Expectation of Momentum in a Classical (Infinite) Potential Well

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SUMMARY

The discussion focuses on calculating the expectation value of momentum,

, for a classical particle in an infinite potential well. The participants derive the expectation formula and confirm that the wavefunction must vanish at the boundaries. They emphasize that while the particle's velocity is uniform, the wavefunction can be represented as a complex exponential, which maintains constant velocity, unlike sine or cosine functions. The conclusion aligns with quantum mechanics, where the expectation value for momentum in this scenario is zero, consistent with the classical interpretation of a particle bouncing between walls.

PREREQUISITES
  • Understanding of quantum mechanics, specifically wavefunctions and expectation values.
  • Familiarity with the concept of infinite potential wells in quantum systems.
  • Knowledge of classical mechanics, particularly the behavior of particles in confined spaces.
  • Basic proficiency in calculus, especially differentiation and integration of functions.
NEXT STEPS
  • Study the Correspondence Principle in quantum mechanics to understand the transition from quantum to classical behavior.
  • Learn about the normalization of wavefunctions in quantum mechanics, focusing on infinite potential wells.
  • Explore the implications of boundary conditions on wavefunctions in quantum systems.
  • Investigate the relationship between wavefunctions and probability density functions in quantum mechanics.
USEFUL FOR

Students and professionals in physics, particularly those studying quantum mechanics and classical mechanics, as well as educators seeking to clarify concepts related to potential wells and wave-particle duality.

  • #31
To elaborate on @jtbell #24. The expectation value, as you know, is an average.

Suppose you performed ##N## measurements of the momentum at random intervals and you got ##N_1## values at ##+p## and ##N_2## at ##-p##, where ##N=N_1+N_2##. What is the average value of the momentum?

Suppose you flip a fair coin ##N## times and you get ##N_1## tails and ##N_2## heads. You assign +1 point to tails and -1 point to heads. How would you express the average of the flips as a number? What do you think the a priori average could be before you do any measurements?
 

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