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I am calculating the expectation value of the position x. I have the wave function

psi(x) = |1> + |2>

so I use the equation <x> = <psi|(a+a^+|psi> to calculate the mean value. So I get

(<1| + <2|)(a+a^+)(|1> + |2>)

which I reduce to

<1|a|1> + <1|a|2> + <2|a|1> + <2|a|2> + <1|a+|1> + <1|a+|2> + <2|a+|1> + <2|a+|2>

if I use the properties of annihilation and creation I get some strange results such fx

sqrt(1) <1|0> or sqrt(3)<2|3> which are totally wrong. What have I done wrong?

Thanks in advance