What Defines the Energy Spectrum in a Hamiltonian System?

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Homework Help Overview

The discussion revolves around understanding the energy spectrum of a Hamiltonian system, specifically one defined by a Hamiltonian that includes both a harmonic oscillator term and perturbative terms. The original poster expresses confusion about the concept of the energy spectrum and its components.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the definition of the energy spectrum and its relation to energy eigenvalues. There are inquiries about the implications of the perturbation parameter (lambda) and its role in determining the nature of the Hamiltonian. Questions arise regarding the number of energy eigenvalues and whether perturbations can alter them.

Discussion Status

The discussion is ongoing, with participants sharing their interpretations and questioning the terminology used in quantum mechanics. Some guidance has been offered regarding the meaning of the perturbation parameter, but there is no explicit consensus on the implications for the energy eigenvalues.

Contextual Notes

Participants are navigating terminology and concepts from quantum mechanics, particularly in the context of perturbation theory. There is a noted lack of clarity regarding the definitions and implications of the energy spectrum and eigenvalues.

degerativpart
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Homework Statement


Find the energy spectrum of a system whose Hamiltonian is
H=Ho+H'=[-(planks const)^2/2m][d^2/dx^2]+.5m(omega)^2x^2+ax^3+bx^4


I gues my big question to begin is what exactly makes up the energy spectrum. I know the equation to the first and second order perturbations but I am not sure exactly what the energy spectrum entails. Please help.

Homework Equations





The Attempt at a Solution


ive figured out that H'=ax^3+bx^4

and Ho==[-(planks const)^2/2m][d^2/dx^2]+.5m(omega)^2x^2
and lambda=1 which mean ita a full perturbation
 
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degerativpart said:
I gues my big question to begin is what exactly makes up the energy spectrum.
In QM, "energy spectrum" is just a stupid word that they use to mean the set of possible energy eigenvalues. I personally hate that terminology; it's so misleading.



degerativpart said:
and lambda=1 which mean ita a full perturbation
I don't know what this means.
 
haha obviously I agree with you and I read that the lambda in the equation for H=Ho+(lambda)H' when equals to zero means its an unperturbed equation and when it is equal to 1 then its fully perturbed. I don't know I read it.
But I guess my next question how many energy eigenvalues are there? Does that mean I should probably only go to the second oreder corrections?
 
degerativpart said:
... how many energy eigenvalues are there?
How many eigenvalues are there for H0? Can the perturbation remove or add any, or does it just shift them?
 
Sorry, I could say that book is the question?. thank you.
 

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