Lecture 3: Solutions of Schrodinger equation for … 3 Schrodinger Eq.pdfLecture 3: Solutions of...
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1 Lecture 3: Solutions of Schrodinger equation for special cases Last time: !ħ ! !! ∇ ! + = = ℏ !! !" Ψ Ψ( ) ! = 1 !! !! The probability of finding it must be unity. This is a critical point and will come up over and over. 1. Free electrons: (1‐D) (Simple but very important one) ‐ We consider electrons that propagate freely in potential free space in the positive x‐direction ‐ ∇ ! → ! ! ! !! ! ‐ ‐ !ħ ! !! ! ! ! !! ! = _____ Eq. 1 ‐ ‐ = !"# ‐
Transcript of Lecture 3: Solutions of Schrodinger equation for … 3 Schrodinger Eq.pdfLecture 3: Solutions of...
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Lecture3:SolutionsofSchrodingerequationforspecialcases
Lasttime:
!ħ!
!!∇!𝜓 + 𝑉𝜓 = 𝐸𝜓 = 𝑖ℏ !!
!"
Ψ 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑎 𝑤𝑎𝑣𝑒𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
Ψ(𝑥) !𝑑𝑥 = 1!!
!!
Theprobabilityoffindingitmustbeunity.Thisisacriticalpointandwillcomeupoverandover.
1. Freeelectrons:(1‐D)(Simplebutveryimportantone)‐ Weconsiderelectronsthatpropagatefreelyinpotentialfreespaceinthepositivex‐direction
‐ ∇!𝜓 → !!!!!!
‐
‐ !ħ!
!! !
!!!!!
= 𝐸𝜓_____Eq.1‐ ‐ 𝜓 𝑥 = 𝑒!"#‐
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‐ !"!"= 𝑖𝛼𝑒!"#
‐
‐ !!!!!!
= 𝑖!𝛼!𝑒!"#‐ ‐ InEq.1‐
‐ ħ!
!!𝛼!𝑒!"# = 𝐸𝜓
‐ ħ!𝛼!
2𝑚 𝑒!"# = 𝐸𝜓 𝑟𝑒𝑚𝑒𝑚𝑏𝑒𝑟 𝜓 = 𝑒!"#
∴ 𝐸 =ħ!𝛼!
2𝑚
‐Sincenoboundaryconditions,allvaluesofenergyareallowed.
α=!!"ħ!
______Eq.2
𝑝!
2𝑚 = 𝐸(𝐾.𝐸. ) 𝑝 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚
𝑝! = 2𝑚𝐸 𝑖𝑛 𝐸𝑞. 1.
α=!!
ħ!
=!ħ= ! ∙ !!
!
!!→ 𝜆 𝑑𝑒 𝐵𝑟𝑜𝑔𝑙𝑖𝑒!𝑠 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
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α=!!!= 𝑘
=k(wavenumber)
Kisavector→ 𝑘!𝑘!𝑘!
λ→ wavevector
Note:k–vectordescribesthewavepropertiesofanelectron.
QuantumMechanics(k)
ClassicalMechanism(p)
Andrememberk=!
ħ
BacktoEnergyE,wecanfind
E=ħ!
!!𝑘!
ψ(x)=𝑒!"#
Separationofvariables
ψ(x,t)=𝑒!"# ∙ 𝑒!"#
Thisisanequationoftravelingwave.Itrepresentsafreeparticle.
2. Boundelectrons[potentialwall]‐ Weconsiderthattheelectroncanmovefreelybetweeninfinitelyhighpotentialbarriers
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Electronscannotescapebecauseofthepotentialwall
Note:Thepotentialinsidethewalliszero.Thedifferencebetweenthiscaseandlastcaseistheboundaryconditions.
−ħ!
2𝑚 𝑑!𝜓𝑑𝑥! = 𝐸𝜓
Assumethesolutionisψ(x)=AeiαL+Be‐iαL
𝛼 = 2𝑚𝐸ħ!
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ApplyingBoundaryConditions
atx=zeroψ=0
atx=Lψ=0 sinceψ=0,x≤0,x≥L
0=A𝑒!"#$ + 𝐵𝑒!"#$
A=‐BorB=‐A
A𝑒!"# − 𝐴𝑒!!"# = 0
A(𝑒!"# − 𝑒!!"#) = 0 A
EULER'SFORMULAISTHEKEYTOUNLOCKINGTHESECRETSOFQUANTUMPHYSICS
FromEuler’sEquation
sin 𝛿 =12𝑖 𝑒
!" − 𝑒!!"
2𝑖𝐴2𝑖 𝑒!"# − 𝑒!!"# = 0
2A*𝑖sin(αL)=0
∴sinαL=0→αL=n𝜋
α= !"!n=0,1,2,3...(integer)
fromα=!!"ħ!
