Can somebody please help me solve this?
$C_n$ = $2 D^{n-1}$ + $\hat{D}*C_{n-2} * \hat{D}$
where $C_1 = I_{2x2}$ and $C_2 = 2*D $ and D is a matrix operator**. I want to find $C_n$ in terms of D and n ONLY.
For those interested, this is actually a quantum mechanics harmonic oscillator problem. Define $C_n = [\hat{a}, \hat{a}_{+}^{n}], D = \hat{a}_{+}$.
where $a$ and $a_{+}$ are annihilation and creation operators.
Update: I found the answer using principles of physics = $C_n = n*D^{n-1} $.
**By Matrix operator, I mean an operator in matrix representation.