Propagator Theory

Given a wave packet which in the remote past represented a particle approaching a potential, what does that wave look like in the remote future?

We will first consider the nonrelativistic propagator. Huygens' principle can be written as

$$\psi(\vec{x}^\prime,t^\prime) = i \int d^3x G(\vec{x}^\prime... ...ec{x},t) \psi(\vec{x},t) \quad\textrm{for}\quad t^\prime > t ,$$ (6.5)

where the integral extends over all space. $\psi(\vec{x}^\prime,t^\prime)$ is the total wave arriving at the point $\vec{x}^\prime$ at time $t^\prime$ and $\psi(\vec{x},t)$ is the original wave amplitude. $G(\vec{x}^\prime,t^\prime;\vec{x},t)$ is the Green's function or propagator. In general one does not consider stationary eigenstates of energy (i.e. stationary waves). Knowledge of $G$ enables us to construct the physical state which develops in time from any given initial state. This is equivalent to a complete solution to Schrödinger equation.

Consider a free-particle solution $\phi$ and its Green function $G_0$. We introduce a potential $V(\vec{x},t)$ which is ``turned on'' for a brief interval of time $\Delta t_1$ about $t_1$ ($V=0$ outside of time $\Delta t_1$). $V(\vec{x}_1,t_1)$ acts as a source of new waves and we can write

$$\left(i\frac{\partial}{\partial t_1} - H_0 \right) \psi(\vec{x}_1,t_1) = V(\vec{x}_1,t_1) \psi(\vec{x}_1,t_1) \quad\textrm{for}\quad t=\Delta t_1 ,$$

$$= 0 \quad\textrm{for}\quad t\ne \Delta t_1 ,$$ (6.6)

where right-hand side is zero outside the interval $\Delta t_1$.

The new wave function can be written as

$$\psi(\vec{x}_1,t_1) = \phi(\vec{x}_1,t_1) + \Delta\psi(\vec{x}_1,t_1) .$$ (6.7)

Substituting this solution into equation 6.6 and using the free particle Schrödinger equation gives

$$\left( i\frac{\partial}{\partial t_1} -H_0\right) \Delta\psi... ...c{x}_1,t_1)[\phi(\vec{x}_1,t_1) + \Delta\psi(\vec{x}_1,t_1)] .$$ (6.8)

The second terms on both sides of the equation are smaller than the first terms on both sides of the equation. This is clear for the righthand side of the equation. To see it for the lefthand side, we drop the second term on the righthand side and write

$$i\frac{\partial}{\partial t_1} \Delta\psi(\vec{x}_1,t_1) = V(\vec{x}_1,t_1) \phi(\vec{x}_1,t_1) + H_0 \Delta \psi(\vec{x}_1,t_1) ,$$ (6.9)

$$i \Delta\psi(\vec{x}_1,t_1+\Delta t_1) = \int_{t_1}^{t_1+\Delta t_1} dt^\prime \left[ V(\vec{x}_1,t^\prime) \phi(\vec{x}_1,t^\prime) + H_0 \Delta \psi(\vec{x}_1,t^\prime) \right] .$$ (6.10)

After dropping the second two terms on both sides of equation 6.8, we have

$$i\frac{\partial}{\partial t_1} \Delta\psi(\vec{x}_1,t_1) = V(\vec{x}_1,t_1) \phi(\vec{x}_1,t_1) .$$ (6.11)

To first order

$$\Delta \psi (\vec{x}_1,t_1+\Delta t_1) = -i V(\vec{x}_1,t_1) \phi(\vec{x}_1,t_1) \Delta t_1 .$$ (6.12)

The integration shows that the potential produces an additional change in $\psi$ during $\Delta t_1$ in addition to that taking place in the absence of $V$. Since the potential $V(\vec{x}_1,t_1)$ vanishes after the time interval $\Delta t_1$, the scattered wave propagates according to the free propagator $G_0$ too. This added wave at a future time $t^\prime$ leads to a new contribution to $\psi(x^\prime,t^\prime)$,

$$\Delta \psi(\vec{x}^\prime,t^\prime) = i\int d^3x_1 G_0(\vec{x}^\prime,t^\prime;\vec{x}_1,t_1) \Delta \psi(\vec{x}_1,t_1) ,$$

$$= \int d^3x_1 G_0(\vec{x}^\prime,t^\prime;\vec{x}_1,t_1) V(\vec{x}_1,t_1) \phi(\vec{x}_1,t_1) \Delta t_1 .$$ (6.13)

Here we have replaced $t_1+\Delta t_1$ by $t_1$ which is justified in the limit.

