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)

• 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?

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ψIII x( ) = Ae−κx for a2

< x < ∞

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κ =2m V0 − E( )

h

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

Horse Liver Alcohol Dehydrogenase (HLADH)

A. Kohen and J. P. Klinman, Chemistry & Biology, 1999, 6, 191-198

Particle in a Well Wavefunctions

Tunneling Through a Barrier