QM of a particle in a 1D Dirac delta potential

A particle in a Dirac delta potential $𝑉 (𝑥) = - 𝛼 𝛿 (𝑥)$ ¹ has exactly one bound state

𝜓 (𝑥) = \sqrt 𝑚 𝛼 ℏ 𝑒 - 𝑚 𝛼 | 𝑥 | / ℏ 2 𝐸 = - 𝑚 𝛼 2 2 ℏ 2

and scattering states²

𝜓 (𝑥) = {𝐴 𝑒 𝑖 𝑘 𝑥 + 𝐵 𝑒 - 𝑖 𝑘 𝑥 𝑥 \leq 0 𝐹 𝑒 𝑖 𝑘 𝑥 + 𝐺 𝑒 - 𝑖 𝑘 𝑥 𝑥 \geq 0

where $𝐹 - 𝐺 = 𝐴 (1 + 2 𝑖 𝛽) - 𝐵 (1 - 2 𝑖 𝛽)$ and $𝛽 = 𝑚 𝛼 ℏ 2 𝑘$

Proof

We begin with bound states (i.e. $𝐸 < 0$ ). Letting $𝜅 = \sqrt - 2 𝑚 𝐸 / ℏ$ the Schrödinger equation for $𝑥 \neq 0$ reads

𝑑 2 𝜓 𝑑 𝑥 2 = 𝜅 2 𝜓

which has the general solution $𝜓 (𝑥) = ˜ 𝐴 𝑒 𝜅 𝑥 + ˜ 𝐵 𝑒 - 𝜅 𝑥$ . Applying the boundary conditions $𝜓 (\pm \infty) = 0$ and continuity of $𝜓$ we conclude $𝜓 (𝑥) = 𝐵 𝑒 - 𝜅 | 𝑥 |$ . Integrating the complete Schrödinger equation over $(- 𝜖, 𝜖)$ gives

𝐸 \int 𝜖 - 𝜖 𝜓 (𝑥) 𝑑 𝑥 = - ℏ 2 2 𝑚 \int 𝜖 - 𝜖 𝑑 2 𝜓 𝑑 𝑥 2 𝑑 𝑥 - 𝛼 \int 𝜖 - 𝜖 𝛿 (𝑥) 𝜓 (𝑥) 𝑑 𝑥 = - ℏ 2 2 𝑚 [𝑑 2 𝜓 𝑑 𝑥 2] 𝑥 = 𝜖 𝑥 = - 𝜖 - 𝛼 𝜓 (0)

and taking the limit $𝜖 \to 0$ gives

- 2 𝑚 𝛼 ℏ 2 𝐵 = 𝑑 𝜓 𝑑 𝑥 (0 +) - 𝑑 𝜓 𝑑 𝑥 (0 -) = 2 𝜅 𝐵

hence $𝜅 = - 𝑚 𝛼 / ℏ 2$ and

𝐸 = - ℏ 2 𝜅 2 2 𝑚 = - 𝑚 𝛼 2 2 ℏ 2

Normalization yields

1 = | 𝐵 | 2 \int \infty - \infty 𝑒 - 2 𝜅 | 𝑥 | 𝑑 𝑥 = 2 | 𝐵 | 2 \int \infty 0 𝑒 - 2 𝜅 𝑥 𝑑 𝑥 = | 𝐵 | 2 𝜅

hence $𝐵 = \sqrt 𝜅 = \sqrt 𝑚 𝛼 / ℏ$ .

For scattering states ( $𝐸 \geq 0$ ), let $𝑘 = \sqrt 2 𝑚 𝐸 / ℏ$ . The Schrödinger equation for $𝑥 \neq 0$ thence becomes

𝑑 2 𝜓 𝑑 𝑥 2 = - 𝑘 2 𝜓

it follows

𝜓 (𝑥) = {˜ 𝐴 𝑒 𝑖 𝑘 𝑥 + ˜ 𝐵 𝑒 - 𝑖 𝑘 𝑥 𝑥 \leq 0 ˜ 𝐹 𝑒 𝑖 𝑘 𝑥 + ˜ 𝐺 𝑒 - 𝑖 𝑘 𝑥 𝑥 \geq 0

where continuity requires $𝐴 + 𝐵 = 𝐹 + 𝐺$ . The derivatives are

𝑑 𝜓 𝑑 𝑥 (0 -) = 𝑖 𝑘 - 𝐴 - 𝐵) 𝑑 𝜓 𝑑 𝑥 (0 -) = 𝑖 𝑘 (𝐹 - 𝐺)

hence

- 2 𝑚 𝛼 ℏ 2 (𝐴 - 𝐵) = 𝑖 𝑘 (𝐹 - 𝐺 - 𝐴 + 𝐵)

which may be reärranged to

𝐹 - 𝐺 = 𝐴 (1 + 2 𝑖 𝛽) - 𝐵 (1 - 2 𝑖 𝛽)

where $𝛽 = 𝑚 𝛼 / ℏ 2 𝑘$

Properties

The reflection and transmission coëfficients (regardless of which side the particle enters) for scattering states are

𝑅 = 1 1 + (2 ℏ 2 𝐸 / 𝑚 𝛼 2) 𝑇 = 1 1 + (2 𝑚 𝛼 2 / 2 ℏ 2 𝐸)

which do not depend on the sign of $𝛼$ . 2. The bound state has the following expectation values - $⟨ ˆ 𝑥 ⟩ = 0$ - $⟨ ˆ 𝑝 ⟩ = 0$ - $⟨ ˆ 𝑥 2 ⟩ = ℏ 2 2 𝑚 2 𝛼 2$ - $⟨ ˆ 𝑝 2 ⟩ = (𝑚 𝛼 ℏ) 2$

Proof of 1–2

Since the velocities of the particle are equal on either side of the potential, it is sufficient to compare the coëfficients of the unnormalized scattering state. Since the setup is symmetric, without loss of generality let the particle scatter from the left, so $𝐺 = 0$ . Thus $𝐴$ corresponds to the incident wave amplitude, $𝐵$ to the reflected wave, and $𝐹$ to the transmitted wave.