Thus the wave $\psi$ developing from an arbitary wave packet $\phi$ in the remote past is

$$\psi(\vec{x}^\prime,t^\prime) = \phi(\vec{x}^\prime,t^\prime) + \Delta \psi(\vec{x}^\prime,t^\prime) ,$$

$$= \phi(\vec{x}^\prime,t^\prime) + \int d^3x_1 G_0(\vec{x}^\prime,t^\prime;\vec{x}_1,t_1) V(\vec{x}_1,t_1) \phi(\vec{x}_1,t_1) \Delta t_1 ,$$ (6.14)

$$= i\int d^3x G_0(\vec{x}^\prime,t^\prime;\vec{x},t) \phi(\vec{x},t)$$

$$+ \int d^3xd^3x_1 \Delta t_1 G_0(\vec{x}^\prime,t^\prime;\vec{x}_1,t_1) V(\vec{x}_1,t_1) G_0(\vec{x}_1,t_1;\vec{x},t) \phi(\vec{x},t) ,$$

$$= i\int d^3x [G_0(\vec{x}^\prime,t^\prime;\vec{x},t)$$

$$+ \int d^3x_1 \Delta t_1 G_0(\vec{x}^\prime,t^\prime;\vec{x}_1,t_1) V(\vec{x}_1,t_1) G_0(\vec{x}_1,t_1;\vec{x},t) ] \phi(\vec{x},t).$$ (6.15)

Therefore the Green's function here is

$$G(\vec{x}^\prime,t^\prime;\vec{x},t) = G_0(\vec{x}^\prime,t^... ...vec{x}_1,t_1) V(\vec{x}_1,t_1) G_0(\vec{x}_1,t_1;\vec{x},t) .$$ (6.16)

The first term represents the propagation for $(\vec{x},t)$ to $(\vec{x}^\prime,t^\prime)$ as a free particle. The second term represents propagation from $(\vec{x},t)$ to $(\vec{x}_1,t_1)$, a scattering at $(\vec{x}_1,t_1)$, and free propagation from $(\vec{x}_1,t_1)$ to $(\vec{x}^\prime,t^\prime)$.

If we turn on another potential $V(\vec{x}_2,t_2)$ for an interval $\Delta t_2$ at time $t_2>t_1$, the additional contribution to $\psi(\vec{x}^\prime,t^\prime)$ for $t^\prime>t_2$ is

$$\Delta\psi(x^\prime) = \int d^3x_2 G_0(x^\prime;2) V(2) \psi(2) \Delta t_2 ,$$ (6.17)

$$= i\int d^3x_2d^3x \Delta t_2 G_0(x^\prime;2) V(2) [G_0(2;x)$$

$$+ \int d^3x_1 \Delta t_1 G_0(2;1) V(1) G_0(1;x) ] \phi(x),$$ (6.18)

where we have used the obvious notation: $(x) \equiv (\vec{x},t)$ and $\psi(2) \equiv \psi(x_2)$. The first term represents a single scattering at time $t_2$. The second term is a double scattering.

The total wave is obtained by inserting (6.14) for $\psi(2)$ into (6.17) and adding it to (6.14)

$$\psi(x^\prime) = \phi(x^\prime) + \int d^3x_1 \Delta t_1 G_0(x^\prime;1) V(1) \phi(1)$$

$$+ \int d^3x_2 \Delta t_2 G_0(x^\prime;2) V(2) \phi(2)$$

$$+ \int d^3x_1 \Delta t_1 d^3x_2 \Delta t_2 G_0(x^\prime;2) V(2) G_0(2;1) V(1) \phi(1) .$$ (6.19)

The four terms of the above equation are depicted in figure 6.1.

Figure 6.1: Space-time diagrams for propagation from ($x$,$t$) to ($x^\prime$,$t^\prime$) as a) a free particle, b) with one scattering at ($x_1$,$t_1$), c) with one scattering at ($x_2$,$t_2$) and d) with double scattering.

If there are $n$ such time intervals when the potential $V$ is turned on

$$\psi(x^\prime) = i\int d^3x G(x^\prime;x) \phi(x) ,$$

$$= \phi(x^\prime) + \sum_i\int d^3x_i\Delta t_i G_0(x^\prime;x_i) V(x_i) \phi(x_i)$$

$$+ \sum_{i,j;t_i>t_j}\int d^3x_i\Delta t_i d^3x_j\Delta t_j G_0(x^\prime;x_i) V(x_i) G_0(x_i;x_j) V(x_j) \phi(x_j)$$

$$+ \sum_{i,j,k;t_i>t_j>t_k}\int d^3x_i\Delta t_i d^3x_j\Delta t_j d^3x_k\Delta t_k$$

$$\cdot G_0(x^\prime;x_i) V(x_i) G_0(x_i;x_j) V(x_j)G_0(x_j;x_k) V(x_k) \phi(x_k) + \cdots .$$ (6.20)

