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Particle in a Well (PIW) (14.5)

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Particle in a Well (PIW) (14.5). A more realistic scenario for a particle is it being in a box with walls of finite depth (I like to call it a well) Particles can escape the well by having enough energy, and then behave like free particles

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particle in a well piw 14 5
Particle in a Well (PIW) (14.5)
  • A more realistic scenario for a particle is it being in a box with walls of finite depth (I like to call it a well)
    • Particles can escape the well by having enough energy, and then behave like free particles
    • When a free particle passes by a well, it is still influenced by the well though it is not trapped
  • The problem is now divided into three regions and the wavefunctions (and their first derivatives) in the three regions must match at the boundaries
    • Regions I and III have a non-zero, but constant, potential energy V0
    • Region II is the well and has no potential energy (length is from –a/2 to a/2)
  • Since it is possible for the particle to exist in regions I and III above the well, it is also possible for the particle to exist there “below” the well
    • The wavefunctions extend beyond the walls of the well into classically forbidden regions
    • The wavefunctions MUST approach zero as one moves deeper into the well walls
quantum mechanical tunneling 14 5
Quantum Mechanical Tunneling (14.5)
  • Inside classically forbidden regions, the wavefunction must decay to zero and do so quickly
    • For PIW, the wavefunctions beyond the well wall decay exponentially
  • How quickly the particle decays outside the well depends on the parameter κ
    • Larger value of κ means faster decay
    • Heavy particles have a more difficult time tunneling into well wall
    • Particles closer to the top of the well (i.e., in higher energy states) have an easier time penetrating the walls
  • Tunneling into a well wall is possible, but leads to the eventual decay of the particle
    • What if the wall had a finite length?
tunneling through a barrier 14 9
Tunneling Through a Barrier (14.9)
  • If the PIW model is inverted, we now have a barrier
    • The barrier has a certain width (a) and height (V0), and the potential everywhere else is zero
    • Classically, a particle can only get from one side of the barrier to the other by going over it (e.g., passing through transition states)
  • Since the wavefunction is nonzero inside the barrier, it is possible for the particle to completely pass through the barrier
    • The width of the barrier dictates whether the particle can pass or not
    • The decay parameter κ also determines whether particles can pass through the barrier
    • The wavefunctions must “connect” between all three regions
  • When tunneling occurs, processes occur faster than one expects
    • Classically, reaction rates depend on the size of the activation barrier
    • Tunneling may make the activation barrier appear smaller
    • Tunneling occurs most often with electron and proton transfer processes
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