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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Presentation Transcript
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)
• 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)
• 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