Finding the expected value of position in a Potential Well

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SUMMARY

The discussion focuses on calculating the expected value of position for an electron trapped in a one-dimensional potential well using the Schrödinger Equation. The user initially derived a complex exponential wave function, which does not satisfy boundary conditions, leading to confusion regarding the symmetric (cosine) and asymmetric (sine) solutions. The expected value is calculated as <(z-)²> = , with the user attempting to compute integrals for both symmetric and asymmetric solutions. Ultimately, the correct normalization and limits of integration are crucial for obtaining the expected value consistent with the textbook solution.

PREREQUISITES
  • Understanding of Schrödinger's Equation and its boundary conditions
  • Familiarity with wave functions and their normalization
  • Knowledge of definite integrals and integration techniques
  • Concept of expected value in quantum mechanics
NEXT STEPS
  • Study the normalization of wave functions in quantum mechanics
  • Learn about boundary conditions in quantum systems
  • Explore the implications of symmetric and asymmetric solutions in potential wells
  • Review integration techniques for solving quantum mechanical problems
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Students and researchers in quantum mechanics, particularly those dealing with potential wells, wave functions, and the application of the Schrödinger Equation in one-dimensional systems.

  • #31
ja07019 said:
Okay you say that this is only half of the solution
And that only for ##|z|\le L/2##. The wave function is nonzero there and contributes to ##<z^2>##.

I'm not a textook. Read this from cover to cover. We (you) can't spend weeks on this topic.
 
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  • #32
BvU said:
And that only for ##|z|\le L/2##. The wave function is nonzero there and contributes to ##<z^2>##.

Okay just one more question then, if the wavefunction is piecewise, how would you come up with the expected values? Does that require a transform?
 

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