1. Operators - are performed on functions -are performed on vector functions and have directional qualities as well. These are referred to as vector operators. - can obey the Eigen equation, and thus have eigen values and eigen functions. - In general we are concerned with the function that obey this equation.

2. Classical Mechanics-Position Example Notice that we are using a function of time to describe the position not some fixed value. This function tells you the position at any point in time.

3. Classical Mechanics-Position-3D Note that the operator is applied to the position function and the result is the quantity associated with the operator. Ie. The x operator give you the x component of r(t), this is know as a projection operator. The vector operator r can be constructed from the projector operators.

4. Classical Mechanics-Position-3D Example

5. Classical Mechanics-Velocity-1D Example

6. Classical Mechanics-Velocity-3D Example

7. Classical Mechanics-Velocity-3D

8. The corresponding vector operator to velocity can be reconstructed from the projector operators of the components:

9. Classical Mechanics-Velocity-3D Example

10. Classical Mechanics-Acceleration-1D Example

11. Classical Mechanics-Acceleration-3D Example

12. Classical Mechanics-Force-1D Example

13. Classical Mechanics-Force-3D Example

14. Impulse and Momentum Momentum Impulse In general For a constant force

15. Momentum-1D Example

16. Momentum-3D Example

17. Impulse-1D Force can be thought of as a change in potential energy with change in position

18. Impulse-1D Examples i) In terms of the Force operator: ii) In terms of the Potential operator:

19. Impulse-3D

20. Angular Momentum

23. Kinetic Energy

24. Potential Energy Hooks Law Coulombs Law

25. Conservation of Energy Total energy remains constant, as long as V is not an explicit function of time. (i.e V(x(t))) Hamiltonian

26. Conservation of Energy-Hook’s Law Since: Newtons Law F – ma = 0