Annihilation creation operator

Hey,

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?

OlderDan
Homework Helper
greisen said:
Hey,

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?

I am not following the notation. If a is an operator and a^ is an operator, then why is there a + after a^ and why are there no a^ in your expansion?

Last edited:
Maybe I have expressed the problem badly. a is the annihilation operator and a+ is the creation operator. I have a two state system |1>, |2> with a wavefunction (psi) = |1> + |2>. My problem is to perform the multiplication in order to find the expectation value:

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

if I expand this calculation I get some strange results. How to multiple this? Any help appreciated - thanks in advance

greisen said:
Hey,

<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?

<1|0>=<2|3>=0

In the above sum, only two terms are nonzero.

OlderDan