WKB Approximation

Also called the Semiclassical Approximation for Quantum Mechanics.

$S (\vec{x}, t) = \pm \int^{x} d x^{'} \sqrt{2 m (E - V (x^{'}))} - E t = W (x) - E t$ .

Continuity of One-Dimension

$\frac{\partial}{\partial t} P + \frac{\partial}{\partial x} j = 0$ .

$j = \frac{P}{m} \frac{\partial S}{\partial x}$ .

Stationary Case

$\frac{\partial P}{\partial t} = 0$ .

So, $\frac{\partial j}{\partial x} = 0$ . Hence, $\frac{\partial}{\partial x} [P \frac{\partial S}{\partial x}] = 0$ . So, $ρ \frac{d W}{d x}$ is a constant, $C$ . Then, $\sqrt{P} = \frac{C}{\sqrt[4]{E - V (x)}}$ , so $P = \frac{C^{2}}{\sqrt{E - V (x)}}$ .

Thus, we are able to get a wavefunction without any solving: $Ψ = \frac{C}{\sqrt[4]{E - V (x)}} \exp (\pm \frac{i}{ℏ} \int^{x} d x^{'} \sqrt{2 m (E - V (x^{'}))} - \frac{i}{ℏ} E t)$ . Wentzel-Kramers-Brillouin solution. Thus, $P \sim \frac{1}{v_{c l a s s i c a l}}$ .

WKB Criteria

$ℏ | \frac{d^{2}}{d x^{2}} W | ≪ {| \frac{d W}{d x} |}^{2}$ . So, $ℏ \frac{2 m | \frac{d V}{d x} |}{2 \sqrt{2 m (E - V (x))}} ≪ | 2 m (E - V (x)) |$ . Hence, $\frac{ℏ}{\sqrt{2 m (E - V (x))}} = \frac{1}{k} = \frac{λ_{B}}{2 π} ≪ \frac{2 (E - V (x))}{| \frac{d V}{d x} |}$ , where $k$ is the wavenumber, related to the momentum by $ℏ$ . This gives us a deBrolie wavelength to compare.

So, WKB is valid in the low wavelength cases.

Problem Areas

Turning points pose problem areas for WKB, but they can be solved as linear wavefunction pieces to splice the region solutions together.

Small Example

Finite Well but with rounded edges, energy such that it is a bound state.

For regions I, III: $Ψ = \frac{C}{\sqrt[4]{2 m (V (x) - E)}} \exp (\pm \frac{1}{ℏ} \int^{x} \sqrt{2 m (V (x^{'}) - E)}) d x^{'}$

For regions II: $Ψ = \frac{C}{\sqrt[4]{2 m (E - V (x))}} \exp (\pm \frac{i}{ℏ} \int^{x} \sqrt{2 m (E - V (x^{'}))}) d x^{'}$

For turning point solutions to splice together the regions: $Ψ = A x + B$ $- \frac{ℏ^{2}}{2 m} Ψ^{″} = (E - V (x)) Ψ (x) \Rightarrow$ Linear solutions.

Alternatively, $- \frac{ℏ^{2}}{2 m} Ψ^{″} + \frac{d V}{d x} |_{x_{1, 2}} (x - x_{1, 2}) Ψ = 0$ . Gives a Bessel equation - Airy function.

Energies

Rigid $\Rightarrow$ $V \to \infty$ .

$\int_{x_{1}}^{x_{2}} d x \sqrt{2 m (E_{n} - V (x))} = π ℏ g_{n}$ .

Case 1 - No Rigid Walls

$g_{n} = n + 1 / 2$

Case 2 - 1 Rigid Wall

$g_{n} = n + 3 / 4$

Case 3 - 2 Rigid Walls

$g_{n} = n + 1$

Notes

If $n$ is large, each $g_{n}$ is approximately equal.

Also, as $n$ increases, we get closer to classical typically.

Examples

Energies of QMHO

$E = \frac{m ω^{2} x_{1}^{2}}{2}$ . $\int_{- x_{1}}^{x_{1}} d x \sqrt{2 m (\frac{m ω^{2} x_{1}^{2}}{2} - \frac{m ω^{2} x^{2}}{2})} = \frac{π E}{ω} = π ℏ (n + 1 / 2)$ . $x_{1} = \pm \sqrt{\frac{2 E}{m ω^{2}}}$ . WKB: $E = ℏ ω (n + 1 / 2)$

Bouncing Neutrons

Hard floor. So, the potential is infinite in the negative region and a linear potential at $x > 0$ .

Tunneling

Use WKB to find the wavefunction for a more complicated potential barrier, use typical solutions (if possible) for outside propogators.

$T \approx \exp (- 2 γ), γ = \frac{1}{ℏ} \int_{x_{1}}^{x_{2}} \sqrt{2 m (V (x) - E)} d x$