better and equivalent substitution uses
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ψ(x,t) = A exp(ι(kx - (ħk²/2m)t)).
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h = ξᵖ L(ξ²).
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u = ξ².
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ξᵖ = ξ²
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h = ξᵖ L(ξ²)
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ψ(x,t) = A exp( ι(kx - k²/4πm ξᵖ L(ξ²) t) )
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= A exp( ι(kx - k²/4πm ξᵖ L(u) t) ).
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This... is probably not the "h" you meant.
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h = ξᵖ L(ξ²).
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dHₙ/dξ = 2nHₙ₋₁(ξ).
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h'(ξ) = dh/dξ = pξᵖ⁻¹ L(ξ²) + ξᵖ L'(ξ²).
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h''(ξ) = pξᵖ⁻¹ L(ξ²) + ξᵖ L'(ξ²) = p(p-1)ξᵖ⁻² L(ξ²) + pξᵖ⁻¹ L'(ξ²) + pξᵖ⁻¹ L'(ξ²) + ξᵖ L''(ξ²).
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p(p-1)ξᵖ⁻² L(ξ²) + pξᵖ⁻¹ L'(ξ²) + pξᵖ⁻¹ L'(ξ²) + ξᵖ L''(ξ²).
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Hₙ in Griffiths maps to hₚ in the problem sheet, I'll assume.
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Hₙ = ξⁿ.
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3279
hw7.motes.ps
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hw7.motes.ps
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@ -11,7 +11,7 @@ covered:
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Correspondence Principle
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This seems to be primarily historical motivation.
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This seems to be primarily historical motivation.
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The correspondence principle, I'm not sure I've ever attempted to compute for even a simple system. Hmm... I see the note here
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"What is the Correspondence principle? Derive the quantum condition from it." I don't know what is meant by the "quantum condition". I guess just that the states of a system are quantized. I have no clue how I would show that using the correspondence principle. Should ask about this.
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@ -1,9 +1,4 @@
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Ehrenfest Theorems
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<p> = m d/dt <x> and <-∂/∂x V> = d/dt <p>
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─────────────
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<p> = m d/dt <x> and <-∂/∂x V> = d/dt <p>
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@ -226,12 +226,6 @@ Regarding group and phase velocities,
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Here, ω represents the phase velocity, and ϕ is narrowly peaked about k₀.
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<<<<<<< HEAD
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*** dispersion notes: derive group velocity from 2 frequencies
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*** dispersion comes from curvature of ω(k)
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=======
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A Taylor expansion helps elucidate the situation
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ω(k) = ω₀ + ω′₀(k-k₀)
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@ -247,4 +241,3 @@ At t=0, ψ(x,0) = 1/√(π) ∫[-∞,∞]ds ϕ(k₀ + s) exp(ι(kₒ+s)x),
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ψ(x,t) = 1/√(π) exp(ι(-ω₀ + kₒ ω′₀)t∫[-∞,∞]ds ϕ(k₀ + s) exp(ι(kₒ+s)x)
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>>>>>>> 9d60318cfe5d81c018be9b1fdbf72e03e8733ea0
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BIN
project/Group3_Documentation.pdf
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project/Group3_Documentation.pdf
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1770
project/HydrogenAtomRadialFunctions.nb
