Particle in a Box: Probability of Ground State & Excited State

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SUMMARY

The discussion centers on a quantum mechanics problem involving a particle confined in a box, initially of length L, which is suddenly expanded to length 2L. The potential energy inside the box remains zero, while it becomes infinite at the walls. The particle's position is measured at the center of the new box, represented by a Dirac delta function. Participants explore how to derive the probability of measuring the ground state and first excited state energies using the wave functions of the initial potential and the Schrödinger equation.

PREREQUISITES
  • Understanding of quantum mechanics principles, specifically wave functions.
  • Familiarity with Schrödinger's equation and its applications in potential wells.
  • Knowledge of Dirac delta functions and their role in quantum mechanics.
  • Concept of energy eigenfunctions in quantum systems.
NEXT STEPS
  • Study the derivation of wave functions for a particle in a box of length 2L.
  • Learn how to apply the Schrödinger equation to different potential energy scenarios.
  • Explore the concept of probability amplitudes and their relation to measurement outcomes in quantum mechanics.
  • Investigate the mathematical properties of Dirac delta functions in quantum contexts.
USEFUL FOR

Students and professionals in physics, particularly those focusing on quantum mechanics, wave functions, and potential energy problems. This discussion is beneficial for anyone seeking to deepen their understanding of particle confinement and energy state probabilities.

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A particle of mass m is confined to move in a box of length L. The potential energy of the particle within the box is zero and rises abruptly to a very large value (i.e. infinity) at the walls of the box. The walls of the box are now pulled out so that the box has length 2L. The walls are pulled out sufficiently quickly that instantaneously the state of the particle doesn’t change. The position of the particle is now measured and the outcome is that the particle is located exactly at the centre of the box (i.e. it is represented by a Dirac delta function located in the middle of the box). The energy of the particle is again measured.

Question
What is the probability that the outcome of this measurement will be the ground state energy? the first excited state?

My input
It actually contained more questions, but the last 2 are the ones i require help in. anyone with any idea? I am not too knowledgeable about dirac delta functions. I am not sure how i can get Schrödinger's equation into something that can measure the probability of the ground state or 1st excited state.
 
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HINT: Can you write the wave function of the second potential (box of width 2L) in terms of the energy eigenfunctions of the initial potential (box of with L)?
 

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