The corresponding Green's function is

$$G(x^\prime;x) = G_0(x^\prime;x) + \sum_i \int d^3x_i\Delta t_i G_0(x^\prime;x_i,t_i) V(\vec{x}_i;t_i) G_0(\vec{x}_i,t_i;x)$$

$$+ \sum_{i,j;t_i>t_j} \int d^3x_i \Delta t_i d^3x_j \Delta t_j$$

$$\cdot G_0(x^\prime;\vec{x}_i,t_i) V(\vec{x}_i;t_i) G_0(\vec{x}_i,t_i;\vec{x}_j,t_j) V(\vec{x}_j,t_j) G_0(\vec{x}_j,t_j;x) + \cdots .$$ (6.21)

$G(x^\prime;x)$ is the probability amplitude for a particle wave originating at $x$ to propagate to $x^\prime$, as depicted in figure 6.2. This amplitude is a sum of amplitudes, the $n$th such term begin a product of factors corresponding to the figure below. Each line in the figure represents the amplitude $G_0(x_i;x_{i-1})$ that a particle wave originating at $x_{i-1}$ propagates freely to $x_i$. At the point $x_i$ it is scattered with probability amplitude per unit space-time volume $V(x_i)$ to a new wave propagating forward in time with amplitude $G_0(x_{i+1};x_i)$ to the next interaction. This amplitude is then summed over all space-time points in which the interactions can occur.

Figure 6.2: $n$th order contribution to $G(x^\prime;x)$.

We may lift the time-ordering restriction $t_i>t_j$, etc, if we define

$$G_0^+(\vec{x}^\prime,t^\prime;\vec{x},t) = \left\{ \begin{array}{lcl} 0 & \textrm{for} & t^\prime < t \\ G_... ...^\prime,t^\prime;\vec{x},t) & \textrm{for} & t^\prime > t \end{array} \right. ,$$ (6.22)

$$G^+(\vec{x}^\prime,t^\prime;\vec{x},t) = \left\{ \begin{array}{lcl} 0 & \textrm{for} & t^\prime < t \\ G(... ...^\prime,t^\prime;\vec{x},t) & \textrm{for} & t^\prime > t \end{array} \right. .$$ (6.23)

$G$ is a retarded propagator since it only propagates waves forward in time. Physically this just means that no Huygens wavelets $\Delta\psi$ from the $i$'th iteration (at time $t_i$) appear until after $t_i$.

In the limit of a continuous interaction $\Delta t\rightarrow dt$ and $n \rightarrow \infty$, so that we can write

$$\sum_i \int d^3x \Delta t_i \rightarrow \int d^4x$$ (6.24)

and obtain

$$G^+(x^\prime;x) = G_0^+(x^\prime;x) + \int d^4x_1 G_0^+(x^\prime;x_1) V(x_1) G_0^+(x_1;x)$$

$$+ \int d^4x_1d^4x_2 G_0^+(x^\prime;x_1) V(x_1) G_0^+(x_1,x_2) V(x_2) G_0^+(x_2;x) + \cdots .$$ (6.25)

This multiple scattering series is assumed to converge and may be summed formally to yield

$$G^+(x^\prime;x) = G_0^+(x^\prime;x) + \int d^4x_1 G_0^+(x^\prime;x_1) V(x_1) G^+(x_1;x) .$$ (6.26)

This is the Lippmann-Schwinger equation. We have ignore the possibility of bound states in the potential $V$.

Similarly the series for the wave function $\psi(x^\prime)$ can be summed, resulting in

$$\psi(x^\prime) = \lim_{t\rightarrow -\infty} i \int d^3x G^+(x^\prime;x) \phi(x) ,$$

$$= \lim_{t\rightarrow -\infty} i\int d^3x \left[ G_0^+(x^\prime;x) + \int d^4x_1 G_0^+(x^\prime;x_1) V(x_1) G^+(x_1;x) \right] \phi(x) ,$$

$$= \phi(x^\prime) + \lim_{t\rightarrow -\infty} \int d^4x_1 G_0^+(x^\prime;x) V(x_1) i \int d^3x G^+(x_1;x) \phi(x) ,$$

$$= \phi(x^\prime) + \int d^4x_1 G_0^+(x^\prime;x_1) V(x_1) \psi(x_1) .$$ (6.27)

This is the integral equation for $\psi(x^\prime)$, where the second term is the scattered wave.