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project/HydrogenAtomRadialFunctions.nb
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project/InfiniteSquareWell.nb
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project/InfiniteSquareWell.nb
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1, StandardForm, "Variables" :> {$CellContext`n$$ = 10},
|
||||
"ControllerVariables" :> {
|
||||
Hold[$CellContext`n$$, $CellContext`n$870100$$, 0]},
|
||||
"OtherVariables" :> {
|
||||
Typeset`show$$, Typeset`bookmarkList$$, Typeset`bookmarkMode$$,
|
||||
Typeset`animator$$, Typeset`animvar$$, Typeset`name$$,
|
||||
Typeset`specs$$, Typeset`size$$, Typeset`update$$, Typeset`initDone$$,
|
||||
Typeset`skipInitDone$$}, "Body" :>
|
||||
Plot3D[Abs[(-1)^(2 $CellContext`n$$) 2^$CellContext`n$$
|
||||
E^(Rational[1, 2] $CellContext`x^2 +
|
||||
Rational[1, 2] $CellContext`y^2)
|
||||
Pi^Rational[
|
||||
1, 2] $CellContext`x^(-$CellContext`n$$) \
|
||||
$CellContext`y^(-$CellContext`n$$) Factorial[$CellContext`n$$]^(-1)
|
||||
HypergeometricPFQRegularized[{
|
||||
Rational[1, 2], 1}, {
|
||||
1 + Rational[1, 2] (-1 - $CellContext`n$$), 1 +
|
||||
Rational[-1, 2] $CellContext`n$$}, -$CellContext`x^2]
|
||||
HypergeometricPFQRegularized[{
|
||||
Rational[1, 2], 1}, {
|
||||
1 + Rational[1, 2] (-1 - $CellContext`n$$), 1 +
|
||||
Rational[-1,
|
||||
2] $CellContext`n$$}, -$CellContext`y^2]]^2, {$CellContext`x, \
|
||||
-6, 6}, {$CellContext`y, -6, 6},
|
||||
AxesLabel -> {
|
||||
"x", "y", "|\[Psi]\!\(\*SuperscriptBox[\(|\), \(2\)]\)"}, PlotRange ->
|
||||
All, PlotPoints -> 50],
|
||||
"Specifications" :> {{{$CellContext`n$$, 10, "Energy"}, 0, 10}},
|
||||
"Options" :> {}, "DefaultOptions" :> {}],
|
||||
ImageSizeCache->{612., {257., 266.}},
|
||||
SingleEvaluation->True],
|
||||
Deinitialization:>None,
|
||||
DynamicModuleValues:>{},
|
||||
SynchronousInitialization->True,
|
||||
UndoTrackedVariables:>{Typeset`show$$, Typeset`bookmarkMode$$},
|
||||
UnsavedVariables:>{Typeset`initDone$$},
|
||||
UntrackedVariables:>{Typeset`size$$}], "Manipulate",
|
||||
Deployed->True,
|
||||
StripOnInput->False],
|
||||
Manipulate`InterpretManipulate[1]]], "Output",
|
||||
CellChangeTimes->{{3.7216870765625362`*^9, 3.7216871038097744`*^9}, {
|
||||
3.7216871405282593`*^9, 3.721687155276568*^9},
|
||||
3.721687187423189*^9},ExpressionUUID->"aad3fb46-5ba0-4c07-9cd1-\
|
||||
f69642993cb6"]
|
||||
}, Open ]]
|
||||
},
|
||||
WindowSize->{1500, 917},
|
||||
WindowMargins->{{-8, Automatic}, {Automatic, -8}},
|
||||
Magnification:>1.5 Inherited,
|
||||
FrontEndVersion->"11.2 for Microsoft Windows (64-bit) (September 10, 2017)",
|
||||
StyleDefinitions->"Default.nb"
|
||||
]
|
||||
(* End of Notebook Content *)
|
||||
|
||||
(* Internal cache information *)
|
||||
(*CellTagsOutline
|
||||
CellTagsIndex->{}
|
||||
*)
|
||||
(*CellTagsIndex
|
||||
CellTagsIndex->{}
|
||||
*)
|
||||
(*NotebookFileOutline
|
||||
Notebook[{
|
||||
Cell[CellGroupData[{
|
||||
Cell[580, 22, 2035, 65, 151, "Input",ExpressionUUID->"4ca179a4-27d3-4c34-ae58-8b94da183330"],
|
||||
Cell[2618, 89, 2222, 45, 606, "Output",ExpressionUUID->"9b3d54d1-47ed-479c-8632-0bd2936168ee"]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[4877, 139, 992, 30, 78, "Input",ExpressionUUID->"810f6902-f7b5-477f-b48e-3c81937d6083"],
|
||||
Cell[5872, 171, 2407, 49, 593, "Output",ExpressionUUID->"2fe19ef6-740d-46c6-a73e-dbf82142124e"]
|
||||
}, Open ]],
|
||||
Cell[CellGroupData[{
|
||||
Cell[8316, 225, 1208, 34, 112, "Input",ExpressionUUID->"30bedbf7-3291-4d2a-a81a-c18d3b6ec891"],
|
||||
Cell[9527, 261, 2651, 54, 551, "Output",ExpressionUUID->"aad3fb46-5ba0-4c07-9cd1-f69642993cb6"]
|
||||
}, Open ]]
|
||||
}
|
||||
]
|
||||
*)
|
||||
|
||||
(* End of internal cache information *)
|
||||
|
||||
@ -18,6 +18,11 @@ double a = 4*pi*ep_0*hbar*hbar/m/e/e;
|
||||
// kappa = sqrt(-2*m*E/hbar)
|
||||
// kappa = m * e^2 / (4 * pi * ep_0 * hbar^2)
|
||||
|
||||
int factorial(int n)
|
||||
{
|
||||
return (n == 1 || n == 0) ? 1 : factorial(n - 1) * n;
|
||||
}
|
||||
|
||||
double kappa(double n)
|
||||
{
|
||||
double result = m * e^2 / (4 * pi * ep_0 * hbar^2);
