Expectation values
The fifth example discusses expectation values.
- Expectation_values.Quanty
-- Using operators and wavefunctions as explained in -- the Operators and Wavefunctions example we can do -- simple calculations of expectation values -- 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 6 one electron wave functions psi0=NewWavefunction(NF,NB,{{"100000",1}}) psi0.Name = "psi(1 -1 1/2 -1/2)" psi1=NewWavefunction(NF,NB,{{"001000",1}}) psi1.Name = "psi(1 0 1/2 -1/2)" psi2=NewWavefunction(NF,NB,{{"000010",1}}) psi2.Name = "psi(1 1 1/2 -1/2)" psi3=NewWavefunction(NF,NB,{{"010000",1}}) psi3.Name = "psi(1 -1 1/2 1/2)" psi4=NewWavefunction(NF,NB,{{"000100",1}}) psi4.Name = "psi(1 0 1/2 1/2)" psi5=NewWavefunction(NF,NB,{{"000001",1}}) psi5.Name = "psi(1 1 1/2 1/2)" -- 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) -- We can look at expectation values by multiplying -- a wavefunction with an operator with a -- wavefunction ExpLz = psi0 * OppLz * psi0 print("The expectation value for psi0 is given as <psi0 | Lz | psi0>=",ExpLz) ExpSz = psi0 * OppSz * psi0 print("The expectation value for psi0 is given as <psi0 | Sz | psi0>=",ExpSz) print("") -- A print of the operator gives the operator in -- second quantization. knowing a basis of -- wavefunctions we can also look at a matrix -- representation of the operators. To simplify -- loops we first create a table of the functions psiList={psi0,psi1,psi2,psi3,psi4,psi5} -- print the Sz operator print("The Sz operator on a basis of psi0 to psi5 looks like:") for i = 1, 6 do for j = 1, 6 do io.write(string.format("%5.2f ",psiList[i] * OppSz * psiList[j])) end io.write("\n") end print("") -- print the Lz operator print("The Lz operator on a basis of psi0 to psi5 looks like:") for i = 1, 6 do for j = 1, 6 do io.write(string.format("%5.2f ",psiList[i] * OppLz * psiList[j])) end io.write("\n") end print("") -- the Lz and Sz operators are represented by -- diagonal matrices. This is so while our basis -- states are chosen (created) to be eigen states -- of Lz and Sz. If one looks at other operators -- this is different. print("The spin orbit coupling operator l.s on a basis of psi0 to psi5 looks like:") psiList={psi0,psi1,psi2,psi3,psi4,psi5} for i = 1, 6 do for j = 1, 6 do io.write(string.format("%5.2f ",psiList[i] * Oppldots * psiList[j])) end io.write("\n"); end -- The matrix form of the operators depends on the -- basis states we take. For two electrons we need -- to define 15 two particle states in the p-shell. print("") print("========== Two electrons in the p-shell ==========") psi0 =NewWavefunction(NF,NB,{{"110000",1}}) psi1 =NewWavefunction(NF,NB,{{"101000",1}}) psi2 =NewWavefunction(NF,NB,{{"100100",1}}) psi3 =NewWavefunction(NF,NB,{{"100010",1}}) psi4 =NewWavefunction(NF,NB,{{"100001",1}}) psi5 =NewWavefunction(NF,NB,{{"011000",1}}) psi6 =NewWavefunction(NF,NB,{{"010100",1}}) psi7 =NewWavefunction(NF,NB,{{"010010",1}}) psi8 =NewWavefunction(NF,NB,{{"010001",1}}) psi9 =NewWavefunction(NF,NB,{{"001100",1}}) psi10=NewWavefunction(NF,NB,{{"001010",1}}) psi11=NewWavefunction(NF,NB,{{"001001",1}}) psi12=NewWavefunction(NF,NB,{{"000110",1}}) psi13=NewWavefunction(NF,NB,{{"000101",1}}) psi14=NewWavefunction(NF,NB,{{"000011",1}}) -- The operator in terms of creation and -- annihilation operators is still the same -- and can thus be used to calculate expectation -- values print("") ExpLz = psi0 * OppLz * psi0 print("The expectation value for psi0 is given as <psi0 | Lz | psi0> =",ExpLz) ExpSz = psi0 * OppSz * psi0 print("The expectation value for psi0 is given as <psi0 | Sz | psi0> =",ExpSz) print("") -- to simplify loops we first create a table of the -- functions psiList = {psi0, psi1, psi2, psi3, psi4, psi5, psi6, psi7, psi8, psi9, psi10, psi11, psi12, psi13, psi14} -- print the Sz operator print("The Sz operator on a basis of psi0 to psi14 looks like:") for i = 1, 15 do for j = 1, 15 do io.write(string.format("%4.1f ",psiList[i] * OppSz * psiList[j])) end io.write("\n") end print("") -- print the Lz operator print("The Lz operator on a basis of psi0 to psi14 looks like:") for i = 1, 15 do for j = 1, 15 do io.write(string.format("%4.1f ",psiList[i] * OppLz * psiList[j])) end io.write("\n") end print("") -- print the Ssqr operator print("The S^2 operator on a basis of psi0 to psi14 looks like:") for i = 1, 15 do for j = 1, 15 do io.write(string.format("%4.1f ",psiList[i] * OppSsqr * psiList[j])) end io.write("\n") end print("")
The output is:
- Expectation_values.out
The expectation value for psi0 is given as <psi0 | Lz | psi0>= -1 The expectation value for psi0 is given as <psi0 | Sz | psi0>= -0.5 The Sz operator on a basis of psi0 to psi5 looks like: -0.50 0.00 0.00 0.00 0.00 0.00 0.00 -0.50 0.00 0.00 0.00 0.00 0.00 0.00 -0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.50 The Lz operator on a basis of psi0 to psi5 looks like: -1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 -1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 The spin orbit coupling operator l.s on a basis of psi0 to psi5 looks like: 0.50 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.71 0.00 0.00 0.00 0.00 -0.50 0.00 0.71 0.00 0.00 0.71 0.00 -0.50 0.00 0.00 0.00 0.00 0.71 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.50 ========== Two electrons in the p-shell ========== The expectation value for psi0 is given as <psi0 | Lz | psi0> = -2 The expectation value for psi0 is given as <psi0 | Sz | psi0> = 0 The Sz operator on a basis of psi0 to psi14 looks like: 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 The Lz operator on a basis of psi0 to psi14 looks like: -2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 The S^2 operator on a basis of psi0 to psi14 looks like: 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
