Schroedinger

An abstract way to state the Schrödinger equation is \[ (H - E) \left| \psi \right\rangle = 0 \] with $ H $ the Hamiltonian operator, $ E $ the energy operator and $\left| \psi \right\rangle$ the state vector of the system. Notice that this equation does not say anything about the wavefunction: it only involve things that live abstractly in the Hilbert space. In order to be more concrete, we introduce a complete basis for the Hilbert space. Given a state vector $ \left| \psi \right\rangle $ we conveniently expand it in terms of a basis vectors $ \left| x \right\rangle $ as \[ \left| \psi \right\rangle = \sum_{x} \psi_{x} \left| x \right\rangle \] Sometimes the sum over $ x $ is discrete, sometimes it is continuous. Usually it is some familiar quantity like position or momentum. But not always! The $\psi_{x}$ are the complex coefficients that determine the state: \[ \psi_{x} = \left\langle x | \psi \right\rangle \] We can also find the matrix elements of an operator: \[ O_{xy} = \left\langle x | O | y \right\rangle \] So once we have choosen a basis for the state vectors and the operators, we can consider matrix-like equations of the form \[ \sum_{y} O_{xy} \psi_{y} = 0 \] In component form, the Schrödinger equation is \[ \sum_{y} (H - E)_{xy} \psi_{y} = 0 \] We need to incorporate the notion of "evolution". There are two useful ways to parameterize evolution: in time or in energy. Time and energy are conjugate quantities. This means that we can do either one but not both simultaneously. If we choose to describe evolution in time, then (in the Schrödinger picture) states will carry a time label $t$. A state $ \left| \psi \right\rangle $ with a time label $t_{A}$ is related to the same state with a different time label $t_{Z}$ via a unitary transformation: \[ \left| \psi, \, t_{Z} \right\rangle = U(t_{Z}, \, t_{A}) \left| \psi, \, t_{A} \right\rangle \] Again, this is an abstract equation. In terms of components this reads \[ \psi_{z}(t_{Z}) = \sum_{a} G_{za} (t_{Z}, \, t_{A}) \psi_{a} (t_{A}) \] We have stumble upon a very important object, the fundamental "matrix" $G_{za}(t_{Z}, \, t_{A})$. This object "propagates" the coefficients of a state vector in time. Let me be more concrete: in the position basis, the fundamental amplitude reads \[ G(z, \, a | t_{Z}, \, t_{A}) = \left\langle z | U(t_{Z}, \, t_{A}) | a \right\rangle \] This is the amplitude that Feynman wrote as a sum over trajectories: a path integral.

Let us note that $ G_{za} (t_{Z}, \, t_{A}) $ will satisfy the same Schrödinger equation that $\psi_{z}(t_{Z})$ does. In the Van Vleck approximation, the fundamental amplitude is approximated with the Van Vleck amplitude $ V_{za} (t_{Z}, \, t_{A})$. I will discuss this soon.