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E=ħ!
!!𝛼!
=ħ!
!!!!!!
!"!
E= ħ!!!
!!!"!𝑛!n=1, 2,3…
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‐ Thesecondcaseissimilartoanelectronboundtoitsatomicnucleus
‐ Onlycertainenergylevelsareallowedfortheelectron‐ Thisis“Energyquantization”‐ Probabilityoffindingtheelectronatinsidethewell:
𝜓! = A(𝑒!"# − 𝑒!!"#)α=n π/a
= 2𝐴𝑖 ∙ sin𝛼𝑥
𝜓𝜓∗ = 4𝐴!𝑠𝑖𝑛!𝛼𝑥
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3. Finitepotentialbarrier:(Tunneleffect)Assumeafreeelectronpropagatinginthepositivex‐directionmeetsapotentialbarrierV0(higherthanthetotalenergyofelectron).
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Writesch.Eq.Foreachregion:
RegionI.V=0
!!!!!!
+ !!ħ!𝐸𝜓
RegionII.
!!!!!!
+ !!ħ! 𝐸 − 𝑉! 𝜓 = 0
RegionIsolutions:
𝜓! = 𝐴𝑒!"# + 𝐵𝑒!!"#
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𝑎 = 2𝑚𝐸ħ!
RegionIIsolution:
𝜓!! = 𝐶𝑒!"# + 𝐷𝑒!!"#
Onlycertainsolutionsexist(forwhichnisinteger)
Let’splottheenergysolutionsforthetwocases:
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Thesecondcaseissimilartoanelectronbondtoanucleus.
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Firstthreelevelsofthestationarystateandprobabilityofthosestates.
RegionIsolutions:
𝜓! = 𝐴𝑒!"# + 𝐵𝑒!!"#
𝑎 = 2𝑚𝐸ħ!
RegionIIsolution:
𝜓!! = 𝐶𝑒!"# + 𝐷𝑒!!"#
β=!!ħ!
𝐸 − 𝑉!
(E–V0)islessthanzero
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β→imaginary
γ =if∴ 𝛾 = !!!! 𝑉! − 𝐸
𝜓!! = 𝐶𝑒!" + 𝐷𝑒!!"
Usingboundaryconditions
X→ ∞
𝜓!! = 𝐶 ∙∞+ 𝐷 ∙ 0
Cmustbezero
∴ 𝜓!! = 𝐷𝑒!!"
ΨdecreasesinregionIIexponentially
Thedecreaseishigherforlargerγ,forlargerpotentialbarrier
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Theelectronwavepropagatesinthefinitepotentialbarrier.
Tunnelingeffect:penetrationofapotentialbarrier
‐ Thisisonlyquantummechanicaleffect.‐ Inclassicalmechanics:IftheelectronkineticenergyissmallerthanV,theelectronwillbeentirelyreflectedand“cannotovercomethebarrier”
Examplesoftunneling:
‐ Tunnelingofelectronsfromonemetaltoanotherthroughanoxidefilm.
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‐ Emissionofalphaparticlesfromnucleibytunnelingthroughthebindingpotentialbarrier.
4. AnotherCase
‐ WecanfindthatelectroncanpenetrateregionIIandpropagatesinregionIII.
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−ħ!
2𝑚 ∇!𝜓 + 𝑉𝜓 = 𝐸𝜓 = 𝑖ℏ𝜕Ψ𝜕𝑡
Partialdiffereq.in!
‐ ∇!𝜓 → !!!!!!
+ !!!!!!
+ !!!!!!
‐ ‐ ! = h / 2!
mass=m
V=potential
𝜓 𝑥,𝑦, 𝑧, 𝑡 2dxdydz
givestheprobablyoffindingtheelectroninvolumedxdydz
!ħ!
!!∇!!!+ 𝑉 = 𝑖ħ !
!!"!"Eq.1
functionr functiontime
𝜓 𝑟, 𝑡 = 𝜓 𝑟 𝜔 𝑡 Eq2
SeparationofvariablessubstituteEq2inEq.1
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ForeachEq.tobecorrectitmustbeequaltoaconstant
‐ Foreachequationtoberight,itmustequaltoconstant!ħ!
!!∇!!!+ 𝑉 = 𝐸 Eisaconstant→Eq.3
iħ!"!"= 𝐸𝜔 Eisthesameconst.
solution:𝜔=exp !!"#ħ
fromeq.3
!ħ!
!!∇! + 𝑉 𝜓 = 𝐸𝜓
timeindependentSchrödingerEquation.
Itwillbeappliedtobecalculationsofstationeryconditions.
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