2 Double Well Potential using the Finite Difference Approxima- tion Vo x=-3 V(x) 0.0 -Vo a=0 x=3 (A) Using the N=3 Finite Difference Approximation, write down the 3x3 matrix representation of the Hamiltonian, H, for the above potential. This should explicitly include the "t" value in the lecture notes. (B) Diagonalize the matrix in (A) by calculating det(H-XI) = 0 by hand. You will arrive at a characteristic polynomial of which you must find the roots. Write down this characteristic polynomial, grouping terms together by the power of A. (C) Now set t = 1 and Vo = 2.0 You can try to find the roots by hand, or you can use Mathematica's Solve[] function to find the three roots of this equation. These are the three eigenvalues of the Hamiltonian. (D) Now, set t = 1 and Vo = 2.0 in your H matrix and calculate the eigenvector corresponding to each eigenvalue from (C). You may do this either by hand or using Mathematica's Eigensystem routine. Make sure that the eigenvectors are normalized. (E) Order the three eigenvectors by energy from lowest to highest and draw the wavefunctions corresponding to these states. (F). Calculate the transition probability between the lowest and second lowest energy states using matrix vector multiplication. You can assume that the transition dipole moment operator takes the form ₂ = . (G) Calculate the expectation value of <2> for the state that is an equal linear combination of the lowest energy and second lowest energy eigenfunctions of the system.
2 Double Well Potential using the Finite Difference Approxima- tion Vo x=-3 V(x) 0.0 -Vo a=0 x=3 (A) Using the N=3 Finite Difference Approximation, write down the 3x3 matrix representation of the Hamiltonian, H, for the above potential. This should explicitly include the "t" value in the lecture notes. (B) Diagonalize the matrix in (A) by calculating det(H-XI) = 0 by hand. You will arrive at a characteristic polynomial of which you must find the roots. Write down this characteristic polynomial, grouping terms together by the power of A. (C) Now set t = 1 and Vo = 2.0 You can try to find the roots by hand, or you can use Mathematica's Solve[] function to find the three roots of this equation. These are the three eigenvalues of the Hamiltonian. (D) Now, set t = 1 and Vo = 2.0 in your H matrix and calculate the eigenvector corresponding to each eigenvalue from (C). You may do this either by hand or using Mathematica's Eigensystem routine. Make sure that the eigenvectors are normalized. (E) Order the three eigenvectors by energy from lowest to highest and draw the wavefunctions corresponding to these states. (F). Calculate the transition probability between the lowest and second lowest energy states using matrix vector multiplication. You can assume that the transition dipole moment operator takes the form ₂ = . (G) Calculate the expectation value of <2> for the state that is an equal linear combination of the lowest energy and second lowest energy eigenfunctions of the system.
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Double Well Potential using the Finite Difference Approxima-
tion
Vo
x=-3
V(x)
0.0
-Vo
a=0
x=3
(A)
Using the N=3 Finite Difference Approximation, write down the 3x3 matrix
representation of the Hamiltonian, H, for the above potential. This should explicitly include the
"t" value in the lecture notes.
(B)
Diagonalize the matrix in (A) by calculating det(H-XI) = 0 by hand. You will
arrive at a characteristic polynomial of which you must find the roots. Write down this characteristic
polynomial, grouping terms together by the power of A.
(C)
Now set t = 1 and Vo = 2.0 You can try to find the roots by hand, or you can
use Mathematica's Solve[] function to find the three roots of this equation. These are the three
eigenvalues of the Hamiltonian.
(D)
Now, set t = 1 and Vo = 2.0 in your H matrix and calculate the eigenvector
corresponding to each eigenvalue from (C). You may do this either by hand or using Mathematica's
Eigensystem routine. Make sure that the eigenvectors are normalized.
(E)
Order the three eigenvectors by energy from lowest to highest and draw the
wavefunctions corresponding to these states.
(F).
Calculate the transition probability between the lowest and second lowest energy
states using matrix vector multiplication. You can assume that the transition dipole moment
operator takes the form ₂ = .
(G)
Calculate the expectation value of <2> for the state that is an equal linear
combination of the lowest energy and second lowest energy eigenfunctions of the system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb924e824-f7b8-486a-bee8-48b1feb67b77%2F6a3104d2-5c72-4c03-b841-58523824cbee%2F9r4qwor_processed.png&w=3840&q=75)
Transcribed Image Text:2
Double Well Potential using the Finite Difference Approxima-
tion
Vo
x=-3
V(x)
0.0
-Vo
a=0
x=3
(A)
Using the N=3 Finite Difference Approximation, write down the 3x3 matrix
representation of the Hamiltonian, H, for the above potential. This should explicitly include the
"t" value in the lecture notes.
(B)
Diagonalize the matrix in (A) by calculating det(H-XI) = 0 by hand. You will
arrive at a characteristic polynomial of which you must find the roots. Write down this characteristic
polynomial, grouping terms together by the power of A.
(C)
Now set t = 1 and Vo = 2.0 You can try to find the roots by hand, or you can
use Mathematica's Solve[] function to find the three roots of this equation. These are the three
eigenvalues of the Hamiltonian.
(D)
Now, set t = 1 and Vo = 2.0 in your H matrix and calculate the eigenvector
corresponding to each eigenvalue from (C). You may do this either by hand or using Mathematica's
Eigensystem routine. Make sure that the eigenvectors are normalized.
(E)
Order the three eigenvectors by energy from lowest to highest and draw the
wavefunctions corresponding to these states.
(F).
Calculate the transition probability between the lowest and second lowest energy
states using matrix vector multiplication. You can assume that the transition dipole moment
operator takes the form ₂ = .
(G)
Calculate the expectation value of <2> for the state that is an equal linear
combination of the lowest energy and second lowest energy eigenfunctions of the system.
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