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# Particle in a Well (PIW) (14.5) - PowerPoint PPT Presentation

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)

• 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

• 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?

• 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