How come energy is conserved if identical Physical experiments can have different results depending on time? I just learned about Noether's theorem. It states that temporal symmetry is the reason for energy conservation. But in quantum mechanics, identical isolated experiments conducted at different times can give different results because of probabilities. Does this mean energy is not conserved in quantum mechanics?
 A: Note that Noether's theorem appeared before quantum mechanics. It is nevertheless true, following the quantum-classical correspondence principle (indeed, quantum mechanics is obtained from the theoretical classical mechanics by replacing all the variables with the corresponding operators). The probabilistic nature of the quantum mechanics however means that the theorem holds for the quantum mechanical averages rather than for the results of particular measurements.
The deviation of measured energies from the mean energy can be viewed as a manifestation of the Heisenberg uncertainty principle. Indeed, the energy operator (appearing in the Schrödinger equation) is 
\begin{equation}
\hat{E} = \imath\hbar \frac{\partial}{\partial t}
\end{equation}
does not commute with time. In fact, the very form of this energy operator - as a generator of time translation - testifies to the quantum definition of energy firmly grounded in Noether's theorem.
Btw, this situation is very similar to momentum conservation, which, according to Noether's theorem, follows from the translational invariance. 
A: 
But in quantum mechanics, identical isolated experiments conducted at different times can give different results because of probabilities. 

The probabilistic nature of quantum mechanics are about the possibility of measuring $(x,y,z,t)$ and $(p_x,p_y,p_z,E)$ for a quantum mechanical particle. The overall energy momentum angular momentum conservation laws between input particles per interaction, i.e.per event, and output particles are strictly conserved. It is the only way the data of scattering experiments in elementary particle physics are gathered. 
Searching one finds a number of papers , for example, discussing Noethers theorem and quantum mechanics.
