8. Show that any linear combination of the 2p orbitals, | Yi = c1|2101 + c2|211i + c3|121-1İ 1= is also an eigenfunction of the hydrogen atom Hamiltonian with the same eigenvalue as the individual orbitals, |210i, |211i, and №21-1i. This allows us to use the more convenient Rx Rx, and orbitals instead of the complex p₁, po, and P-1 forms. The property that any linear combination of degenerate eigenfunctions is also an eigenfunction holds true in general, so we can use it other problems too (e.g. for the d and forbitals). Rz 9. (Adapted from McQuarrie 7.24) Atoms are spherical—they have no discernible orientation. However, the Rx Rx, and pz orbitals have definite orientation. Show that the combined Homework 7, Chem 113, Spring 2024, Prof. Beran3 probability density of the 2p orbitals, Prob |210|2 + |211|2 + |21-1|2| is spherically symmetric (i.e. it has the same value for any angles and p). This result implies that while an individual 2p orbital is not spherically symmetric, the set of all three 2p orbitals is spherically symmetric. Combining this with the results from the previous question, we can see that any reasonable linear combination of these orbitals will maintain the overall spherical symmetry.

8. Show that any linear combination of the 2p orbitals, | Yi = c1|2101 + c2|211i + c3|121-1İ 1= is also an eigenfunction of the hydrogen atom Hamiltonian with the same eigenvalue as the individual orbitals, |210i, |211i, and №21-1i. This allows us to use the more convenient Rx Rx, and orbitals instead of the complex p₁, po, and P-1 forms. The property that any linear combination of degenerate eigenfunctions is also an eigenfunction holds true in general, so we can use it other problems too (e.g. for the d and forbitals). Rz 9. (Adapted from McQuarrie 7.24) Atoms are spherical—they have no discernible orientation. However, the Rx Rx, and pz orbitals have definite orientation. Show that the combined Homework 7, Chem 113, Spring 2024, Prof. Beran3 probability density of the 2p orbitals, Prob |210|2 + |211|2 + |21-1|2| is spherically symmetric (i.e. it has the same value for any angles and p). This result implies that while an individual 2p orbital is not spherically symmetric, the set of all three 2p orbitals is spherically symmetric. Combining this with the results from the previous question, we can see that any reasonable linear combination of these orbitals will maintain the overall spherical symmetry.

Principles of Modern Chemistry

8th Edition

ISBN:9781305079113

Author:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Publisher:David W. Oxtoby, H. Pat Gillis, Laurie J. Butler

Chapter4: Introduction To Quantum Mechanics

Section: Chapter Questions

Problem 39P: Chapter 3 introduced the concept of a double bond between carbon atoms, represented by C=C , with a...

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