[sjastro]

Here is a non maths version of the QM explanation as to why time is not an observable.

As Carl mentioned in a previous post one can define a quantum wavefunction as superimposed state containing all the possible outcomes for the observable.

Examples of observables are position, linear momentum, angular momentum, energy and spin. If the wavefunction is time independent the superimposed state does not change.
When a measurement is performed the wavefunction collapses and a measurement or observable is the outcome.
The same test can be performed over and over. The same superimposed state collapses each time to give an observable. From this you can calculate the average and standard deviation or variance of the measurements.

Now let's bring time into the picture. Suppose the wavefunction is now time dependent. The superimposed state now changes with time along with any of the observables.
To perform a measurement involving a time dependent function one would have to so at the same specific time for each measurement.

It's for this reason time is not an observable. If it was each different time measurement would correspond to a different collapsing wavefuction.

For example if a particle decays, it has an intial and final quantum state.
If time was observable, the first few measurements for say energy may correspond to the initial state, all others may be for the final state.
You can't mix the different states of superimposition.

There is however an important relationship between the energy difference in 2 different quantum states and the time taken to go from one state to another. If the energy difference is large the time interval is small and vice versa.

Regards

Steven