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Three Dimensional Problem - Advanced Quantum Chemistry and Spectroscopy - Problem, Exercises of Chemistry

Aligarh Muslim University Chemistry

Three Dimensional Problem, The Two Matrices Commute, Determinant, The Matrix, Eigenvalues, Orthonormal Basis the Hamiltonian, Represented By the Matrix, The Exact Eigenvalues, Non Degenerate Perturbation. This file contains some practice problems of Advanced Quantum Chemistry and Spectroscopy.

Typology: Exercises

2011/2012

Uploaded on 11/21/2012

sonia.anum 🇮🇳

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Chemistry

Problem Set 3

1.) Given:

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛−

310

012

001

054

203

121

~BA

Find BA

and AB

. Do the two matrices commute?

2.) Evaluate the following determinant:

1000

1100

1110

1111

−

−−

3.) Given the matrix:

⎟

⎠

⎞

⎜

⎝

⎛

−=

001

a) Is

~ Hermitian?

b) Find the eigenvalues of

4.) Consider a three-dimensional problem. In a given orthonormal basis the Hamiltonian is

represented by the matrix:

⎟

⎠

⎞

⎜

⎝

⎛

⎟

⎠

⎞

⎜

⎝

⎛

−

=+=

HHH

200

030

001

~~~ )1()0(

c is a constant << 1.

a) Find the exact eigenvalues of

b) Determine the eigenvalues by non-degenerate perturbation to second order.

c) Compare the results of part a) and b)

Note: you will find the binomial expansion useful for part c): (1+x)n ~ 1+nx; x < 1.

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Problem Set 3Chemistry 1.) Given:

⎟⎟

~^100

A B

Find A ~^ B ~ and B ~ A^ ~. Do the two matrices commute? 2.) Evaluate the following determinant:

3.) Given the matrix:

⎟⎟

i A a) Is A^ ~^ Hermitian? b) Find the eigenvalues of A^ ~ 4.)represented by the matrix: Consider a three-dimensional problem. In a given orthonormal basis the Hamiltonian is

c H H H 0 0

~ ~( 0 ) ~( 1 )^100

c is a constant << 1. a) Find the exact eigenvalues of H^ ~ b) Determine the eigenvalues by non-degenerate perturbation to second order. c) Compare the results of part a) and b) Note: you will find the binomial expansion useful for part c): (1+x)n^ ~ 1+nx; x < 1.