|
||||
@ -43,24 +48,26 @@ double v(double n, double l, double rho)
|
||||
double R(double n, double l,
|
||||
double r)
|
||||
{
|
||||
double rho = r/a*n;
|
||||
double rho = r*a/n;
|
||||
double result = 1/r
|
||||
* pow(rho,l+1)
|
||||
* exp(-rho)
|
||||
* v(n,l,rho);
|
||||
}
|
||||
|
||||
normalize()
|
||||
|
||||
|
||||
double Y(double l, double m,
|
||||
double theta, double phi);
|
||||
|
||||
double psi(double n, double l, double m,
|
||||
double r, double theta, double phi)
|
||||
= R(n,l,r) * Y(l,m,theta,phi);
|
||||
|
||||
|
||||
{
|
||||
norm(n,l) * R(n,l,r) * Y(l,m,theta,phi);
|
||||
}
|
||||
double norm(double n, double l)
|
||||
{
|
||||
double result_squared = pow((2*a/n),3) * factorial(n-l-1) / (2*n) / pow(factorial(n+1),3);
|
||||
return sqrt(result_squared);
|
||||
}
|
||||
|
||||
|
||||
int main(int argc, char const *argv[])
|
||||
|
||||
@ -1,3 +1,20 @@
|
||||
PHY 520: Group Project – Written Documentation
|
||||
Steven Wallbrown, Henry Colburn, Bruce He, Otho Ulrich
|
||||
|
||||
We present descriptions of our models in three parts. In this document, the infinite square well and 1/r potential models are presented. The simple harmonic oscillator is described in word document, accompanying this submission. In each case, a model was built in mathematica that parallels the developments in "Introduction to Quantum Mechanics" [Griffiths] and Dr. Chris Crawford's course notes for the PHY 520 course at UK, 2017.
|
||||
|
||||
The states were easy to model without normalization. We found that normalizing the states was a difficult consideration. We were pleased to see the qualitative behaviour we expected, at least, and look forward to developing similar models more fully. The simple harmonic oscillator is plotted in 1d and 2d, and the infinite square well and 1/r potential solutions are plotted in 1d. Mathematica programs are included in this submission.
|
||||
|
||||
|
||||
Modeling the infinite square well potential
|
||||
─────────────
|
||||
|
||||
The concept behind the simulation created is that of the infinite square well potential of a one-dimensional wave. The basics behind the idea is that a particle trapped in an area bordered by infinite potential must be composed of waves whose nodes correspond to the distance from one infinite potential barrier to another. The waves must be zero at the boundaries but are not required to have their derivatives be zero since there is no exponential decay of the wave, due to the infinite potential. Thus, the classical equation, sin(nπx/a) (where a is the well width and n is an integer) works for this situation where as the full quantum equation is not neccesary. This gives rise to the quantization of the energy levels so only discrete values are allowed. The program in Mathematica is designed so that up to three different waves can be plotted of different energy levels. The Y axis is the energy levels of the waves and the x axis is the width of the well. The program demonstrates the relationship between the number of nodes of a wave (entered in as n1, n2, and n3) and the energy levels associated with those waves. The simulation also draws attention to the spreading out of each successive wave from its predecessor in terms of separation energy, e.g. to go from n=5 to n=6 requires less energy input than n=100 to n=101. Within each wave function being plotted there is a horizontal line that represents the energy level of the wave that is on top of it. This is for a clearer representation of the energy level of the wave. The plotting range was also modified within each function so as to guarantee the plot would be large enough to fit up to the highest energy wave. With continued playing, the relationship between energy increases and node variation would become apparent to a user.
|
||||
|
||||
|
||||
Modeling the 1/r potential - Hydrogen Atom
|
||||
─────────────
|
||||
|
||||
To model a quantum particle in a radial potential, we addressed the derivation of the wave function parallel with the development in Griffiths, pg. 145, and in Dr. Crawford's course notes, and some various online sources.
