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documentation:tutorials:introduction_to_quanty:eigenstates [2016/10/07 20:26] – created Maurits W. Haverkortdocumentation:tutorials:introduction_to_quanty:eigenstates [2021/06/25 00:36] (current) – Zhaoyuan Yang
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 +{{indexmenu_n>6}}
 +====== Eigenstates ======
  
 +###
 +The sixth example discusses eigenstates.
 +###
 +
 +###
 +<code Quanty Eigenstates.Quanty>
 +-- Using operators and wavefunctions as explained in 
 +-- the Operators and Wavefunctions example
 +-- and being able to multiply them to get
 +-- expectation values we can continue and look
 +-- at eigenstates of operators
 +
 +-- define the basis
 +  -- For a p-shell we would like the have 6 
 +  -- spinorbitals, with the quantum numbers 
 +  -- spin up ml=-1,ml=0,ml=1 and
 +  -- spin down with ,ml=-1, ml=0, ml=1
 +NF=6
 +NB=0
 +IndexDn={0,2,4}
 +IndexUp={1,3,5}
 +
 +-- Define spin and angular momentum operators.
 +OppSx   =NewOperator("Sx"   ,NF,IndexUp,IndexDn)
 +OppSy   =NewOperator("Sy"   ,NF,IndexUp,IndexDn)
 +OppSz   =NewOperator("Sz"   ,NF,IndexUp,IndexDn)
 +OppSsqr =NewOperator("Ssqr" ,NF,IndexUp,IndexDn)
 +OppSplus=NewOperator("Splus",NF,IndexUp,IndexDn)
 +OppSmin =NewOperator("Smin" ,NF,IndexUp,IndexDn)
 +
 +OppLx   =NewOperator("Lx"   ,NF,IndexUp,IndexDn)
 +OppLy   =NewOperator("Ly"   ,NF,IndexUp,IndexDn)
 +OppLz   =NewOperator("Lz"   ,NF,IndexUp,IndexDn)
 +OppLsqr =NewOperator("Lsqr" ,NF,IndexUp,IndexDn)
 +OppLplus=NewOperator("Lplus",NF,IndexUp,IndexDn)
 +OppLmin =NewOperator("Lmin" ,NF,IndexUp,IndexDn)
 +
 +OppJx   =NewOperator("Jx"   ,NF,IndexUp,IndexDn)
 +OppJy   =NewOperator("Jy"   ,NF,IndexUp,IndexDn)
 +OppJz   =NewOperator("Jz"   ,NF,IndexUp,IndexDn)
 +OppJsqr =NewOperator("Jsqr" ,NF,IndexUp,IndexDn)
 +OppJplus=NewOperator("Jplus",NF,IndexUp,IndexDn)
 +OppJmin =NewOperator("Jmin" ,NF,IndexUp,IndexDn)
 +
 +Oppldots=NewOperator("ldots",NF,IndexUp,IndexDn)
 +
 +-- Define the coulomb operator
 +-- We here define the part depending on F0 
 +-- separately from the part depending on F2.
 +-- When summing we can put in the numerical values 
 +-- of the slater integrals.
 +OppF0 = NewOperator("U",NF,IndexUp,IndexDn,{1,0})
 +OppF2 = NewOperator("U",NF,IndexUp,IndexDn,{0,1})
 +OppU = 5.0 * OppF0 + 4.0 * OppF2
 +
 +-- Note that the previous definition is the same as
 +-- OppU = NewOperator("U", NF, IndexUp, IndexDn, 
 +--                         {5.0,4.0})
 + 
 +-- Define the Hamiltonian as a numerical sum of the 
 +-- previous defined operators.
 +Hamiltonian = 5.0 * OppF0 + 4.0 * OppF2 + 0.000001 * (2*OppSz + OppLz)
 +Hamiltonian2 = 6.0 * OppF0 + 4.0 * OppF2 + 0.000001 * (2*OppSz + OppLz)
 +
 +-- For large systems we do not need to know all
 +-- eigenstates, but can restrict ourselves to the
 +-- Npsi lowest states:
 +Npsi=15
 +
 +-- In order to make sure we have a filling of 2
 +-- electrons we need to define some restrictions
 +StartRestrictions = {NF, NB,  {"111111",2,2}}
 +
 +-- We now can create the lowest Npsi eigenstates:
 +psiList = Eigensystem(Hamiltonian, StartRestrictions, Npsi)
 +
 +oppList={Hamiltonian, OppSsqr, OppLsqr, OppJsqr, OppSz, OppLz, Oppldots, OppF0, OppF2}
 +
 +-- after we've created the eigen states we can look 
 +-- at a list of their expectation values
 +print(" <E>     <S^2>   <L^2>   <J^2>   <S_z>   <L_z>   <l.s>   <F[0]>  <F[2]>");
 +for key,value in pairs(psiList) do
 +  expvalue = value * oppList * value
 +  for k,v in pairs(expvalue) do
 +    io.write(string.format("%7.4f ",v))
 +  end;
 +  io.write("\n")
 +end
 +</code>
 +###
 +
 +###
 +The output is:
 +<file Quanty_Output Eigenstates.out>
 +Start of BlockGroundState. Converge 15 states to an energy with relative variance smaller than  1.490116119384766E-06
 +
 +Start of BlockOperatorPsiSerialRestricted
 +Outer loop   1, Number of Determinants:        15        15 last variance  3.608398351428264E-01
 +Start of BlockOperatorPsiSerialRestricted
 +Start of BlockGroundState. Converge 15 states to an energy with relative variance smaller than  1.490116119384766E-06
 +
 +Start of BlockOperatorPsiSerial
 + <E>     <S^2>   <L^2>   <J^2>   <S_z>   <L_z>   <l.s>   <F[0]>  <F[2]>
 + 4.2000  2.0000  2.0000  6.0000 -1.0000 -1.0000  0.5000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  4.0000 -1.0000  0.0000  0.0000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  4.0000  0.0000 -1.0000  0.0000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  2.0000 -1.0000  1.0000 -0.5000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  4.0000  0.0000  0.0000 -0.0000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  4.0000  0.0000  1.0000  0.0000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  2.0000  1.0000 -1.0000 -0.5000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  4.0000  1.0000  0.0000  0.0000  1.0000 -0.2000 
 + 4.2000  2.0000  2.0000  6.0000  1.0000  1.0000  0.5000  1.0000 -0.2000 
 + 5.1600  0.0000  6.0000  6.0000  0.0000 -2.0000  0.0000  1.0000  0.0400 
 + 5.1600  0.0000  6.0000  6.0000  0.0000 -1.0000  0.0000  1.0000  0.0400 
 + 5.1600  0.0000  6.0000  6.0000  0.0000  0.0000  0.0000  1.0000  0.0400 
 + 5.1600  0.0000  6.0000  6.0000  0.0000  1.0000  0.0000  1.0000  0.0400 
 + 5.1600  0.0000  6.0000  6.0000  0.0000  2.0000  0.0000  1.0000  0.0400 
 + 6.6000  0.0000 -0.0000 -0.0000  0.0000  0.0000  0.0000  1.0000  0.4000 
 +</file>
 +###
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort}}

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