Discussion Overview
The discussion revolves around the concept of eigenfunctions in quantum mechanics, specifically in the context of a particle in a three-dimensional cubic box. Participants explore whether a specific linear combination of eigenfunctions is itself an eigenfunction of the Hamiltonian and the implications of boundary conditions on the solutions.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant questions if the expression 1/\sqrt{2} * (ψ(1,1,5)+ψ(3,3,3)) is an eigenfunction of the Hamiltonian and expresses confusion about how to relate it to the Hamiltonian.
- Another participant clarifies that if ψ is an eigenfunction of H, then Hψ should equal Eψ, where E is the eigenvalue, and suggests applying H to the linear combination.
- There is a discussion about the notation, with participants suggesting that ψ(1,1,5) and ψ(3,3,3) should be interpreted as functions of position, ψ1,1,5(x,y,z) and ψ3,3,3(x,y,z).
- One participant notes that the length L can be arbitrary, leading to a more general solution, and explains how the boundary conditions are satisfied with the sine functions.
- Another participant raises a concern that if sin(nπ) = 0, then the entire wave function would be zero, questioning the nature of the eigenvalue solution.
- It is discussed that the actual wave function is non-zero except at the boundaries, and the application of the Hamiltonian should yield a consistent eigenvalue relationship.
- A participant shares their results after applying the Hamiltonian, noting that the resulting terms have different coefficients, raising the question of whether this means the linear combination is not an eigenfunction.
- Another participant suggests that the issue may stem from incorrectly multiplying derivatives instead of adding them, emphasizing the need to apply the Laplacian correctly.
- In a later post, one participant concludes that they needed to recognize the degeneracy of the energy functions to affirm that the linear combination is indeed an eigenfunction of the Hamiltonian.
Areas of Agreement / Disagreement
Participants express differing views on the interpretation of the wave functions and the implications of boundary conditions. While some agree on the need to apply the Hamiltonian to determine eigenfunction status, there is no consensus on the specific calculations or interpretations of results.
Contextual Notes
Participants mention potential confusion regarding the notation and the implications of boundary conditions, as well as the need for careful application of mathematical operations when dealing with eigenfunctions and the Hamiltonian.