I TISE solutions should be combinations-of-eigenstates, why this is not?

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The discussion centers on the Time-Independent Schrödinger Equation (TISE) and the nature of its solutions. It highlights that while TISE is a linear equation, linear combinations of solutions do not yield valid solutions unless the eigenvalues are degenerate. The example of a free particle solution, expressed as exp[-ikx], illustrates that combining solutions with Gaussian coefficients does not produce a valid wave packet in the TISE context. Participants emphasize the distinction between TISE and the Time-Dependent Schrödinger Equation (TDSE), noting that temporal dependence allows for valid combinations. The conversation ultimately clarifies the limitations of forming solutions in TISE compared to TDSE.
JackeTheDog132
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Why a wave pocket is not a solution of the TISE?
I would really appreciate some help with a question I have aboute the TISE (Sch. tipe indipendent equation). This is a linear equation and linear combination of the solution should be solution too. The problem is that for the free particle, which solution can be written like exp[-ikx], a linear combination using gaussian coefficent is not anymore a solution (we should get a wave pocket this way). Of course taking a combination considering the temporal dipendence give a solution to the TDSE. My question is why that does not appen in the TISE case.
 
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Can you express your question more mathematically? I'm not sure what you are asking.
 
PeroK said:
Can you express your question more mathematically? I'm not sure what you are asking.
I agree. It also sounds like you are trying to model a traveling wave with a time-independent model, which of course will not work.
 
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JackeTheDog132 said:
This is a linear equation and linear combination of the solution should be solution too.
Careful. What you are calling the "TISE" is not a single equation. It is many different equations, one for each different eigenvalue. So the only case where you can form a linear combination of solutions to get another solution is degeneracy, i.e., there are multiple eigenvectors with the same eigenvalue.
 

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