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Chapter 36: Problem 63

A selection rule for the infinite square well allows only those transitions in which \(n\) changes by an odd number. Suppose an infinite square well of width 0.200 nm contains an electron in the \(n=4\) state. (a) Draw an energy-level diagram showing all allowed transitions that could occur as this electron drops toward the ground state, including transitions from lower levels that could be reached from \(n=4 .\) (b) Find all the possible photon energies emitted in these transitions.

Short Answer

Expert verified
The possible transitions as the electron drops toward the ground state are from \(n=4\) to \(n=3\) and \(n=1\), and from \(n=3\) to \(n=1\). The respective photon energies emitted in these transitions, using the energy formula for an infinite square well, are \((4^2-3^2)h^2/8mL^2\), \((4^2-1^2)h^2/8mL^2\), and \((3^2-1^2)h^2/8mL^2\).

Step by step solution

01

Understand transition rules

The deduction rule for the infinite square well provides that \(n\) can only change by an odd number. This depicts that an electron in the \(n=4\) state could make transitions to \(n=1\) or \(n=3\) states, but not to \(n=2\), since 2 is not an odd number.
02

Diagram of energy-level transitions

A diagram should be drawn based on the transition rules explained in Step 1, showing potential transitions from \(n=4\) to \(n=3\) and \(n=1\), and any further transitions from \(n=3\) to \(n=1\). Lower levels that could be reached from \(n=4\) are therefore \(n=3\) and \(n=1\).
03

Find the possible photon energies

The possible photon energies can be found using the energy formula \(E=n^2h^2/8mL^2\), where \(h\) is Planck's constant, \(m\) is the mass of the electron and \(L\) is the well width. The transitions from \(n=4\) to \(n=3\) and \(n=1\), and from \(n=3\) to \(n=1\) would therefore emit photons with energies equal to the difference in energies of the respective levels. For transitions from \(n=4\) to \(n=3\) and \(n=1\), those would be \((4^2-3^2)h^2/8mL^2\) and \((4^2-1^2)h^2/8mL^2\) respectively. The transition from \(n=3\) to \(n=1\) would emit photons with energy \((3^2-1^2)h^2/8mL^2\).

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Most popular questions from this chapter

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