Stationary States: Definition & Free Particle Incident

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SUMMARY

The discussion focuses on the definition of stationary states in quantum mechanics and their application to free particles incident on finite potential steps. A stationary state is characterized by wavefunctions that do not change in time, represented mathematically as y(x,t) = Bsin(kx +/- wt), where B is arbitrary, k is the wave number, and w is the angular frequency. The relationship between stationary states and standing wave patterns is emphasized, particularly in the context of potential wells. This understanding is crucial for solving problems involving wave equations and particle interactions.

PREREQUISITES
  • Understanding of wave equations and their properties
  • Familiarity with quantum mechanics concepts, specifically stationary states
  • Knowledge of potential wells and their implications in quantum systems
  • Basic proficiency in mathematical concepts such as partial derivatives
NEXT STEPS
  • Study the mathematical formulation of stationary states in quantum mechanics
  • Explore the concept of standing waves and their relation to potential wells
  • Learn about the implications of wavefunctions in quantum mechanics
  • Investigate the behavior of particles at finite potential steps
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics, wave phenomena, and potential theory. This discussion is beneficial for anyone looking to deepen their understanding of stationary states and their applications in particle physics.

Sirius24
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Homework Statement



dJ(x,t)/dx = -dY^2 / dt , where y is the wave equation, and the d's represent partial derivatives. I want to make an assumption that I can describe the wave equation as a stationary state, so my question is the following:

What is the definition of a stationary state and can it be used to describe a free particle incident on a finite potential step from the left? This is not a specific question for the problem, but I need to know in order to make an assumption to solve it.
 
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I'm not that clued up on this myself, but stationary states appear to be similar to solving for standing waves on a string.

The assumption is made that the wavefunction y(x,t) = Bsin(kx +/- wt) (B arbitrary, k the wave number, w the angular frequency, x position and t time) and the wave is bouncing back and forth within a potential well "box". The stationary states correspond to the particular circumstances that produce standing wave patterns.
 

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