restart; Digits:=14: # Completeness Relation - how much is complete? # we look at a subset of states -> how much do they contribute towards unity in add(|n><n|, n=1..N)? # How do we approach the question? # think about powers of operators -> take their matrix representation, compare that against direct calculations. # Use A=-I*d/dx, and A^2=-d/dx^2 as an operator example, and a subset of the harmonic oscillator states. with(orthopoly);

NyhJIkdHNiJJIkhHRiRJIkxHRiRJIlBHRiRJIlRHRiRJIlVHRiQ=

w:=0.75;

JCIjdiEiIw==

alpha:=sqrt(w);

JCIvVyV5LmEtbSkhIzk=

phi:=n->exp(-0.5*(alpha*x)^2)*H(n,alpha*x);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSRleHBHRiU2IywkKiYkIiImISIiIiIiKiZJJmFscGhhR0YlIiIjSSJ4R0YlRjVGMkYxRjItSSJIR0YlNiQ5JComRjRGMkY2RjJGMkYlRiVGJQ==

NI:=n->int(phi(n)^2,x=-infinity..infinity);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50R0YlNiQqJC1JJHBoaUdGJTYjOSQiIiMvSSJ4R0YlOywkSSlpbmZpbml0eUclKnByb3RlY3RlZEchIiJGNkYlRiVGJQ==

NI(0);

JCIvSSplVGBtLyMhIzg=

NT:=[seq(1/sqrt(NI(n)),n=0..99)]: # Now we have to pick some states |mu> and |nu>, which will be used to define the A^2 matrix element. # For simplicity we will choose as reference states harmonic oscillator eigenstates with some other oscillator constant: beta:=0.5;

JCIiJiEiIg==

psi:=n->exp(-0.5*(beta*x)^2)*H(n,beta*x);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywkKiYkIiImISIiIiIiKiZJJWJldGFHRiUiIiNJInhHRiVGOEY1RjRGNS1JIkhHRiU2JDkkKiZGN0Y1RjlGNUY1RiVGJUYl

NIb:=n->int(psi(n)^2,x=-infinity..infinity);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJC1JJHBzaUdGJTYjOSQiIiMvSSJ4R0YlOywkSSlpbmZpbml0eUdGLCEiIkY5RiVGJUYl

n1:=0: n2:=2: NC1:=1/sqrt(NIb(n1));

JCIvaDhnJ2Y3SiYhIzk=

NC2:=1/sqrt(NIb(n2));

JCIvQztoUSJ5KD0hIzk=

ME:=-NC1*NC2*int(psi(n1)*diff(psi(n2),x$2),x=-infinity..infinity);

JCEvaydIJnB3bjwhIzk=

# Now let us see how we get this by inserting a completeness relation (however, a truncated one!) # WE COMPUTE THE FIRST ME by ACTING with -I*diff on the bra, so we change it to +I*diff ! ME1:=n->I*NC1*NT[n+1]*int(phi(n)*diff(psi(n1),x),x=-infinity..infinity);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKipeIyIiIkYrSSROQzFHRiVGKyZJI05UR0YlNiMsJjkkRitGK0YrRistSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JComLUkkcGhpR0YlNiNGMUYrLUklZGlmZkdGNTYkLUkkcHNpR0YlNiNJI24xR0YlSSJ4R0YlRisvRkM7LCRJKWluZmluaXR5R0Y1ISIiRkdGK0YlRiVGJQ==

ME1(0); # derivative of an even function is odd !

XiMkIiIhIiIh

ME1(1);

XiMkIS9OPkBRUVxHISM5

ME2:=n->-I*NC2*NT[n+1]*int(phi(n)*diff(psi(n2),x),x=-infinity..infinity);

Zio2I0kibkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKipeIyEiIiIiIkkkTkMyR0YlRiwmSSNOVEdGJTYjLCY5JEYsRixGLEYsLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQqJi1JJHBoaUdGJTYjRjJGLC1JJWRpZmZHRjY2JC1JJHBzaUdGJTYjSSNuMkdGJUkieEdGJUYsL0ZEOywkSSlpbmZpbml0eUdGNkYrRkhGLEYlRiVGJQ==

ME2(1);

XiMkIS8/ZnheJz0wKCEjOQ==

ME_tr:=N->add(ME1(n)*ME2(n),n=0..N);

Zio2I0kiTkc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkYWRkRyUqcHJvdGVjdGVkRzYkKiYtSSRNRTFHRiU2I0kibkdGJSIiIi1JJE1FMkdGJUYwRjIvRjE7IiIhOSRGJUYlRiU=

ME_tr(0);

JCIiIUYj

ME_tr(1);

JCEvb2lhcU00PyEjOQ==

ME_tr(2);

JCEvb2lhcU00PyEjOQ==

ME_tr(3);

JCEvIVJvayEqcDYjISM5

ME_tr_T:=[seq(ME_tr(n),n=0..30)]: plot([seq([j-1,ME_tr_T[j]],j=1..31)]);

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

ME_tr_T[10],ME;

NiQkIS9dYzROJ1J6IiEjOSQhL2snSCZwd248RiU=

ME_tr_T[20],ME;

NiQkIS9iYC5EJXl3IiEjOSQhL2snSCZwd248RiU=

ME_tr_T[30],ME;

NiQkIS8heSw0bnh3IiEjOSQhL2snSCZwd248RiU=

# Exercises: # change the states |mu> and |nu>. For example: decrease their beta, so that they are more delocalized. Then repeat the convergence analysis. # # Change the states |mu> and |nu> by choosing different n1 and n2, then repeat the convergence analysis. # This is sending the message: a truncated completeness relation -> there is no universal statement possible as to what happens when N is xx or yy.