Step potential

In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modeled as a Heaviside step function.

In quantum mechanics and scattering theory, the one-dimensional step potential is an idealized system used to model incident, reflected and transmitted matter waves. The problem consists of solving the time-independent Schrödinger equation for a particle with a step-like potential in one dimension. Typically, the potential is modeled as a Heaviside step function. \hat H\psi(x) = \left[-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x)\right]\psi(x) = E\psi(x), where Ĥ is the Hamiltonian, ħ is the reduced Planck constant, m is the mass, E the energy of the particle. The step potential is simply the product of V 0 , the height of the barrier, and the Heaviside step function: V(x) = \begin{cases} 0, & x The barrier is positioned at x = 0, though any position x 0 may be chosen without changing the results, simply by shifting position of the step by −x 0 . The first term in the Hamiltonian, -\frac{\hbar^2}{2m} \frac{d^2}{dx^2}\psi is the kinetic energy of the particle. \psi_1(x)= \left(A_\rightarrow e^{i k_1 x} + A_\leftarrow e^{-ik_1x}\right)\quad x \psi_2(x)= \left(B_\rightarrow e^{i k_2 x} + B_\leftarrow e^{-ik_2x}\right)\quad x>0 where subscripts 1 and 2 denote the regions x 0 respectively, the subscripts (→) and (←) on the amplitudes A and B denote the direction of the particle's velocity vector: right and left respectively. The wave vectors in the respective regions being k_1=\sqrt{2m E/\hbar^2}, k_2=\sqrt{2m (E-V_0)/\hbar^2} both of which have the same form as the De Broglie relation (in one dimension) p=\hbar k . The coefficients A, B have to be found from the boundary conditions of the wave function at x = 0. The wave function and its derivative have to be continuous everywhere, so: \psi_1(0)=\psi_2(0), \left.\frac{d\psi_1}{dx}\right|_{x=0} = \left.\frac{d\psi_2}{dx}\right|_{x=0}. Inserting the wave functions, the boundary conditions give the following restrictions on the coefficients (A_\rightarrow+A_\leftarrow)=(B_\rightarrow+B_\leftarrow) k_1(A_\rightarrow-A_\leftarrow)=k_2(B_\rightarrow-B_\leftarrow) To find the amplitudes for reflection and transmission for incidence from the left, we set in the above equations A → = 1 (incoming particle), A ← = (reflection), B ← = 0 (no incoming particle from the right) and B → = (transmission ). We then solve for T and R. The result is: \sqrt{T}=\frac{2\sqrt{k_1 k_2}}{k_1+k_2} \sqrt{R}=\frac{k_1-k_2}{k_1+k_2}. The model is symmetric with respect to a parity transformation and at the same time interchange k 1 and k 2 . For incidence from the right we have therefore the amplitudes for transmission and reflection \sqrt{T'}=\sqrt{T}=\frac{2\sqrt{k_1k_2}}{k_1+k_2} \sqrt{R'}=-\sqrt{R}=\frac{k_2-k_1}{k_1+k_2}. T=|T'|=\frac{4k_1 k_2}{(k_1+k_2)^2} R=|R'|=1-T=\frac{(k_1-k_2)^2}{(k_1+k_2)^2} In the limit of large energies E ≫ V 0 , we have k 1 ≈ k 2 and the classical result T = 1, R = 0 is recovered. Thus there is a finite probability for a particle with an energy larger than the step height to be reflected.

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