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## Homework Statement

Consider a two-dimensional space spanned by two orthonormal state vectors [itex] \mid \alpha \rangle [/itex] and [itex] \mid \beta \rangle [/itex]. An operator is expressed in terms of these vectors as

[tex] A = \mid \alpha \rangle \langle \alpha \mid + \lambda \mid \beta \rangle \langle \alpha \mid + \lambda^* \mid \alpha \rangle \langle \beta \mid + \mu \mid \beta \rangle \langle \beta \mid [/tex]

Determine the eigenstates of A for the case where (i) [itex] \lambda = 1, \mu = \pm 1 [/itex], (ii) [itex] \lambda = i, \mu = \pm 1 [/itex]. Do this problem also by expressing A as a 2 X 2 matrix with eigenstates as the column vectors.

## Homework Equations

Just linear algebra rules.

## The Attempt at a Solution

I started with [itex] \lambda = 1, \mu = 1 [/itex]. Then A is:

[tex] A = \mid \alpha \rangle \langle \alpha \mid + \mid \beta \rangle \langle \alpha \mid + \mid \alpha \rangle \langle \beta \mid + \mid \beta \rangle \langle \beta \mid [/tex]

[tex] A \mid \alpha \rangle = \mid \alpha \rangle \langle \alpha \mid \alpha \rangle + \mid \beta \rangle \langle \alpha \mid \alpha \rangle + \mid \alpha \rangle \langle \beta \mid \alpha \rangle + \mid \beta \rangle \langle \beta \mid \alpha \rangle = \mid \alpha \rangle + \mid \beta \rangle [/tex]

[tex] A \mid \beta \rangle = \mid \alpha \rangle \langle \alpha \mid \beta \rangle + \mid \beta \rangle \langle \alpha \mid \beta \rangle + \mid \alpha \rangle \langle \beta \mid \beta \rangle + \mid \beta \rangle \langle \beta \mid \beta \rangle = \mid \alpha \rangle + \mid \beta \rangle [/tex]

The eigenstate is [itex] \mid a_n \rangle [/itex] with eigenvalue [itex] a_n [/itex]. Then the following holds:

[tex] A \mid a_n \rangle = a_n \mid a_n \rangle [/tex]

The eigenstate [itex] \mid a_n \rangle [/itex] can be expressed in the basis vectors [itex] \mid \alpha \rangle [/itex] and [itex] \mid \beta \rangle [/itex]:

[tex] \mid a_n \rangle = c_1 \mid \alpha \rangle + c_2 \mid \beta \rangle [/tex]

Then the earlier equation becomes:

[tex] A \mid a_n \rangle = A( c_1 \mid \alpha \rangle + c_2 \mid \beta \rangle ) = a_n (c_1 \mid \alpha \rangle + c_2 \mid \beta \rangle [/tex]

But this is also:

[tex] A( c_1 \mid \alpha \rangle + c_2 \mid \beta \rangle ) = c_1 A \mid \alpha \rangle + c_2 A \mid \beta \rangle = c_1 (\mid \alpha \rangle + \mid \beta \rangle) + c_2 (\mid \alpha \rangle + \mid \beta \rangle) \\ = (c_1 + c_2) \mid \alpha \rangle + (c_1 + c_2) \mid \beta \rangle [/tex]

This gives the equations :

[tex] a_n c_1 = c_1 + c_2 [/tex]

[tex] a_n c_2 = c_1 + c_2 [/tex]

The only solution is if [itex] c_1 = c_2 = 0 [/itex]. Obviously I am doing something wrong but I can't see it.