Assume that for a real system, a real wave function is a linear combination of two orthonormal basis functions where energy integrals are H11 = –15, H22 = –4, H12 = H21 = –1. What would be the ground state and excited wave functions?a)G.S.() = 0.996 1 + 0.08962 and E.S.() = –0.996 1 + 0.08962b)G.S.() = 0.996 1 + 0.08962 and E.S.() = –0.0896 1 + 0.9962.c)G.S.() = –15.09 1 – 3.912 and E.S.() = –3.91 1 – 15.092d)G.S.() = 15.09 1 + 3.912 and E.S.() = –3.91 1 – 15.092Correct answer is option 'B'. Can you explain this answer?

Question

Assume that for a real system, a real wave function is a linear combination of two orthonormal basis functions where energy integrals are H11 = &ndash;15, H22 = &ndash;4, H12 = H21 = &ndash;1. What would be the ground state and excited wave functions?a)G.S.() = 0.996 1 + 0.08962 and E.S.() = &ndash;0.996 1 + 0.08962b)G.S.() = 0.996 1 + 0.08962 and E.S.() = &ndash;0.0896 1 + 0.9962.c)G.S.() = &ndash;15.09 1 &ndash; 3.912 and E.S.() = &ndash;3.91 1 &ndash; 15.092d)G.S.() = 15.09 1 + 3.912 and E.S.() = &ndash;3.91 1 &ndash; 15.092Correct answer is option 'B'. Can you explain this answer?

Answer

E11, H22 = E22, and H12 = H21 = 0. In other words, the Hamiltonian matrix for this system is given by:

H = | E11 0 |
| 0 E22 |

Here, E11 and E22 represent the energies associated with the two basis functions, and the off-diagonal elements of the Hamiltonian matrix are zero.

To find the energy eigenvalues and eigenfunctions for this system, we need to solve the Schrödinger equation:

Hψ = Eψ

Plugging in the Hamiltonian matrix, we get:

| E11 0 | | ψ1 | = E | ψ1 |
| 0 E22 | | ψ2 | | ψ2 |

This equation can be written as two separate equations:

E11ψ1 = Eψ1
E22ψ2 = Eψ2

Simplifying, we see that the eigenvalues E are simply equal to the energies E11 and E22 associated with the basis functions. The corresponding eigenfunctions ψ1 and ψ2 are the coefficients of the basis functions in the linear combination.

In other words, the energy eigenvalues for this system are E11 and E22, and the corresponding eigenfunctions are the two orthonormal basis functions.

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