Originally Posted by **nairbdm** Say you have an arbitraty vector:
/w> = {a
b
c}
where a, b, c are complex numbers. Calculate the "expectation value"
<M>*w = <w*/*M*/*w>*
and show that <M>w is always a real number, no matter what a,b,c are. Hint - Dont forget how to get the dual vector, <w/.
Thanks. |

Originally Posted by **nairbdm** I'm sorry, M is the 3X3 matrix:
M= {1,-*i*(2^(1/2)),0} {*i(*2^(1/2)),0,0} {0,0,2} Thanks. |

$\displaystyle <M> = <w|M|w> = < a^* b^* c^* | \left ( \begin{matrix} 1 & -i\sqrt{2} & 0 \\ i\sqrt{2} & 0 & 0 \\ 0 & 0 & 2 \end{matrix} \right ) \left | \begin{matrix} a \\ b \\ c \end{matrix} \right >$

$\displaystyle = < a^* b^* c^* | \left | \begin{matrix} a - ib\sqrt{2} \\ ia\sqrt{2} \\ 2c \end{matrix} \right >$

$\displaystyle = a^*(a - ib\sqrt{2}) + b^*(ia\sqrt{2}) + c^*2c$

$\displaystyle = a^*a + 2c^*c + i(ab^* - a^*b)\sqrt{2}$

Now, given any two complex numbers a and b, the number $\displaystyle ab^* - a^*b$ will be pure imaginary. Thus the expectation value is a real number.

-Dan