Propagator

Time-Evolution Revisited

Start from one-dimensional time-evolution, and it's easy to extend to three-dimensional circumstance.

Consider time-dependent Schrödinger Equation with a common Hamiltonian

$$ i\hbar \frac{\partial}{\partial t}\psi(x,t)=H\psi(x,t),\quad\text{where}\ H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x, t), $$

and we introduce time-evolution operator $U(t,t_0)$ which is unitary and satisfies composition property

$$ |\psi(t)\rangle=U(t,t_0)|\psi(t_0)\rangle,\quad U(t_2,t_1)U(t_1,t_0)=U(t_2,t_0). $$

We can also derive a differential equation for $U(t,t_0)$ with the initial condition $U(t_0,t_0)=1$:

$$ i\hbar \frac{\partial}{\partial t}U(t,t_0)=H(t)U(t,t_0) $$

With a time-independent Hamiltonian, this equation's solution is

$$ U(t,t_0)=\exp\Big[-\frac{i}{\hbar}H(t-t_0)\Big], $$

where $H$ commutes with $U(t,t_0)$. In this case, consider a basis constituted with eigenkets $\{|a'\rangle\}$ of an observable $A$, which commutes with $H$ (can be $\{|\psi'\rangle\}$ for $H$). A quantum state's expansion under this basis is

$$ |\alpha,t;t_0\rangle=\exp\Big[-\frac{i}{\hbar}H(t-t_0)\Big]|\alpha,t_0\rangle\\ =\sum_{a'}|a'\rangle\langle a'|\alpha,t_0\rangle\exp\Big[-\frac{i}{\hbar}E_{a'}(t-t_0)\Big]. $$

General Form

Then we can define propagator. For generality (both time-independent and time-dependent cases), put its definition first. Propagator with initial position $(x_0,t_0)$ and final position $(x,t)$ is defined by

$$ K(x,t;x_0,t_0)=\langle x|U(t,t_0)|x_0\rangle, $$

where $U(t,t_0)$ is time-evolution operator.

Remarks

1. Obviously, $K$ is the $x$-space matrix element of $U(t,t_0)$. We can say $K$ is the amplitude of finding a particle at $(x,t)$ with the initial condition $(x_0,t_0)$​.
2. Fix $(x_0,t_0)$, then $K=K(x,t)$ satisfies Schrödinger Equation (as if it's a wave function).

3. Initially localized ($\delta$-function initial conditions):

   $$ \lim_{t\to t_0^+}K(x,t;x_0,t_0)=\langle x|x_0\rangle = \delta(x-x_0). $$

4. Impose

   $$ K(x,t;x_0,t_0)=0\quad \text{if}\quad t<t_0. $$

Then, with a general initial condition $\psi(x,t_0)$, the final wave function is

$$ \psi(x,t)=\langle x|\psi(t)\rangle=\langle x|U(t,t_0)|\psi(t_0)\rangle=\int dx_0K(x,t;x_0,t_0)\psi(x_0,t_0). $$

The propagator is the kernel of the integral transform!

Free Particle Propagator

To be continued.