approximation methods
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8/13/2019 Approximation Methods
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Quantum Mechanics — Approximation Methods
Although approximation methods are not required for a 1D particle in a box problem (since theexact solution is known) this model ser!es as a good example of the application ofapproximation methods" #or real s$stems where exact solutions are either unknown or not possible approximation methods ser!e as the onl$ means b$ which scientists can estimate s$stemstates and energies" %n real s$stems onl$ the h$drogen atom and & '
molecule are sol!ed exactly"All other atomic and molecular s$stems require approximation methods"
article in a 1D box *
+ariation Model *
#ind φ that is an approximate function representing the state of the s$stem that is exactl$
represented b$ ψ (which howe!er is usuall$ unknown if this s$stem is being applied)" %t can be
pro!en that """""""
∫ φ,Ĥφ dτ ≥ -gs if φ is normali.ed otherwise """""""
∫ φ,Ĥφ dτ////////// ≥ -gs
∫ φ,φ dτ
An approximate function for the 1D particle in a box is """" φ = x(a-x)
#or this function do the following """"""""""
1" 0how that it meets the boundar$ conditions"
if x = 0 then φ = 0 & if x = a then φ = 0 : this checks out.'" ormali.e the function (find the normali.ation constant)"
A2 ∫0a φ2 dx = 1 & φ2 = x2a2 – 2ax3 x!
A2 ∫0a φ2 dx = x3a2"3 – ax!"2 x#"#$0a = a#"3 – a#"2 a#"# = 1 = A2(a#"30)
A = (30"a#)1"2
2" 0how that -φn 3 -ψ n 4 find the 5 error in the
calculated -"
%φn = -1#'2"a# ∫0a φ d2φ"dx2) dx = -1#'2"a# 2x3"3 –ax2$0
a = #'2"a2 = #h2"(!π2a2)= 0.12*+(h2"a2) , 0.12#(h2"a2)
eo = 1.!6" lot the function along with the
Ψ 7 ('8a)18' sin(nπx8a) and compare"
9he plots of φ and φ' (red) compared to ψ 8ψ ' (black)
9he !ariational functions are a bit lower at the maxima"
8/13/2019 Approximation Methods
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erturbation Model *9his is often applied when the exact wa!e function is known but it is impossible to sol!e
the &almitonian for the s$stem" :ften this is due to the potential energ$ term" %n such a case the&amiltonian can be separated into two (or more) parts"""""""
Ĥ ≈ Ĥ
;
Ĥ′ Ĥ′′ """""""" etc"
where Ĥ′ makes a small but significant contribution to the energ$" 9o get the approximate energ$
of the s$stem one sol!es for -; (exact) and adds to it -′ (an expectation !alue)" <nlike the
+ariation Method the resultant energ$ need not be greater than the actual energ$=
9o use the particle in a 1D box as a model let>s modif$ the s$stem b$ introducing a potentialenerg$ term inside the box such that + 7 kx" (k 7 kg s/') ote that the wa!efunction is
unchanged Ψ 7 ('8a)18' sin(nπx8a) but this cannot be sol!ed when + is included in the potential
energ$ term of Ĥ"
9herefore let Ĥ;
7 the same as the &amiltonian for the 1D box while Ĥ′ 7 onl$ the potentialenerg$ term (kx)" 9hus -; 7 n'h'8?ma' "
-′ 7 ∫ ψ n,&′ψ n 7 ∫ 0a (2/a) • kx • sin'(nπx8a) dx
1" 0ol!e for -′ using a table of integrals"
'k8a ∫ x sin'cx dx 7 x'86 @ x sin('cx)8(6c)B @ cos('cx)8(?c')B where c 7 (nπ8a)
7 k a'86 @ ; @ a'8(?n'π') @ ; ; a'8(?n'π')B 7 ka8'
- C -o -′ 7 n'h'8?ma' ka8'
9he !alue of - obtained represents a better result than simpl$ ignoring the potential energ$ term but does not gi!e the actual energ$ !alue of the s$stem" <nless $ou ha!e a wa$ to measure theactual energ$ of the s$stem it cannot be determined how close $ou reall$ are" %f $ou do ha!e anexperimental> !alue for - $ou can modif$ the perturbation operators b$ a trial and error processto obtain a better result" ote that with the !ariation method $ou don>t ha!e to ha!e anexperimental target> to know if $ou are getting closer to the true result" &ow significant the potential energ$ term is depends on two parameters the potential energ$ force constant k andthe si.e of the box a" ote that as the box gets larger the potential energ$ term makes a greatercontribution to the total energ$"