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

Answer:

Given that:

$a = (\frac{x+ip}{2})^{\frac{1}{2}}$ and $a+= (\frac{x-ip}{2})^{\frac{1}{2}}$

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

First find:

$aa^+ = (\frac{x+ip}{2})^{\frac{1}{2}} (\frac{x-ip}{2})^{\frac{1}{2}}$

                = $(\frac{(x)^2-(ip)^2}{4})^{\frac{1}{2}}=(\frac{(x)^2+(p)^2}{4})^{\frac{1}{2}}$

Now;

$a^+a =(\frac{x-ip}{2})^{\frac{1}{2}} (\frac{x+ip}{2})^{\frac{1}{2}} = (\frac{(x)^2-(ip)^2}{4})^{\frac{1}{2}}$

              =$(\frac{(x)^2-(ip)^2}{4})^{\frac{1}{2}}=(\frac{(x)^2+(p)^2}{4})^{\frac{1}{2}}$

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