Motion in One Dimension

We study the one dimensional problems in quantum theory, not only because the interest of study the simplest cases to learn about the general properties. Actually there are many cases in two and three dimensions that can be reduced to one-dimensional problem like the cases in central potentials.

Let's consider the motion in 1 direction of a particle in the potential V(x). Supposing that V(x) has finite values when x goes to infinity:

$\lim _{x\to -\infty }V(x)=V_{-},\lim _{x\to +\infty }V(x)=V_{+}$

and assuming that: $V_{-}<V_{+}\!$

The Schrodinger equation becomes:

$\left[-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)\right]\psi (x)=E\psi (x)$

$\rightarrow {\frac {d^{2}}{dx^{2}}}\psi (x)+{\frac {2m}{\hbar ^{2}}}(E-V(x))\psi (x)=0$

From this equation we can discuss some general properties of 1-D motion as follows:

If $E>V_{+}\!$:

$E-V(x)>0\!$ at both $-\infty$ and $+\infty$. Therefore, the solutions of Schrodinger equation can be express as linear combinations of trigonometric functions (sine or cosine). The wave function is oscillating at both $-\infty$ and $+\infty$. The particle is in an unbound state. The energy spectrum is continous. Both oscillating solutions are allowed, and the energy levels are two-fold degenerate.

If $V_{-}\leq E\leq V_{+}$:

$E-V(x)>0\!$ at $-\infty$ but $E-V(x)<0$ at $+\infty$. Therefore, the wave function is oscillating at $-\infty$ but decaying exponentially at $+\infty$. The energy spectrum is still continous but no longer degenerate.

If $E<V_{-}\!$:

$E-V(x)\!<0$ at both $-\infty$ and $+\infty$. Therefore, the wave function decays exponentially at both $-\infty$ and $+\infty$. The particle is in a bound state. The energy spectrum is discrete and non-degenerate.

Example