Mixed State Eigenfuntion Equations

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Discussion Overview

The discussion revolves around the possibility of calculating eigenvalues E1 and E2 from a mixed state eigenfunction equation involving a Hamiltonian operator H, wave functions u1 and u2, and normalization constants a1 and a2. The focus is on the theoretical aspects of eigenvalue problems in quantum mechanics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions whether it is possible to calculate the eigenvalues E1 and E2 from the given equation, noting that there are two unknowns and only one equation.
  • Another participant suggests that splitting the equation into two separate equations could facilitate solving for E1 and E2, proposing the forms H(a1u1)=E1(a1u1) and H(a2u2)=E2(a2u2).
  • A later reply clarifies that 'h' was a typo and confirms that the normalization constants would remain the same since the wave functions are not being altered, only the terms are being considered separately.

Areas of Agreement / Disagreement

Participants express differing views on the approach to solving for the eigenvalues, with some suggesting splitting the equation while others raise concerns about the implications of normalization constants. The discussion remains unresolved regarding the feasibility of calculating the eigenvalues.

Contextual Notes

There are assumptions regarding the properties of the Hamiltonian operator and the wave functions that are not fully explored. The dependence on the definitions of normalization constants and eigenvalues is also noted but not resolved.

kd001
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H(a1u1 + a2u2) = a1E1 u1 + a2 E2u2

H is the Hamiltonian energy operator, a1 and a2 are normalisation constants, u1 and u2 are wave functions, E1 and E2 are the eigenvalues. Is it possible to calculate the values of E1 and E2 from the above equation if everything else is given? It should be possible to calculate the two eigenvalues given the two eigenfunctions shouldn't it? But there are two unknowns and just one equation. I'm not asking for a solution, whether it is possible to do it or not and if yes some hints as how to go about it.

Thanks.
 
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yes it does look possible perhaps splitting it into 2 equations would help:
H(a1u1)=E1(a1u1)
h(a2u2)=E2(a2u2)
and then solve the eigenvalue problem for E1 and E2
 
VanOosten said:
yes it does look possible perhaps splitting it into 2 equations would help:
H(a1u1)=E1(a1u1)
h(a2u2)=E2(a2u2)
and then solve the eigenvalue problem for E1 and E2

Thanks. But Would 'h' be the same as 'H'? Also I believe the normalisation constants wouldn't be the same when the equation is split.
 
yes i did mean H, that was a typo
and the normalization constants will still be the same because the wave function itself is not being changed you are just looking at the u1 terms first then the u2 terms second
 

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