Question

Determine the commutators of the operators a and a+,where a = (x + ip)/2 ^1/2 and a+ = (x - ip)/ 2 ^1/2

Answer 1

Given that:

`a = ((x+ip)/(2))^{(1)/(2)}` and
`a+= ((x-ip)/(2))^{(1)/(2)}`

if a , a+ commutator, it obeys
`aa^+ = a^+a`

First find:

`aa^+ = ((x+ip)/(2))^{(1)/(2)} ((x-ip)/(2))^{(1)/(2)}`

=
`(((x)^2-(ip)^2)/(4))^{(1)/(2)}=(((x)^2+(p)^2)/(4))^{(1)/(2)}`

Now;

`a^+a =((x-ip)/(2))^{(1)/(2)} ((x+ip)/(2))^{(1)/(2)} = (((x)^2-(ip)^2)/(4))^{(1)/(2)}`

=
`(((x)^2-(ip)^2)/(4))^{(1)/(2)}=(((x)^2+(p)^2)/(4))^{(1)/(2)}`

therefore,
`aa^+ = a^+a` which implies the operators a and a+ are commutators.