QM in 1D position-space

QM of a particle in a 1D Dirac delta potential

A particle in a Dirac delta potential ¹ has exactly one bound state

and scattering states^[yet to be dirac normalized]

where and

Proof

We begin with bound states (i.e. ). Letting the Schrödinger equation for reads

which has the general solution . Applying the boundary conditions and continuity of we conclude . Integrating the complete Schrödinger equation over gives

and taking the limit gives

hence and

Normalization yields

hence .

For scattering states (), let . The Schrödinger equation for thence becomes

it follows

where continuity requires . The derivatives are

hence

which may be reärranged to

where

Properties

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

which do not depend on the sign of . 2. The bound state has the following expectation values ^P2 - - - -

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 . Thus corresponds to the incident wave amplitude, to the reflected wave, and to the transmitted wave.

thus

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

First we compute the necessary derivatives

where is the Heaviside function. Thus

proving ^P2.

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tidy | en | sembr

Footnotes

1. 2018. Introduction to quantum mechanics, §2.5.2, pp. 63ff. ↩