What is an example of a quantum mechanics practice problem?
Measuring the real direction and speed of a subatomic particle.
Because the actual probes do use the same type of particles to perform their measurement, it is actually like trying to measure the position and speed of a baseball ball by throwing at it another baseball ball. When the 'measuring' ball will hit the 'measured' ball the first time, the latter will be forced to change its trajectory and speed. And this means that if we measure a particle, we can only know its immediate position, but from that time ahead we can't anymore know what its natural course would have been. Thus, a second measurement will retrieve an information that has already been altered by the first measurement.
This is like asking for an example of a classical mechanics practice problem: many topics exist under the subject, so there are many, many examples.
To provide a good example of a practice problem would heavily depend on the level of QM you are looking for. It could be something as easy as normalizing a simple quantum state vector, something a little more complicated involving time-dependent systems with time evolution, harmonic perturbation...etc.
Here is a one about operators.
I took this problem from the book: Physical Chemistry 6th edition, Ira N. Levine, pg. 613, Example 17.4. You can buy the book here.
Let the operators
If you want me to write the solution, don't hesitate to leave a comment.