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Diffstat (limited to 'content/know/concept/wkb-approximation')
| -rw-r--r-- | content/know/concept/wkb-approximation/index.pdc | 27 |
1 files changed, 13 insertions, 14 deletions
diff --git a/content/know/concept/wkb-approximation/index.pdc b/content/know/concept/wkb-approximation/index.pdc index 985bcec..b8ace29 100644 --- a/content/know/concept/wkb-approximation/index.pdc +++ b/content/know/concept/wkb-approximation/index.pdc @@ -29,17 +29,17 @@ $$\begin{aligned} m^2 (x')^2 = 2 m (E - V(x)) \end{aligned}$$ -The left-hand side of the rearrangement is simply the momentum squared, +The left-hand side of the rearranged version is simply the momentum squared, so we define the magnitude of the momentum $p(x)$ accordingly: $$\begin{aligned} p(x) = \sqrt{2 m (E - V(x))} \end{aligned}$$ -Note that this is under the assumption that $E > V$, which is always the -case in classical mechanics, but not necessarily so in quantum -mechanics, but we stick with it for now. We rewrite the Schrödinger -equation: +Note that this is under the assumption that $E > V$, +which is always true in classical mechanics, +but not necessarily in quantum mechanics. +We rewrite the Schrödinger equation: $$\begin{aligned} 0 @@ -172,19 +172,18 @@ $$\begin{aligned} } \end{aligned}$$ -What if $E < V$? In classical mechanics, this is not allowed; a ball +What if $E < V$? In classical mechanics, this is just not allowed; a ball cannot simply go through a potential bump without the necessary energy. -However, in quantum mechanics, particles can **tunnel** through barriers. +On the other hand, in quantum physics, particles can **tunnel** through barriers. -Conveniently, all we need to change for the WKB approximation is to let -the momentum take imaginary values: +Luckily, the only thing we need to change for the WKB approximation +is to let the momentum take imaginary values: $$\begin{aligned} p(x) = \sqrt{2 m (E - V(x))} = i \sqrt{2 m (V(x) - E)} \end{aligned}$$ -And then take the absolute value in the appropriate place in front of -$\psi(x)$: +And then take the absolute value in the appropriate place in front of $\psi(x)$: $$\begin{aligned} \boxed{ @@ -193,9 +192,9 @@ $$\begin{aligned} \end{aligned}$$ In the classical region ($E > V$), the wave function oscillates, and -in the quantum-mechanical region ($E < V$) it is exponential. Note that for -$E \approx V$ the approximation breaks down, due to the appearance of -$p(x)$ in the denominator. +in the quantum-physical region ($E < V$) it is exponential. +Note that for $E \approx V$ the approximation breaks down, +because of the appearance of $p(x)$ in the denominator. ## References |