Discussion Overview
The discussion focuses on the formulation of the Schrödinger equation for a mixture of hydrogen (H2) and helium (He) molecules, specifically addressing how to write the full equation pre-approximation, including the specification of potentials and the Hamiltonian. The scope includes theoretical aspects of quantum mechanics and the challenges of solving the equation for multi-particle systems.
Discussion Character
- Exploratory, Technical explanation, Debate/contested
Main Points Raised
- One participant seeks to write the full Schrödinger equation for a mixture of 75% H2 and 25% He, asking about the specifics of specifying potentials and the Hamiltonian.
- Another participant suggests that the general approach involves each particle moving in the potential of all other particles, noting the need for multiple position vectors corresponding to nuclei and electrons.
- A different viewpoint argues against writing the Schrödinger equation for a mixture, proposing that one should sum the electrons and nuclei to write a Hamiltonian with Coulomb potentials between pairs.
- There is a question about the number of position vectors needed, with one participant suggesting a discrepancy in counting nuclei and electrons.
- Concerns are raised about the implications of having discrete mixtures and whether different mixtures would yield identical properties.
- Participants discuss the challenges of solving the Schrödinger equation for multi-particle systems, emphasizing the use of perturbation theory and the intractability of finding solutions for more than one particle without approximations.
- One participant expresses a desire to see the full form of the equation to understand the approximations that follow, despite recognizing the complexity involved.
- Another participant compares the situation to the three-body problem in classical mechanics, noting that while solutions exist, they cannot be derived from the full equations.
- There is a discussion about the role of the Hamiltonian in the Schrödinger equation and the absence of wave-functions in the provided document, leading to confusion about transitioning from the Hamiltonian to the equation form.
Areas of Agreement / Disagreement
Participants express differing views on how to approach the formulation of the Schrödinger equation for mixtures, with no consensus on whether a discrete number of mixtures leads to identical properties. The discussion remains unresolved regarding the specifics of writing the equation and the implications of the Hamiltonian's form.
Contextual Notes
Participants acknowledge the limitations of directly solving the Schrödinger equation for complex systems and the necessity of approximations, but they do not reach a consensus on the best approach to formulate the equation or the implications of the Hamiltonian.