Linear algebra, eigenvalues, linear combinations, complex numbers.

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SUMMARY

The discussion centers on demonstrating that a linear combination of eigenstates remains an eigenstate with the same eigenvalue in quantum mechanics. Specifically, it addresses the scenario where eigenvalues a1 and a2 are equal, leading to the conclusion that any linear combination (psi) = a1(psi)1 + a2(psi)2 is also an eigenstate, satisfying the equation A(psi) = a(psi). Participants emphasize the importance of showing full workings and independent thought to facilitate assistance in solving the problem.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly eigenvalues and eigenvectors.
  • Familiarity with quantum mechanics principles, especially the role of eigenstates.
  • Knowledge of complex numbers and their application in quantum states.
  • Proficiency in mathematical notation and integrals, particularly in the context of quantum operators.
NEXT STEPS
  • Study the properties of eigenstates in quantum mechanics, focusing on linear combinations.
  • Learn how to apply the eigenvalue equation A(psi) = a(psi) in various quantum systems.
  • Explore the implications of complex numbers in quantum state representations.
  • Review linear algebra techniques, particularly those related to eigenvalues and eigenvectors.
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Students and professionals in physics, particularly those studying quantum mechanics, as well as mathematicians focusing on linear algebra applications in physical systems.

lavenderblue
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Help with Quantum please!

If a1=a2=a then show that any linear combination
(psi)=a1(psi)1 + a2(psi)2, where a1 and a2 re arbitrary complex numbers, is also an eigenstate with th same eigenvalue i.e. show that A(psi)=a(psi)

Could you please show your full workings I don't know where to start :(. Thanks!

Here's what I thought might be right, but I'm not quite sure...

where A(psi)1=a(psi)1 and A(psi)2=a(psi)2
<A> = the integral (psi)*(A)(psi)
 
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You need to show work in order to get help here. Your "work" doesn't have to be calculations, just something to show that you have spent some independent thought and effort on this problem.

What concepts apply?

What formulas or relation do you think you'll need?

Answer these questions and we'll be able to help you.
 

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