Plane waves

The simplest case is the one where the potential $V(x)$ is zero throughout: the two independent solutions of the Schrödinger equation then are plane waves

$$-\frac{\hbar^2\partial_x^2}{2m}\psi(x)= E\psi(x)\leadsto \psi_+(x)=e^{i kx}, \quad \psi_-(x)= e^{-i kx}, \quad k= \sqrt{(2m/\hbar^2)E}$$ (114)

with positive energy $E>0$. We denote both solutions as $\psi_k(x)=e^{ikx}$ with $k$ either positive or negative. They are plane waves with fixed wave vector $k$ and therefore fixed momentum $p=\hbar k$. We can have no physically meaningful solutions with negative energy $E<0$ because in this case the wave function would become infinite either for $x\to \infty$ or $x\to -\infty$.

A problem arises, however, because $\psi_k(x)$ can not be normalized over the whole $x$-axis according to

$$\int_{-\infty}^{\infty}dx \vert\psi_k(x)\vert^2 =1,$$ (115)

because this integral is infinite: the probability density, i.e. the square $\vert\psi_k(x)\vert^2$ is constant, i.e. $1$ everywhere.

In particular, we have for the mean square deviation of the momentum and the position

$$\langle \Delta p ^2 \rangle = \lim_{L \to \infty} \left( \frac{[\int_{-L/2}^{L/2}\vert a\vert^2... ...hbar\partial_x)e^{ikx}} {\int_{-L/2}^{L/2}dx \vert a\vert^2} \right]^2 \right )$$

$$= \hbar^2k^2-\hbar^2k^2=0$$

$$\langle \Delta x ^2 \rangle = \lim_{L \to \infty} \left( \frac{[\int_{-L/2}^{L/2}\vert a\vert^2... ...vert^2e^{-ikx}xe^{ikx}} {\int_{-L/2}^{L/2}dx \vert a\vert^2} \right]^2 \right )$$

$$= \infty.$$ (116)

There is no uncertainty in the momentum of the particle, but there is maximum uncertainty in its position: the wave function $\psi_k(x)$ describes a particle with fixed momentum $p=\hbar k$ which is completely delocalized (spread) over the $x$-axis.

A practical solution is to consider a large, but finite interval $[-L/2,L/2]$ instead of the total $x$-axis, and to normalize the wave functions according to

$$\psi_k=\frac{1}{\sqrt{L}}e^{ikx},\quad \int_{-L/2}^{L/2}dx \vert\psi_k(x)\vert^2 =1.$$ (117)

Then, the boundary conditions at $x=\pm L/2$ have to be specified. Again, a convenient (but not necessary the only) choice are periodic boundary conditions: we bend the interval $[L/2,L/2]$ into a ring such that the points $\pm L/2$ coincide. Demanding continuity of $\psi_k$, i.e. $\psi_k(L/2)= \psi_k(-L/2)$, we obtain a quantization condition for the possible $k$-values,

$$1=e^{ikL} \leadsto k_n = \frac{2\pi n}{L}, \quad n=0, \pm 1, \pm 2, ...,$$ (118)

i.e. a discrete set of possible $k$ values and therewith discrete energies $E(k)= \hbar^2k^2/2m$. To each energy $E=E_n>0$ there are two linearly independent plane waves with wave vectors $2\pi n/L$ and $-2\pi n/L$. One says the energy value $E_n$ is two-fold degenerate. If $L$ is very large, the possible values for $k$ are still discrete but subsequent $k$-values get very close to each other.