In classical mechanics, we have the following relation based on the kinematics with a constant acceleration (namely driven by a constant force):
$$v(t_2)^2-v(t_1)^2 = 2 a x \qquad \qquad (1)$$
and also we all know the dynamical evolution of classical objects governed by the Newtonian's law:
$$F=ma \qquad \qquad (2)$$
combining the formula $(1)$ and $(2)$ one can get the following equation
$$\dfrac{1}{2}mv(t_2)^2-\dfrac{1}{2}mv(t_1)^2 = Fx \qquad \qquad (3)$$
of course, we can generalize the $eq.(3)$ to a more general situation (driven by a variable force and living in a higher dimension).Finally one can get work-energy theorem:
$$\dfrac{1}{2}mv(t_2)^2-\dfrac{1}{2}mv(t_1)^2 = \int_A^B \textbf{F} \cdot d \textbf{X} \qquad \qquad (4)$$
If the force $\textbf{F}$ is a conservative force then one can obtain the following famous $\textbf{law of conservation of energy}$:
$$K_1+U_1 = K_2+ U_2 \qquad \qquad (5)$$
where $K=\dfrac{1}{2}mv^2$ and $\textbf{F} = -\nabla U$.Everything is so fine.
So my question is about can we follow the same logic in quantum mechanics? I mean can we derive the law of conservation of energy from Schrodinger equation? Or what's the exact meaning for the conservation of energy in quantum world ?
