(Extra) Example: Harmonic Oscillator
Classical situation: Mass attached to a spring.
- The spring exerts a force on the mass which is proportional to the distance that the spring is stretched or compressed. This force then produces an acceleration of the mass which leads to an oscillating motion of the mass. The frequency of this oscillation is determined by the stiffness of the spring and the amount of mass.
Quantum situation: suppose F is proportional to distance, then potential energy is proportional to distance squared. Solutions to Schrodinger Eqn:
What is shown here?
Possible wave functions ?(x) at a fixed time t!
How does this change in time?
They oscillate with the classical frequency!
What distinguishes the different solutions?
The Energy! (Classically this corresponds to the amplitude of the oscillation) Note: not all energies are possible! They are quantized!