In Quantum Optics and Quantum Mechanics, the time evolution operator
$$U(t,t_i) = \exp\left[\frac{-i}{\hbar}H(t-t_i)\right]$$
is used quite a lot.
Suppose $t_i =0$ for simplicity, and say the eigenvalue and eigenvectors of the hamiltionian are $\lambda_i, \left|\lambda_i\right>$.
Now, nearly every book i have read and in my lecture courses the following result is given with very little or no explanation:
$$U(t,0) = \sum\limits_i \exp\left[-\frac{i}{\hbar}\lambda_it\right]\left|\lambda_i\right>\left<\lambda_i\right|$$
This is quite a logical jump and I can't see where it comes from, could anyone enlighten me?
Best Answer
Starting with:
$$U(t,t_i) = e^{\frac{-i}{\hbar }H(t-t_i)}$$
If $t_i=0$:
$$U(t,0) = e^{\frac{-i}{\hbar }Ht}$$
Using the identity: $\sum\limits_i \left|\lambda_i\right>\left<\lambda_i\right|=\mathbb{I}$
$$U(t,0) = \sum\limits_i e^{\frac{-i}{\hbar }Ht}\left|\lambda_i\right>\left<\lambda_i\right|$$
Since the exponential of an operator is (by Taylor expanding): $e^H=\mathbb{I}+H+\frac{1}{2}H^2+\dots$
And: $H\left| \lambda_i \right> =\lambda_i \left| \lambda_i \right>$
You should be able to see that:
$$U(t,0) = \sum\limits_i e^{\frac{-i}{\hbar }\lambda_it}\left|\lambda_i\right>\left<\lambda_i\right|$$