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A Hamiltonian represented by a matrix, find the eigevalues
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[QUOTE="rwooduk, post: 4968529, member: 493651"] I'll try contribute something, it could be wrong but might give you some ideas. for [tex]H = \begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix} [/tex] we can write [tex]\begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix} \Psi = E\Psi[/tex] we can introduce the identity operator [tex]\begin{pmatrix} 3 & 0 & -1\\ 0 & a & 0\\ -1 & 0 & 3 \end{pmatrix} \Psi = \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix} E\Psi[/tex] put E in the matrix and minus from the left hand side so [tex]\begin{pmatrix} 3-E & 0 & -1-E\\ 0 & a-E & 0\\ -1-E & 0 & 3-E \end{pmatrix} \Psi = 0[/tex] we can then insert the matrix form of the eigen state [tex]\begin{pmatrix} 3-E & 0 & -1-E\\ 0 & a-E & 0\\ -1-E & 0 & 3-E \end{pmatrix} \begin{pmatrix} a_{1}\\ a_{2} \\ a_{3} \end{pmatrix} = 0[/tex] then take the determinant and this will give you the possible energy values. for each energy value put it back into the matrix and it will tell you what a[SUB]1[/SUB], a[SUB]2[/SUB] and a[SUB]3[/SUB] are in relation to each other. once you know this then it should be clear where the given wave function comes from. i.e. a[SUB]1[/SUB]=a[SUB]3[/SUB] and a[SUB]2[/SUB] = 0 and what the other possible eigenstates are. [/QUOTE]
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A Hamiltonian represented by a matrix, find the eigevalues
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