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Hevander Da Costa

Wolfram University of Maryland: Quantum Mechanics

Hevander Da Costa

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Quantum Mechanics Normalization and Probabilities Relative Statistics Expectation value ( μ ): 〈 k x 〉 = ∫ * ψ k x ψ dxStandard Deviation ( σ ): σ x = 〈 2 x 〉-〈x 2 〉 Patterns (complexconjugate) In[]:= * φ_ :=φ/.Complex[u_,v_]Complex[u,-v] In[]:= (innerprodutwithoptionaloperatorandbounds) InnerProduct[〈φ_,k_:1,ψ_〉,lower_:-Infinity,upper_:Infinity]:=Integrate[ * φ kψ,{x,lower,upper}] In[]:= (momentumoperator) In[]:=  p [k_,var_]:= k ℏ I D[#,{var,k}]&;('I':complex,'#':derivativeofvarwithrespecttok) In[]:=  p [1,x] In[]:= 1 ℏ  ∂ {x,1} #1&(impliesneedstobefeadanotherfunction)  p [1,x]@ψ[x] In[]:= -ℏ ′ ψ [x] Out[]= Examples Time Independent Wave Function 0 x < 0 ψ( x ) = { A 2 x 2 (x-c) 0 ≤ x ≤ c 0 c < x Determine normalization constant “A” Evaluate 〈 x 〉, 〈 2 x 〉 and σ x Evaluate 〈 p 〉, 〈 2 p 〉 and σ p Compute σ x , σ p $Line=0; In[]:= (definewavefunction) In[]:= ψ[x_/;x<0]:=0ψ[x_]:=A 2 x 2 (x-c) In[]:= DownValues@ψ In[]:= {HoldPattern[ψ[x_/;x<0]]0,HoldPattern[ψ[x_]]A 2 x 2 (x-c) } Out[]= (solvefornormalizationconstantandupdateDownValue) In[]:= Solve[InnerProduct[〈ψ[x],ψ[x]〉,0,c]1,A][[2,1]] In[]:= A 3 70 9/2 c Out[]= ψ[x_]=ψ[x]/.%;(replaceAwiththepreviouslycomputedoutput) In[]:= DownValues@ψ In[]:= HoldPattern[ψ[x_/;x<0]]0,HoldPattern[ψ[x_]] 3 70 2 x 2 (-c+x) 9/2 c  Out[]= (showthatthecorrectconstanthasbeendetermined) In[]:= InnerProduct[〈ψ[x],ψ[x]〉,0,c]==1 In[]:= True(correctnormalizationconstantfound) In[]:= (Evaluating〈x〉,〈 2 x 〉, σ x ) In[]:= #,#2, #2- 2 # &@@(InnerProduct[〈ψ[x],#,ψ[x]〉,0,c]&/@{x, 2 x }) In[]:=  c 2 , 3 2 c 11 , 2 c 2 11  Out[]= (Evaluating〈p〉,〈 2 p 〉, σ p ) In[]:= #,#2, #2- 2 # &@@InnerProductψ[x],  p [#,x]@ψ[x],0,c&/@{1,2} In[]:= 0, 12 2 ℏ 2 c ,2 3 2 ℏ 2 c  Out[]= (* σ x σ p >ℏ/2) In[]:= Out[10][[3]]Out[14][[3]] In[]:= 3 11 2 c 2 ℏ 2 c Out[]= unc=%Assuming[c>0&&ℏ>0,Refine@(unc>ℏ/2)] In[]:= 3 11 2 c 2 ℏ 2 c Out[]= True Out[]= Time dependent Wave Function Determine the normalization constant ‘n’ Is the probability of finding the particle in the interval (L, ∞) and t = 0 different from the probability at t = 10? What is the probability of finding the particle in the interval ( 〈 x 〉 - σ x , 〈 x 〉 + σ x )? (SetGlobalassumption) $Assumptions=L∈Reals&&L>0; In[]:= (Definingthewavefunction) Ψ[x_,t_]:=nExp- Abs@x L - eneI ℏ t In[]:= DownValues@Ψ In[]:= HoldPattern[Ψ[x_,t_]]nExp- Abs[x] L - (ene)t ℏ  Out[]= (Normalizationcoefficient) Solve[InnerProduct[〈Ψ[x,t],Ψ[x,t]〉]1,n][[2,1]]Ψ[x_,t_]=Ψ[x,t]/.%;(findpattern{Ψ[x,t]}replace{/.}'n'withthepreviousoutput{%}) n enet ℏ  L Out[]= DownValues@Ψ In[]:= HoldPattern[Ψ[x_,t_]] - Abs[x] L  L  In[]:= (innerproductoverthespecifiedinterval) {#,N@#}&@InnerProduct[〈Ψ[x,t],Ψ[x,t]〉,L,Infinity]({#,N@#}symbolic_value,numeric_value;&endoffunction,@affix) In[]:=  1 2 2  ,0.0676676 Out[]= (<x>,< 2 x >, σ x ) {lower,upper}={#-#3,#+#3}&@@#,#2, #2- 2 # &@@(InnerProduct[〈Ψ[x,t],#,Ψ[x,t]〉]&/@{x, 2 x }) In[]:= - 2 L 2 , 2 L 2  Out[]= {#,N@#}&@InnerProduct[〈Ψ[x,t],Ψ[x,t]〉,lower,upper] In[]:=  - 2  (-1+ 2  ),0.756883 Out[]= (Potting) Plot[Evaluate[ Ψ[x,t] Ψ[x,t]/.L1.],{x,-3.,3.},FillingAxis,GridLinesAutomatic,GridLinesAutomatic,GridLinesStyleDirective[Dashed,GrayLevel[0.8]],FrameLabel{"x","φ(x)"},PlotLabelHoldForm[ * Ψ[x,t] Ψ[x,t];L=1],PlotTheme"Scientific"] In[]:= Out[]= Credit: https://www.youtube.com/channel/UCjI34lMnq_4ifnNoOsaVfzA/videos Attachments: QuantumMechanics.nb

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Wolfram University of Maryland: Quantum Mechanics

Quantum Mechanics

Relative Statistics

Patterns

Examples

Time Independent Wave Function

Time dependent Wave Function