Question 9. Consider the following system with a potential function: E V(x) = S 0 (0

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Chapter1: Chemical Foundations
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Question 9.
Consider the following system with a potential function:
V(x) = }.
0 (0 < x < L, 2L < x < 3L)
l00(x> 0, L < x< 2L,x > 3L)
E
V(x)=0
(barrier)
V(x)=0
(well A)
V(x)=0
(barrier)
V(x)=0
(well B)
V(x)=0
(barrier)
2L
3L
The diagram above is a sketch of this potential V(x). Compared to the "particle in a box" (PIB)
system, this system can be called "particle in two boxes" (PITB) and its solution largely
resembles that of the PIB.
(a) Where would the wavefunction solutions exist (i.e., be non-zero)? Explain your reasoning.
(b) How might we modify this potential energy diagram to become a "double-well" potential?
Sketch such a potential. Where would the wavefunction solutions exist in your potential?
(c) Now let us be more specific. Where would the wavefunction solutions exist for small
quantum numbers, n? What happens in the range L<x<2L? Where would the wavefunction
solutions exist for large quantum numbers?
Transcribed Image Text:Question 9. Consider the following system with a potential function: V(x) = }. 0 (0 < x < L, 2L < x < 3L) l00(x> 0, L < x< 2L,x > 3L) E V(x)=0 (barrier) V(x)=0 (well A) V(x)=0 (barrier) V(x)=0 (well B) V(x)=0 (barrier) 2L 3L The diagram above is a sketch of this potential V(x). Compared to the "particle in a box" (PIB) system, this system can be called "particle in two boxes" (PITB) and its solution largely resembles that of the PIB. (a) Where would the wavefunction solutions exist (i.e., be non-zero)? Explain your reasoning. (b) How might we modify this potential energy diagram to become a "double-well" potential? Sketch such a potential. Where would the wavefunction solutions exist in your potential? (c) Now let us be more specific. Where would the wavefunction solutions exist for small quantum numbers, n? What happens in the range L<x<2L? Where would the wavefunction solutions exist for large quantum numbers?
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