12.7 Green’s functions and solutions to the Schr¨odinger equations

12.7 Green’s functions and solutions to the Schr¨odinger equations

The method of Green’s functions provides solutions $\psi$ to diferential equations

$$D\psi =J\text{\hspace{1em}(12.14)}$$

where $D$ is a diferential operator and $J$ is a fixed function, by finding an inverse $D^{-1}\mathrm{to}D$ and then setting $\psi =D^{-1}J$ . For D a constant coeficient diferential operator, the Fourier transform will take $D$ to multiplication by a polynomial $\hat{D}$ and we define the Green’s function of D to be the function (or distribution) with Fourier transform 1

$$\hat{G}=\frac{1}{\hat{D}}\text{\hspace{1em}(12.15)}$$

Since

$$\hat{D}\hat{G}\hat{J}=\hat{J}$$

the inverse Fourier transform of $\hat{G}\hat{J}$ will be a solution to 12.14.

Note that $\hat{G}$ and $G$ are not uniquely determined by the condition 12.15 since $D$ may have a kernel, and then solutions to 12.14 are only determined up to a solution ${\psi }_0$ of the homogeneous equation $D{\psi }_0=0$ . In terms of Fourier transforms, $\hat{D}$ may have zeros, and then $\hat{G}$ is ambiguous up to functions on the zero set.

For the case of the Schr¨odinger equation, we take

$$D=i\frac{\partial }{\partial t}+\frac{1}{2m}\frac{{\partial }^2}{\partial q^2}$$

and then (Fourier transforming in $q$ and t as above)

$$\hat{D}=i(-i\omega )+\frac{1}{2m}(ik{)}^2=\omega -\frac{k^2}{2m}$$

and

$$\hat{G}=\frac{1}{\omega -\frac{k^2}{2m}}$$

$\mathrm{A}$ solution $\psi$ of 12.14 will be given by computing the inverse Fourier transform of $\hat{G}\hat{J}$

$$\psi (q,t)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\frac{1}{\omega -\frac{k^2}{2m}}\hat{J}(\omega ,k)e^{-i\omega t}{e}^{ikq}d\omega dk\text{\hspace{1em}(12.16)}$$

Here $\hat{D}$ is zero on the set $\begin{array}{r}\omega =\frac{k^2}{2m}\end{array}$ and the non-uniqueness of the solution to $D\psi =J$ is reflected in the ambiguity of how to treat the integration through the points $\begin{array}{r}\omega =\frac{k^2}{2m}\end{array}$

For solutions $\psi (q,t)$ of the Schr¨odinger equation with initial data $\psi (q,0)$ at time $t=0$ , if we define ${\psi }_{+}(q,t)=\theta (t)\psi (q,t)$ we get the “retarded” solution

$${\psi }_{+}(q,t)={\int }_{-\infty }^{+\infty }{U}_{+}(t,q-q_0)\psi (q_0,0)d{q}_0$$

where $U+(t,q-q_0)$ is the retarded propagator $\mathrm{gi}$ ven by equations 12.12 and 12.13. Since

$$D{\psi }_{+}(q,t)=(D\theta (t))\psi (q,t)+\theta (t)D\psi (q,t)=i\delta (t)\psi (q,t)=i\delta (t)\psi (q,0)$$

${\psi }_{+}(q,t)$ is a solution of 12.14 with

$$J(q,t)=i\delta (t)\psi (q,0),\hat{J}(\omega ,k)=\frac{i}{\sqrt{2\pi }}\tilde{\psi }(k,0)$$

Using 12.16 to get an expression for ${\psi }_{+}(q,t)$ in terms of the Green’s function we have

$$\begin{array}{rl}{\psi }_{+}(q,t)& =\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\hat{G}(\omega ,k)\frac{i}{\sqrt{2\pi }}\tilde{\psi }(k,0)e^{-i\omega t}{e}^{ikq}d\omega dk\\ & ={\left(\frac{1}{2\pi }\right)}^2{\int }_{-\infty }^{+\infty }\left({\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }i\hat{G}(\omega ,k)e^{-i\omega t}{e}^{ik(q-q^{\prime })}d\omega dk\right)\psi (q^{\prime },0)d{q}^{\prime }\end{array}$$

Comparing this to equations 12.12 and 12.13, we find that the Green’s function that will give the retarded solution ${\psi }_{+}(q,t)$ is

$${\hat{G}}_{+}(\omega ,k)=\underset{ϵ\to 0^{+}}{\lim }\frac{1}{\omega -\frac{k^2}{2m}+iϵ}$$

and is related to the retarded propagator by

$$\hat{U}(\omega ,k)=\frac{i}{2\pi }{\hat{G}}_{+}(\omega ,k)$$

One can also define an “advanced” Green’s function by

$${\hat{G}}_{-}=\underset{ϵ\to 0^{+}}{\lim }\frac{1}{\omega -\frac{k^2}{2m}-iϵ}$$

and the inverse Fourier transform of ${\hat{G}}_{-}\hat{J}$ will also be a solution to 12.14. Taking the diference between retarded and advanced Green’s functions gives an operator

$$\hat{\Delta }=\frac{i}{2\pi }({\hat{G}}_{+}-{\hat{G}}_{-})$$

with the property that, for any choice of $J,\Delta J$ will be a solution to the Schr¨odinger equation (since it is the diference between two solutions of the inhomogeneous equation 12.14). The properties of $\Delta$ can be understood by using 12.11 to show that

$$\hat{\Delta }=\delta (\omega -\frac{k^2}{2m})$$

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