Consider a three-dimensional vector space spanned by an orthonormal basis |1〉, |2〉,|3〉. Kets |α〉 and |β〉 are given by |α〉 = i |1〉 - 2 |2〉 - i |3〉, |β〉 = i |1〉 + 2 |3〉. (a) Construct 〈α| and 〈β| (in terms of the dual basis 〈1|, 〈2|, 〈3|). (b) Find 〈α|β〉 and 〈β|α〉 ,and confirm that 〈β|α〉 = 〈α|β〉^*.(c)

Question

Consider a three-dimensional vector space spanned by an orthonormal basis [latex] |1 angle,|2 angle,|3 angle . ext { Kets }|\alpha angle ext { and }|\beta angle [/latex] are given by

(b) Find [latex] \langle\alpha \mid \beta angle [/latex] and [latex] \langle\beta \mid \alpha angle [/latex], and confirm that [latex] \langle\beta \mid \alpha angle=\langle\alpha \mid \beta angle^{*} [/latex].

(c) Find all nine matrix elements of the operator [latex] \hat{A} \equiv|\alpha angle\langle\beta| [/latex], in this basis, and construct the matrix A. Is it hermitian?

Accepted Answer

[latex] \langle\beta \mid \alpha angle=\left(-i\left\langle 1|+2\langle 3|)(i|1 angle-2|2 angle-i|3 angle)=(-i)(i)\langle 1 \mid 1 angle+(2)(-i)\langle 3 \mid 3 angle=1-2 i=\langle\alpha \mid \beta angle^{*} ight. ight.[/latex].

(c)

[latex] A_{11}=\langle 1 \mid \alpha angle\langle\beta \mid 1 angle=(i)(-i)=1 ; A_{12}=\langle 1 \mid \alpha angle\langle\beta \mid 2 angle=(i)(0)=0 ; A_{13}=\langle 1 \mid \alpha angle\langle\beta \mid 3 angle=(i)(2)=2 i [/latex];

[latex] A_{21}=\langle 2 \mid \alpha angle\langle\beta \mid 1 angle=(-2)(-i)=2 i ; A_{22}=\langle 2 \mid \alpha angle\langle\beta \mid 2 angle=(-2)(0)=0 ; A_{23}=\langle 2 \mid \alpha angle\langle\beta \mid 3 angle=(-2)(2)=-4 [/latex];

[latex] A_{31}=\langle 3 \mid \alpha angle\langle\beta \mid 1 angle=(-i)(-i)=-1 ; A_{32}=\langle 3 \mid \alpha angle\langle\beta \mid 2 angle=(-i)(0)=0 ; A_{33}=\langle 3 \mid \alpha angle\langle\beta \mid 3 angle=(-i)(2)=-2 i [/latex].

[latex] A =\left(\begin{array}{ccc} 1 & 0 & 2 i \ 2 i & 0 & -4 \ -1 & 0 & -2 i \end{array} ight) [/latex]. No, it's not hermitian.

Question 3.26: Consider a three-dimensional vector space spanned by an orth...

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