|
||||
|
||||
The potential V(r) = 1/r in spherical coordinates describes the system as having constants of motion in the angular coordinates theta and phi, thus conserving angular momentum. Still, it admits an effective potential that includes a centrifugal factor,
|
||||
@ -46,9 +63,9 @@ The complete solution is then constructed assuming a power series v(ρ),
|
||||
The recursion relationship for the cⱼ coefficients is given by
|
||||
|
||||
|
||||
⎧ 2(j + l + 1) - ρ₀ ⎫
|
||||
cⱼ₊₁ = ⎨─────────────────⎬ cⱼ,
|
||||
⎩(j + 1)(j + 2l + 2)⎭
|
||||
⎧ 2(j + l + 1) - ρ₀ ⎫
|
||||
cⱼ₊₁ = ⎨───────────────────⎬ cⱼ,
|
||||
⎩(j + 1)(j + 2l + 2)⎭
|
||||
|
||||
where c₀ is normalized by total probability equal to one.
|
||||
|
||||
@ -92,9 +109,9 @@ The model we've developed is nearly capable of handling the associated LaGuerre
|
||||
|
||||
Substituting our new information back into the recursion relationship gives us the recursion in terms of the quantum numbers,
|
||||
|
||||
⎧ 2(j + l + 1 - n) ⎫
|
||||
cⱼ₊₁ = ⎨────────────────⎬ cⱼ.
|
||||
⎩(j + 1)(j + 2l + 2)⎭
|
||||
⎧ 2(j + l + 1 - n) ⎫
|
||||
cⱼ₊₁ = ⎨───────────────────⎬ cⱼ.
|
||||
⎩(j + 1)(j + 2l + 2)⎭
|
||||
|
||||
This is sufficient information to model the radial component of the wave function, except for normalizing the coefficient c₀. This is done using the expression of probability conservation
|
||||
|
||||
@ -102,19 +119,31 @@ This is sufficient information to model the radial component of the wave functio
|
||||
∫ │Rₙₗ│² r² dr = 1.
|
||||
o
|
||||
|
||||
This is difficult to model numerically in the general case, and while interesting, the more practical approach is probably to normalize any states we're interested in by hand. The normalization of c₀ is different for each set of values {n,l}.
|
||||
We want to simulate the hydrogen atom, where V(r) ≠ -1/r, but instead V(r) = -e²/(4πε₀) 1/r. This changes some of our parameters, i.e.,
|
||||
|
||||
A model has (nearly) been written in c++ to compute the wavefunction resultant from this potential. The data is output to tab-separated data tables in the format
|
||||
ρ = κr = r/an and ρ₀ = m e²/(2π ε₀ ħ² κ),
|
||||
|
||||
r R₁₀ R₂₀ R₃₀ ...
|
||||
where a now represents the Bohr Radius,
|
||||
|
||||
Still deciding how to output the different values of l.
|
||||
a = 5.29 × 10⁻² nm.
|
||||
|
||||
The rest of the development is essentially the same.
|
||||
|
||||
In general, the normalization is given by
|
||||
|
||||
⎛ (2/na)³ (n-l-1)!⎞ 1/2
|
||||
⎜ ────────────── ⎟
|
||||
⎝ 2n [(n+l)!]³ ⎠.
|
||||
|
||||
We will notate this normalization as α. If we also notate the generalized Laguerre polynomials of x as L[q,p](x), then our model is
|
||||
|
||||
α exp(-r/na) (2r/na)ˡ L[n-l-1,2l+1](2r/na) Yₗᵐ(θ,ϕ).
|
||||
|
||||
We have not plotted this, but here Yₗᵐ refers to the spherical harmonics, which have the form (to within a normalization constant)
|
||||
|
||||
Yₗᵐ(θ,ϕ) = exp(ιmϕ) Pₗᵐ (cosθ),
|
||||
|
||||
where Pₗᵐ(x) are the Legendre Polynomials in x.
|
||||
|
||||
This model is implemented in mathematica, and will be presented to the Physics 520 class on Dec. 8, 2017.
|
||||
|
||||
|
||||
3855
project/writeup.mth.ps
Normal file
3855
project/writeup.mth.ps
Normal file
File diff suppressed because it is too large
Load Diff
BIN
project/writeup_othoulrich.pdf
Normal file
BIN
project/writeup_othoulrich.pdf
Normal file
Binary file not shown.
15
scratch.motes
Normal file
15
scratch.motes
Normal file
@ -0,0 +1,15 @@
|
||||
Schrodinger Equation
|
||||
|
||||
ι ħ ∂/∂t ψ(𝐫,t) = Ĥ ψ(𝐫,t)
|
||||
|
||||
|
||||
ψ(x,t) = A exp(ι(kx - (ħk²/2m)t))
|
||||
|
||||
|
||||
|
||||
ξᵖ = ξ⁰ = 1
|
||||
|
||||
|
||||
|
||||
|
||||
where L is the set of Laguerre polynomials,
|
||||
Loading…
Reference in New Issue
Block a user