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A molecule is made up of three identical atoms at the corners of an equilateral triangle as showin in Fig. 7.12. We consider its ion to be nade by adding one clectron with some amplitude on cach sitc. Suppose the matrix element of the Hamiltonian for the electron on two adjacent sites i, j is (i|H\j) = -a for i +j. o' o? 03 Fig. 7.12 (a) Calculate the energy splittings. (b) Suppose an electric field in the z direction is applied, so that tlhe potential energy for the electron on top is lowered by b witlh b| < [aļ. Now calculate the levels. (c) Suppose the electron is in the gronnd state. Suctdeuly the field is rotated by 120° and points toward site 2. Calculate the probability for the electron to remain in the ground state.

A molecule is made up of three identical atoms at the corners of an equilateral triangle as showin in Fig. 7.12. We consider its ion to be nade by adding one clectron with some amplitude on cach sitc. Suppose the matrix element of the Hamiltonian for the electron on two adjacent sites i, j is (i|H\j) = -a for i +j. o' o? 03 Fig. 7.12 (a) Calculate the energy splittings. (b) Suppose an electric field in the z direction is applied, so that tlhe potential energy for the electron on top is lowered by b witlh b| < [aļ. Now calculate the levels. (c) Suppose the electron is in the gronnd state. Suctdeuly the field is rotated by 120° and points toward site 2. Calculate the probability for the electron to remain in the ground state.

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A molecule is made up of three identical atoms at the corncrs of an equilateral triangle as showi in Fig. 7.12. We consider its ion to be nade by adding one clectron with some amplitude on cach site. Suppose the matrix element of the Hamiltonian for the electron on two adjacent sites i, j is (i|H|j) = -a for i # j. 1 o' O? Fig. 7.12 (a) Calculate the energy splittings. (b) Suppose an electric field in the z direction is applied, so that tlhe potential energy for the electron on top is lowered by b witli b < Ja|. Now calculate the levels. (c) Suppose the electron is in the ground state. Sucdeuly the field is rotated by 120° and points toward site 2. Calculate the probability for the electron to remain in the ground state.
Transcribed Image Text:A molecule is made up of three identical atoms at the corncrs of an equilateral triangle as showi in Fig. 7.12. We consider its ion to be nade by adding one clectron with some amplitude on cach site. Suppose the matrix element of the Hamiltonian for the electron on two adjacent sites i, j is (i|H|j) = -a for i # j. 1 o' O? Fig. 7.12 (a) Calculate the energy splittings. (b) Suppose an electric field in the z direction is applied, so that tlhe potential energy for the electron on top is lowered by b witli b < Ja|. Now calculate the levels. (c) Suppose the electron is in the ground state. Sucdeuly the field is rotated by 120° and points toward site 2. Calculate the probability for the electron to remain in the ground state.
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