I Hamiltonian matrix - Eigenvectors

87
1
Hello everybody,

From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix
$$
\left( \begin{array}{cccccc}
\langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\
\langle2|H|1\rangle & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . & . &. &.&. \\
. & . &. &. &.& \langle n|H|n\rangle \end{array} \right)
$$

After diagonalization, I obtain diagonal ##n\times n## matrix that represent the eigenvalues of the Hamiltonian, and another ##n\times n## matrix ##V## composed of scalars that represent the eigenvectors of the same Hamiltonian.

My question is, what is the link between the scalar matrix ##V## and the complete set of orthogonal basis vector ##|i\rangle## that I choose in the beginning ?

Thank you very much everybody.

Konte
 

BvU

Science Advisor
Homework Helper
12,106
2,672
As far as I know the ##\left | {i} \right \rangle ## are the base vectors for ##V##.

so ##V_{ni}## are the coefficients ##\left\langle e_n \middle | i \right \rangle## ; in other words:

Column n of ##V## are the coordinates of eigenvector ##e_n## on the basis ##\left | {i} \right \rangle ##
 
87
1
Column n of ##V## are the coordinates of eigenvector ##e_n## on the basis ##\left | {i} \right \rangle ##
Thanks.

Konte
 

Want to reply to this thread?

"Hamiltonian matrix - Eigenvectors" You must log in or register to reply here.

Related Threads for: Hamiltonian matrix - Eigenvectors

  • Posted
Replies
2
Views
3K
  • Posted
Replies
4
Views
1K
  • Posted
Replies
1
Views
6K
  • Posted
Replies
4
Views
4K
Replies
1
Views
1K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top