Particle in 1-D Box: Pressure Force on Wall

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SUMMARY

The discussion centers on calculating the pressure force exerted on the walls of a one-dimensional box containing a particle of mass m. The problem involves determining the energy levels of the particle, which can be calculated using quantum mechanics principles. The pressure is derived from the relationship P = -dE/dL, where E represents the energy levels and L is the length of the box. This approach is analogous to the three-dimensional case, where pressure is calculated as P = -dE/dV.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically the concept of a particle in a potential well.
  • Familiarity with the Schrödinger equation and its application to one-dimensional systems.
  • Knowledge of thermodynamic pressure and its relationship to energy changes.
  • Basic calculus, particularly differentiation, to compute dE/dL.
NEXT STEPS
  • Study the derivation of energy levels for a particle in a one-dimensional infinite potential well.
  • Explore the implications of the pressure formula P = -dE/dL in various dimensions.
  • Investigate the relationship between quantum mechanics and thermodynamics in particle systems.
  • Learn about the three-dimensional extension of this problem, including the calculation of pressure in a three-dimensional box.
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Students and professionals in physics, particularly those focusing on quantum mechanics and thermodynamics, as well as researchers exploring particle behavior in confined systems.

libai
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This seems a simple problem, but managed to confuse me and my friends a bit, it's definitely not a standard problem from the text-books.
Regard a particle of mass m in a 1 dimensional box of length L (which can be seen as a infinitely deep potential well). What is the 'pressure force' on the wall? (please forgive me my vagueness, I translated this question from chinese).
Thanks for your help!
 
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Calculate the energy levels. (This is pretty standard). Then calculate the pressure as P=-dE/dL. (In 3 dimensions it would be -dE/dV, where V=L3).
 

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