In general the transformation \(A\) of the vectors \(x\) leads to
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| \begin{equation*} y = A x \end{equation*} | (1.20) |
where \(y\) is rotated against \(x\) and the lengths of both
vectors differ.
Now we are looking for special vectors \(v\) with
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| \begin{equation*} A v = \lambda v = \lambda E v \label{Eigeneq} \end{equation*} | (1.21) |
These ”Eigenvectors” only change their length, when the operator \(A\)
is applied to them.
Eq. (1.21)
can be rewritten as
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| \begin{equation*} (A - \lambda E) v = 0 \end{equation*} | (1.22) |
This equations has only nontrivial solutions if
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| \begin{equation*} \det(A - \lambda E) = 0 \end{equation*} | (1.23) |
The zeros of this characteristic polynom are the Eigenvalues \(\lambda_i\)
of the operator \(A\).
For each Eigenvalue \(\lambda_i\) we can calculate
the Eigenvectors \(v_i\) by solving the equation
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| \begin{equation*} (A - \lambda_i E) v_i = 0 \end{equation*} | (1.24) |
Example:
The matrix \(\left(\begin{array}{cc}3&-1\\ -1 &3\end{array}\right)\)
hasthe characteristic polynom \(\lambda^2 - 6 \lambda + 8 = 0\) with its solutions \(\lambda_1
= 2\) and \( \lambda_2 = 4\).
The corresponding Eigenvectors are \(\vec{v}_1
= \frac{1}{\sqrt{2}} \left(\begin{array}{c}1 \\ -1\end{array}\right)\) and \(\vec{v}_2 = \frac{1}{\sqrt{2}}
\left(\begin{array}{c}1 \\ 1\end{array}\right)\).
The adjacent unitary transformation is therefore
\(U = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1&1\\ -1 &1\end{array}\right)\).
An illustrative
example for an Eigenvalue problem you can find in the math-script.
Eigenvalues of linear Hermitian Operators are always real
Let
\(A\) be a Hermitian operator with the Eigenfunction \(f_1\) and the Eigenvalue \(a_1\);
it follows that
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| \begin{equation*} a_1\langle f_1|f_1\rangle = \langle f_1|A f_1\rangle = \langle A f_1|f_1\rangle = a_1^*\langle f_1|f_1\rangle \end{equation*} | (1.25) |
thus \(a_1\) is real.
Eigenvectors
belonging to different Eigenvalues are orthogonal
Let \(A\) be a Hermitian operator
with Eigenfunctions \(f_1\) and \(f_2\). Let \(a_1 \neq a_2\) be the corresponding
Eigenvalues. We find
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| \begin{equation*} a_2\langle f_1|f_2\rangle = \langle f_1|A f_2\rangle = \langle A f_1|f_2\rangle = a_1^*\langle f_1|f_2\rangle \end{equation*} | (1.26) |
Consequently
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| \begin{equation*} (a_2 - a_1^*) \langle f_1|f_2\rangle = 0 \qquad ; \end{equation*} | (1.27) |
since \((a_2 - a_1^*) \neq 0\) we find that \(f_1\) and \(f_2\) are orthogonal.
Degenerated Eigenvalues
Let \(A\) be a Hermitian
operator with the Eigenfunctions \(f_1\) and \(f_2\). Let \(a_1 = a_2 = a\) be the
corresponding Eigenvalue, i.e.
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| \begin{equation*} \begin{split} A |f_1\rangle & = a |f_1\rangle \\ A |f_2\rangle & = a |f_2\rangle \end{split} \end{equation*} | (1.28) |
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| \begin{equation*} A\left(x|f_1\rangle + y|f_2\rangle \right) = xa |f_1\rangle + ya |f_2\rangle = a \left(x|f_1\rangle + y|f_2\rangle \right) \qquad . \end{equation*} | (1.29) |
Thus degenerated Eigenfunctions form a sub vector space. This sub space is perpendicular
to all other Eigenvectors. The orthonormalization procedure of Schmidt allows to find a set of orthogonal vectors of length
1 in this subspace.
Orthonormal base of vectors
The system of Eigenvectors for every
Hermitian matrix can thus be transformed to a set of orthonormal vectors \(e_i\), with represent a basis in
the vector space:
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| \begin{equation*} \left\langle e_i| e_j \right\rangle = \delta_{i,j} \qquad . \end{equation*} | (1.30) |
Each vector can be written as
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| \begin{equation*} \langle a| = \sum_i \langle a|e_i\rangle \langle e_i| \qquad . \end{equation*} | (1.31) |
which is called the Closure-relation. The \(a_i = \langle a|e_i\rangle \)
are called the components of the vector for the basis \(e_i\). In component representation the vector is just
written as \(\langle a| = (a_1, a_2, a_3, ...,a_n)\).
REMARK: Remember the Bra-, Ket-
representation of vectors; for complex vectors the components have to be chosen complex conjugated when changing from Bra-
to Ket- representation or vice versa.
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© J. Carstensen (Quantum Mech.)