𝐵 = 𝑖 𝛽 1 - 𝑖 𝛽 𝐴 𝐹 = 1 1 - 𝑖 𝛽 𝐴

thus

𝑅 = | 𝐵 | 2 | 𝐴 | 2 = 𝛽 2 1 + 𝛽 2 = 1 1 + (2 ℏ 2 𝐸 / 𝑚 𝛼 2) 𝑇 = | 𝐹 | 2 | 𝐴 | 2 = 1 1 + 𝛽 2 = 1 1 + (2 𝑚 𝛼 2 / 2 ℏ 2 𝐸)

as claimed by ^P1. Note that these are unchanged for negative $𝛼$ .

First we compute the necessary derivatives

𝑑 𝑑 𝑥 𝜓 (𝑥) = \sqrt 𝑚 𝛼 ℏ {𝑚 𝛼 ℏ 2 𝑒 𝑚 𝛼 𝑥 / ℏ 2 𝑥 \leq 0 - 𝑚 𝛼 ℏ 2 𝑒 - 𝑚 𝛼 𝑥 / ℏ 2 𝑥 \geq 0 = (𝑚 𝛼 ℏ 2) 3 / 2 (Θ (- 𝑥) 𝑒 𝑚 𝛼 𝑥 / ℏ 2 - Θ (𝑥) 𝑒 - 𝑚 𝛼 𝑥 / ℏ 2) 𝑑 2 𝑑 𝑥 2 𝜓 (𝑥) = (𝑚 𝛼 ℏ 2) 3 / 2 (- 𝛿 (𝑥) (𝑒 𝑚 𝛼 𝑥 / ℏ + 𝑒 - 𝑚 𝛼 𝑥 / ℏ) + 𝑚 𝛼 ℏ 2 Θ (- 𝑥) (𝑒 𝑚 𝛼 𝑥 / ℏ 2 + Θ (𝑥) 𝑒 - 𝑚 𝛼 𝑥 / ℏ 2)) = (𝑚 𝛼 ℏ 2) 3 / 2 (- 2 𝛿 (𝑥) + 𝑚 𝛼 ℏ 2 𝑒 - 𝑚 𝛼 | 𝑥 | / ℏ 2)

where $Θ$ is the Heaviside function. Thus

⟨ 𝜓 | ˆ 𝑥 | 𝜓 ⟩ = 𝑚 𝛼 ℏ 2 \int \infty - \infty 𝑥 𝑒 - 2 𝑚 𝛼 | 𝑥 | / ℏ 2 ⏟__⏟__⏟ o d d 𝑑 𝑥 = 0 ⟨ 𝜓 | ˆ 𝑥 2 | 𝜓 ⟩ = 𝑚 𝛼 ℏ 2 \int \infty - \infty 𝑥 2 𝑒 - 2 𝑚 𝛼 | 𝑥 | / ℏ 2 ⏟__⏟__⏟ e v e n 𝑑 𝑥 = 2 𝑚 𝛼 ℏ 2 \int \infty 0 𝑥 2 𝑒 - 2 𝑚 𝛼 𝑥 / ℏ 2 𝑑 𝑥 = 2 𝑚 𝛼 ℏ 2 (2!) (ℏ 2 2 𝑚 𝛼) 3 = ℏ 4 2 𝑚 2 𝛼 2 ⟨ 𝜓 | ˆ 𝑝 | 𝜓 ⟩ = - 𝑖 ℏ \int \infty - \infty 𝜓 (𝑥) 𝑑 𝑑 𝑥 𝜓 (𝑥) 𝑑 𝑥 = - 𝑖 ℏ 𝑚 𝛼 ℏ 2 \int \infty - \infty (Θ (- 𝑥) - Θ (𝑥)) 𝜓 (𝑥) 2 ⏟____⏟____⏟ o d d 𝑑 𝑥 = 0 ⟨ 𝜓 | ˆ 𝑝 2 | 𝜓 ⟩ = - ℏ 2 \int \infty - \infty 𝜓 (𝑥) 𝑑 2 𝑑 𝑥 2 𝜓 (𝑥) 𝑑 𝑥 = - (𝑚 𝛼 ℏ) 2 \int \infty - \infty 𝑒 - 𝑚 𝛼 | 𝑥 | / ℏ 2 (- 2 𝛿 (𝑥) + 𝑚 𝛼 ℏ 2 𝑒 - 𝑚 𝛼 | 𝑥 | / ℏ 2) 𝑑 𝑥 = (𝑚 𝛼 ℏ) 2 [2 - 𝑚 𝛼 ℏ 2 2 \int \infty - \infty 𝑒 - 2 𝑚 𝛼 | 𝑥 | / ℏ 2 𝑑 𝑥] = (𝑚 𝛼 ℏ) 2 [2 - 𝑚 𝛼 ℏ 2 ℏ 2 2 𝑚 𝛼] = (𝑚 𝛼 ℏ) 2 [2 - 𝑚 𝛼 ℏ 2 ℏ 2 𝑚 𝛼] = (𝑚 𝛼 ℏ) 2

proving ^P2.

#state/tidy | #lang/en | #SemBr

2018. Introduction to quantum mechanics, §2.5.2, pp. 63ff. ↩
yet to be dirac normalized ↩

QM of a particle in a 1D Dirac delta potential

Properties

Footnotes