Attention : This is a strictly qualitative problem-- no calculations allowed! Co
ID: 1756635 • Letter: A
Question
Attention: This is a strictly qualitative problem-- no calculations allowed! Consider a double square wellpotential.----- -----
| ------- |
| | | |
| |<-b->| |
| <- a -> | | <- a ->|
----------- ----------- -V0
Suppose the depth V0 are and width a are fixed, largerenough so that several bound states occur.
(a) Sketch the ground state wave function Psi1(t) andthe first excited state for that case that b = 0, b = a, and b>> a.
(b) Qualitatively, how do the corresponding energies (E1and E2) vary, as b grows from 0 to ? SketchE1(b) and E2(b) on the same graph.
(c) The double well is a very primitive one-dimensional model forthe potential experienced by an electron in a diatomic molecule. Ifthe nuclei are free to move, they will adopt the configuration ofminimum energy. In view of your conclusions in (b), does theelectron tend to draw the nuclei together, or push them apart?
How do you do this?
Explanation / Answer
Ground state is even.Exponential decay outside, sinusoidal inside the wells, hyperboliccosine inbarrier. First excited state is odd– hyperbolic sine inbarrier. No nodes for 1, one node for2.
In the (even) ground state the energyis lowest in configuration (i), with b 0, so the electron tendsto
draw the nucleitogether, promoting bonding of the atoms. In the (odd) firstexcited state, by contrast,
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