Shrodinger Equation for Multiple Particles

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SUMMARY

The discussion focuses on solving the Schrödinger equation for five identical non-interacting electrons in an infinite square well of width L = 1 nm. The time-independent Schrödinger equation is provided, emphasizing the need to treat the potential energy as V = V1(x1) + V2(x2) due to non-interaction. The energy levels are derived using the formula En = -13.6(Z²/n²)eV, applicable for each particle. The solution approach for multiple particles is clarified, indicating that the methodology for two particles can be extended to five particles without additional complexity.

PREREQUISITES
  • Understanding of the time-independent Schrödinger equation
  • Familiarity with quantum mechanics concepts, specifically infinite square wells
  • Knowledge of energy quantization in quantum systems
  • Basic calculus, particularly partial derivatives
NEXT STEPS
  • Study the derivation of the time-independent Schrödinger equation for multiple particles
  • Explore the concept of non-interacting particles in quantum mechanics
  • Learn about energy level calculations in infinite potential wells
  • Investigate the implications of identical particles in quantum statistics
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Students and professionals in physics, particularly those focused on quantum mechanics, as well as researchers working on multi-particle systems and energy quantization in confined spaces.

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Homework Statement



Five identical non-interacting particles are placed in an infinite square well with width L = 1 nm. I am asked to find the energy of the system if the particles are electrons.

Homework Equations



The time-independent Schrödinger equation for a system of two particles, both with the same mass 'm' is:

-h(bar)2/2m * (d2\psi(x1x2 / (d2x21)) - h(bar)2/2m * (d2\psi(x1x2) / (d2x22)) + V\psi(x1x2) = E\psi(x1x2)

All derivatives are partials.

Also, energy En = -13.6(Z2 / n2)eV.

The Attempt at a Solution



Alright so since the question specifies the particles do not interact, V = V1(x1) + V2(x2). As in regular Schrödinger equations, I want to solve where V = 0 and I can write \psinm(x12) = \psin(x1)\psim(x2). From here I'm stuck -- how do I solve for a system of five particles, and how do I find the energy from here?

(sorry for the typo in the title, by the way)
 
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Do you know how to solve it for a system of 2 noninteracting particles? It is treated exactly the same way for 5 particles.
 

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