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| -rw-r--r-- | latex/know/concept/wentzel-kramers-brillouin-approximation/source.md | 16 |
1 files changed, 8 insertions, 8 deletions
diff --git a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md index f862004..a50302c 100644 --- a/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md +++ b/latex/know/concept/wentzel-kramers-brillouin-approximation/source.md @@ -5,8 +5,8 @@ In quantum mechanics, the *Wentzel-Kramers-Brillouin* or simply the *WKB approximation* is a method to approximate the wave function $\psi(x)$ of -the one-dimensional time-independent Schrödinger equation. It is also -known as the *semiclassical approximation*, because it tries to find a +the one-dimensional time-independent Schrödinger equation. It is an example +of a *semiclassical approximation*, because it tries to find a balance between classical and quantum physics. In classical mechanics, a particle travelling in a potential $V(x)$ @@ -49,7 +49,7 @@ This form is reminiscent of the generator of translations. In practice, $V(x)$ and $p(x)$ vary with $x$, but we can still salvage this solution by assuming that $V(x)$ varies slowly compared to the wavelength $\lambda(x) = 2 \pi / k(x)$, where $k(x) = p(x) / \hbar$ is the -wavenumber. The solution is then of the form: +wavenumber. The solution then takes the following form: $$\begin{aligned} \psi(x) @@ -95,7 +95,7 @@ $$\begin{aligned} \end{aligned}$$ Next, we expand this as a power series of $\hbar$. This is why it is -called *semiclassical*: so far we have been using full quatum mechanics, +called *semiclassical*: so far we have been using full quantum mechanics, but now we are treating $\hbar$ as a parameter which controls the strength of quantum effects: @@ -103,9 +103,9 @@ $$\begin{aligned} \chi(x) = \chi_0(x) + \frac{\hbar}{i} \chi_1(x) + \frac{\hbar^2}{i^2} \chi_2(x) + ... \end{aligned}$$ -The heart of the WKB approximation is to assume that quantum effects are +The heart of the WKB approximation is its assumption that quantum effects are sufficiently weak (i.e. $\hbar$ is small enough) that we only need to -consider the first two terms, or, more generally, that we only go up to +consider the first two terms, or, more specifically, that we only go up to $\hbar$, not $\hbar^2$ or higher. Inserting the first two terms of this expansion into the equation: @@ -182,7 +182,7 @@ $$\begin{aligned} } \end{aligned}$$ -In the classical region ($E > V$), the wave function oscillates, whereas -in the quantum region ($E < V$) it is exponential. Note that for +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. |