Solve Schrodinger equation for an eletron in a box. Why discrete?

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SUMMARY

The discussion centers on solving the Schrödinger equation for an electron confined in a one-dimensional box, specifically between x=0 and x=L. The energy levels are derived as En=n²*h²/(8*m*L²) and the wave function as Ψ(x)=√(2/L) * Sin(n*π*x/L). The quantization of energy, velocity, and frequency is explained through the boundary conditions of the wave function, which leads to discrete energy states due to the reflection of the wave at the box's walls.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly wave functions and energy quantization.
  • Familiarity with differential equations, specifically second-order ordinary differential equations.
  • Knowledge of the Schrödinger equation and its applications in quantum systems.
  • Basic concepts of wave-particle duality and boundary conditions in quantum mechanics.
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  • Study the derivation of the time-independent Schrödinger equation in one-dimensional systems.
  • Explore the implications of boundary conditions on wave functions in quantum mechanics.
  • Learn about the relationship between frequency, energy, and wave functions in quantum systems.
  • Investigate the concept of quantization in other quantum systems, such as particles in three-dimensional boxes.
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Students of quantum mechanics, physicists, and educators seeking to deepen their understanding of wave functions and energy quantization in confined systems.

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Homework Statement



An eletron is moving along one axis between x=0 and x=L. It's mass is given by m. We want to know the energy and wave function of its possible states given by the quantic number n.

eq sch.png


Show that the solution to the above equation is

En=n^2*h^2/(8*m*L^2)

[itex]\Psi(x)=({2/L})^{1/2}*Sin(n*Pi*x/L)[/itex]

Homework Equations


The Attempt at a Solution



I see that it is a second order ordinary differencial equation, I should have no problem solving that. However, I can't fully grasp the quantization thing. I mean, I understand why the wavelength is quantized, because of the refletction on its walls, so the wavelength must be 2L/n. But why does the velocity and the frequency must be quanticized? For the energy to be quantized, the velocity must be quanticized and thus, both the wavelength and the frequency must be quanticized. Right? But why is the frequency quantized?
 
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Frequency and energy are related via the time-dependent Schrödinger equation: for static solutions, they are proportional to each other.
 
I suggest you actually solve the differential equation. The answers to your questions come out in the math.
 

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