Schrödinger We already know how to find the momentum eigenvalues of a system. How about the energy and the evolution of a system? Schrödinger Representation: The evolution of a state of a system is given by the application of the Hamiltonian operator to the state wavefunction.

What is the Hamiltonian?

Separation of variables If V(q i ) is t-independent, then the Hamiltonian is also t-independent, and we can proposed a separation of variables

Time and coordinate separation

Cont.

The phase function Even if the wavefunction evolves in time, the probability density remains constant!

Expectation value The expectation value of the Hamiltonian does not depend on time. A superposition of eigenstsates has constant E (E≠t). This is TRUE for any observable whose [A,H]=0

Heisenberg Representation In the Heisenberg (interaction) representation, the evolution of the system is given by the evolution of an operator But… how do we get to this equation? Consider the time derivative of an expectation value: We can evaluate the partial derivatives for each state  and for the operator A We recognize that 2 of these terms correspond to the evolution of states

cont

Schrödinger equation of a free particle Consider again a FREE PARTICLE. We know how to evaluate the momentum, by solving the eigenvalue-eigenvector eq. …what about the Energy of a free particle? We need to construct the Hamiltonian operator to learn about the energy of the system There are 2 ways to solve this problem HARD WAY!!!!

How useful are []? Since we already know the eigenfunction of P, we can try to see if it is also an eigenfunction of H The momentum eigenfunctions are also eigenfunctions of the Hamiltonian, with eigenvalues = classical kinetic energy Even though the quantum and classical energies coincide, the particle behavior is very different, i.e. the Quantum particle can never be at rest, because  p  x ≽ h/4  EASY WAY!!

Good solutions of the Schrödinger eq. Solving the Schrödinger eq  (most times) solve a differential eq., but not all mathematical solutions will be good physical solutions for the probability density.