====== Braket ======

###
//Braket(psi1, O, psi2)// calculates the expectation value or matrix element $\left\langle \psi_1 \mid O \mid \psi_2 \right\rangle$. In Quanty //Braket(psi1, O, psi2)// is the same as // psi1 * O * psi2//. The difference is that the function //Braket// can be faster. //Braket// can work on lists of functions and then returns a matrix or vector with all possible expectation values
###

===== Input =====

  * psi1 or psiList1 : Wavefunction or list of Wavefunctions
  * O : Operator
  * psi2 or psiList2 : Wavefunction or list of Wavefunctions

===== Output =====

  * real or complex number, or a list of these or a matrix of these

===== Example =====

###
The example calculates the expectation values of a few states
###

==== Input ====
<code Quanty Example.Quanty>
NF=2
NB=0
IndexDn={0}
IndexUp={1}

psi0=NewWavefunction(NF,NB,{{"10",1}})
psi1=NewWavefunction(NF,NB,{{"01",1}})

OppSx   = NewOperator("Sx"   ,NF,IndexUp,IndexDn)
OppSy   = NewOperator("Sy"   ,NF,IndexUp,IndexDn)
OppSz   = NewOperator("Sz"   ,NF,IndexUp,IndexDn)

print("The expectation value of a single state")
print(Braket(psi0,OppSz,psi0))

print("The expectation value of two states with a single state")
print(Braket(psi0,OppSx,{psi0,psi1}))

-- note that I made one list of length 3, the other of length 2 so that the order is clear
print("The expectation value of a three states with two other states")
print(Braket({psi0,psi1},OppSy,{psi0,psi1,psi1}))
</code>

==== Result ====
<file Quanty_Output>
The expectation value of a single state
-0.5
The expectation value of two states with a single state
{ 0 , 0.5 }
The expectation value of a three states with two other states
{ { 0 , (0 + 0.5 I) , (0 + 0.5 I) } , 
  { (0 - 0.5 I) , 0 , 0 } }
</file>

===== Table of contents =====
{{indexmenu>.#1}}