A particle of mass moves in the delta-function well with

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Wave mechanics in one dimension

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Stationary states and bound states in one dimension

Problem

A particle of mass mm moves in the delta-function well V(x)=−αδ(x)V(x) = -\alpha\delta(x) with α>0\alpha > 0, where δ(x)\delta(x) is the unit-area spike at the origin. The potential is zero everywhere else, so a bound state has E<0E < 0. Parts ((a)) to (d) concern an electron with α=0.500\alpha = 0.500 eV nm; take ℏ2/2me=0.0381\hbar^2/2m_e = 0.0381 eV nm2^2. (a) By integrating the time-independent Schrödinger equation from x=−ϵx = -\epsilon to x=+ϵx = +\epsilon and letting ϵ→0\epsilon \to 0, derive the jump condition ψ′(0+)−ψ′(0−)=−(2mα/ℏ2)ψ(0)\psi'(0^+) - \psi'(0^-) = -(2m\alpha/\hbar^2)\psi(0). ((b)) Away from the origin the bound state must be ψ=Be−κ∣x∣\psi = Be^{-\kappa|x|} with κ=−2mE/ℏ\kappa = \sqrt{-2mE}/\hbar. Use the jump condition to find κ\kappa, hence EE, and normalise ψ\psi (take BB real and positive). ((c)) Evaluate EE in eV for the electron. ((d)) Find the probability that the electron is found within one decay length of the origin, that is in ∣x∣<1/κ|x| < 1/\kappa. ((e)) Compare with a finite square well of the same strength: V=−V0V = -V_0 for ∣x∣<a|x| < a with a=0.10a = 0.10 nm and V0=α/2a=2.5V_0 = \alpha/2a = 2.5 eV. Compute z0=(a/ℏ)2meV0z_0 = (a/\hbar)\sqrt{2m_eV_0} and state how many bound states this finite well has, given that even states satisfy tan⁡z=(z0/z)2−1\tan z = \sqrt{(z_0/z)^2 - 1} and odd states −cot⁡z=(z0/z)2−1-\cot z = \sqrt{(z_0/z)^2 - 1}, with z=kaz = ka.

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