Linear Combination of Wavefunctions

In summary, the question asks if PsiPlus, a combination of two eigenfunctions, is also an eigenfunction of the Hamiltonian operator. The given equations are for the eigenfunctions Psi12 and Psi21. The solution is not provided due to a lack of understanding of the core concepts.
  • #1
anduril66
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Homework Statement


psiplus = 1/sqrt(2) (psi(1,2) + psi(2,1))
Is PsiPlus an eigenfunction of the Hamiltonian operator? If so, what are the eigenvalues for the energy corresponding to PsiPlus in units of (h^2/(8ma)^2)?


Homework Equations


psiplus = 1/sqrt(2) (psi(1,2) + psi(2,1))

psi12 =2/a sin(Pi*x/a) *sin (2Pi*y/a) (2D Particle in Box)
psi21 =2/a sin(2*Pi*x/a) *sin (Pi*y/a)

The Attempt at a Solution


I don't understand some of the core concepts yet, so I don't know how to approach this problem.
 
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  • #2
I think that because PsiPlus is a combination of two eigenfunctions (Psi12 and Psi21), it must be an eigenfunction. However, I don't know how to find the eigenvalues for the energy corresponding to PsiPlus. Can someone help me?
 

FAQ: Linear Combination of Wavefunctions

1. What is a linear combination of wavefunctions?

A linear combination of wavefunctions is a mathematical concept used to describe the behavior of a quantum system. It involves combining multiple wavefunctions, each representing a possible state of the system, to create a new wavefunction that represents the overall state of the system.

2. Why is linear combination of wavefunctions important in quantum mechanics?

Linear combination of wavefunctions is important because it allows us to describe the behavior of quantum systems that have multiple possible states. By combining these states, we can better understand the overall behavior of the system and make predictions about its future behavior.

3. How is a linear combination of wavefunctions calculated?

A linear combination of wavefunctions is calculated by multiplying each wavefunction by a coefficient and then adding them together. These coefficients can be adjusted to create a linear combination that best fits the observed behavior of the system.

4. What is the significance of the coefficients in a linear combination of wavefunctions?

The coefficients in a linear combination of wavefunctions represent the probability amplitudes for each state of the system. This means that the squared magnitude of each coefficient gives us the probability of observing the system in that particular state.

5. Can a linear combination of wavefunctions accurately describe all quantum systems?

No, a linear combination of wavefunctions can only accurately describe quantum systems that are in a superposition of multiple states. Systems that are not in a superposition can still be described by a single wavefunction.

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