Solution to the Schrodinger equation

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SUMMARY

The discussion focuses on the solution to the Schrödinger equation represented as Aexp(i(kx - ωt)), specifically for a free particle. It highlights the relationship between the wave vector k and the angular frequency ω, emphasizing that this solution is valid under the condition that the energy is purely kinetic. The momentum p is defined as p = ħk, and the energy is expressed as E = ħω, where ħ denotes the reduced Planck's constant.

PREREQUISITES
  • Understanding of quantum mechanics principles
  • Familiarity with the Schrödinger equation
  • Knowledge of wave functions and their properties
  • Basic grasp of kinetic energy and momentum relations
NEXT STEPS
  • Explore the derivation of the dispersion relation for free particles
  • Study the implications of the Schrödinger equation in different potential scenarios
  • Learn about the role of Planck's constant in quantum mechanics
  • Investigate the concept of wave-particle duality in quantum physics
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Students and professionals in physics, particularly those specializing in quantum mechanics, as well as researchers interested in wave functions and their applications in theoretical physics.

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I know that one solution to the Schrödinger equation is

Aexp(i(kx - ωt))

Is there a special relationship between k and ω for this to be a solution, or is it always a solution?
 
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That's the solution for a free particle. You can work out the dispersion relationship easily.

The energy is only kinetic, p = hk (that should be h bar), E = hω (again, h bar).
 

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