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

http://slidepdf.com/reader/full/approximation-methods 2/2

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$"