Particle in a Well (PIW) (14.5)

1. 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

2. 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?

3. 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

4. Particle in a Well Wavefunctions

5. Tunneling Through a Barrier