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Question: Quantum perturbation theory: energy denominators aren't evaluated properly

Question:Quantum perturbation theory: energy denominators aren't evaluated properly

w-Teilchen 80 Maple 18
physics denominator perturbation
October 13 2014
4

Hi!

I seem to run into problems with (quantum) perturbation theory. In the following minimal working example, an energy denominator, as occuring in perturbation theory is not evaluated if I previously assume that the quantum number is a positive integer. It's supposed to return an error, as the energy denominator is 0.

> 

restart; with(Physics)
> 

a := Annihilation(N, 1, notation = explicit):
> 

psi := Ket(N, m);

Physics:-Ket(N, m)

 (1)
> 

E := proc (g) options operator, arrow; (1/2)*g*(g-1)-mu*g end proc:
> 

simplify(value(Typesetting:-delayDotProduct(Typesetting:-delayDotProduct(Dagger(psi), 1/(E(m)-H)), psi)))

-I/(Im(m)*(2*Re(m)-1-2*mu))

 (2)
> 

assume(`in`(m, nonnegint), m > 0);
> 

simplify(value(Typesetting:-delayDotProduct(Typesetting:-delayDotProduct(Dagger(psi), 1/(E(m)-H)), psi)))

-2*Physics:-Bracket(Physics:-Bra(N, m), 1/(-2*mu*Physics:-`*`(`a-`[N[1]], `a+`[N[1]])+2*mu*m-m^2+Physics:-`*`(Physics:-`^`(`a-`[N[1]], 2), Physics:-`^`(`a+`[N[1]], 2))-4*Physics:-`*`(`a-`[N[1]], `a+`[N[1]])+2*mu+m+2), Physics:-Ket(N, m))

 (3)
> 

Parse:-ConvertTo1D, "invalid input %1", `.`(Dagger(psi), 1/(E-H), psi)

2/(2*mu*m-m^2+2*E+m)

 (4)
> 

``

Download energy_denominator.mw

 

Best regards,

Sören
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