## Position and Momentum operators derived from annihilation and creation operators

*By: Levine T. Poralan, MS Physics, MSU-IIT*

Given the annihilation and creation operators below,

Show that

**(a) **

**(b) **

**(c)** and are Hermitian but not and

**Solution:**

**(a) **Adding the annihilation and creation operators

this results to

and this becomes

.

**(b) **Now, finding the difference of the two operators

we can obtain

multiplying the whole equation by (-i)

and this can be expressed as

.

**(c) **Checking whether the operators are Hermitian,

Therefore,

is Hermitian.

Now,

Therefore,

is Hermitian.

Lastly,

From this, we can see that

and

Therefore,

and are not Hermitian.