11.5 Solutions of the Schr¨odinger equation in momentum space

11.5 Solutions of the Schr¨odinger equation in momentum space

Equation 11.7 shows that under Fourier transformation the derivative operator $\frac{d}{dq}$ becomes the multiplication operator $ik,$ and this property will extend to distributions. The Fourier transform takes constant coeficient diferential equations in $q$ to polynomial equations in $k,$ which can often much more readily be solved, including the possibility of solutions that are distributions. The free particle Schr¨odinger equation

$$i\frac{\partial }{\partial t}\psi (q,t)=-\frac{1}{2m}\frac{{\partial }^2}{\partial q^2}\psi (q,t)$$

becomes after Fourier transformation in the $q$ variable the simple ordinary differential equation

$$i\frac{d}{dt}\tilde{\psi }(k,t)=\frac{1}{2m}{k}^2\tilde{\psi }(k,t)$$

with solutions

$$\tilde{\psi }(k,t)=e^{-i\frac{1}{2m}{k}^{2}t}\tilde{\psi }(k,0)$$

Solutions that are momentum and energy eigenstates will be distributions, with initial value

$$\tilde{\psi }(k,0)=\delta (k-k^{\prime })$$

These will have momentum $k^{\prime }$ and energy $\begin{array}{r}E=\frac{{k^{\prime }}^2}{2m}\end{array}$ . The space of solutions can be identified with the space of initial value data $\psi (k,0)$ , which can be taken to be in $S(R)$, $L^2(\mathbf{R})$ or $S^{\prime }(\mathbf{R})$ ).

Instead of working with time-dependent momentum space solutions $\tilde{\psi }(\mathbf{k},t)$ one can Fourier transform in the time variable, defining

$$\hat{\psi }(k,\omega )=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{i\omega t}\tilde{\psi }(k,t)dt$$

Just as the Fourier transform in $q$ takes $\frac{d}{dq}$ to multiplication by $ik,$ , here the Fourier transform in $t$ takes $\frac{d}{dt}$ to multiplication by $-i\omega$ . Note the opposite sign convention in the phase factor from the spatial Fourier transform, chosen to agree with the opposite sign conventions for spatial and time translations in the definitions of momentum and energy.

One finds for free particle solutions

$$\begin{array}{rl}\hat{\psi }(k,\omega )& =\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{i\omega t}{e}^{-i\frac{1}{2m}{k}^{2}t}\tilde{\psi }(k,0)dt\\ & =\delta (\omega -\frac{1}{2m}{k}^2)\sqrt{2\pi }\tilde{\psi }(k,0)\end{array}$$

so $\hat{\psi }(\mathbf{k},\omega )$ will be a distribution on $k-\omega$ space that is non-zero only on the parabola $\begin{array}{r}\omega =\frac{1}{2m}{k}^2\end{array}$ . The space of solutions can be identified with the space of functions (or distributions) supported on this parabola. Energy eigenstates of energy E will be distributions with a dependence on $\omega$ of the form

$${\hat{\psi }}_E(k,\omega )=\delta (\omega -E){\tilde{\psi }}_E(k)$$

For free particle solutions one has $\begin{array}{r}E=\frac{k^2}{2m}.{\tilde{\psi }}_E(k)\end{array}$ will be a distribution in $k$ with a factor $\begin{array}{r}\delta (E-\frac{k^2}{2m})\end{array}$

For any function $f(k)$ , the delta function distribution $\delta (f(k))$ depends only on the behavior of $f$ near its zeros. If $f^{\prime }\ne 0$ at such zeros, one has (using linear approximation near zeros of f)

$$\delta (f(k))=\sum _{k_j:f(k_j)=0}\delta (f^{\prime }(k_j)(k-k_j))=\sum _{k_j:f(k_j)=0}\frac{1}{|f^{\prime }(k_j)|}\delta (k-k_j)\text{\hspace{1em}(11.9)}$$

Applying this to the case of $\begin{array}{r}f(k)=E-\frac{k^2}{2m}\end{array}$ , with a graph that has two zeros, at $k=\pm \sqrt{2mE}$ and looks like

Figure 11.1: Linear approximations near zeros of $f(k)=E-\frac{k^2}{2m}$

we find that

$$\delta (E-\frac{k^2}{2m})=\sqrt{\frac{m}{2E}}(\delta (k-\sqrt{2mE})+\delta (k+\sqrt{2mE}))$$

and

$${\tilde{\psi }}_E(k)=c_{+}\delta (k-\sqrt{2mE})+c_{-}\delta (k+\sqrt{2mE})\text{\hspace{1em}(11.10)}$$

The two complex numbers $c_{+},c_{-}$ give the amplitudes for a free particle solution of energy E to have momentum $\pm \sqrt{2mE}$

In the physical case of three spatial dimensions, one gets solutions

$$\tilde{\psi }(\mathbf{k},t)=e^{-i\frac{1}{2m}|\mathbf{k}{|}^{2}t}\tilde{\psi }(\mathbf{k},0)$$

and the space of solutions is a space of functions (or distributions) $\tilde{\psi }(\mathbf{k},0)$ on ${\mathbf{R}}^3$ . Energy eigenstates with energy E will be given by distributions that are non-zero only on the sphere $|\mathbf{k}{|}^2/2m=E$ of radius $\sqrt{2mE}$ in momentum space (these will be studied in detail in chapter 19).

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