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.
Calculation
Schrödinger equation and potential function
The time-independent Schrödinger equation for the wave function ψ ( x ) {\displaystyle \psi (x)} is
H ^ ψ ( x ) = [ − ℏ 2 2 m d 2 d x 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) , {\displaystyle {\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 V0, the height of the barrier, and the Heaviside step function:
V ( x ) = { 0 , x < 0 V 0 , x ≥ 0 {\displaystyle V(x)={\begin{cases}0,&x<0\\V_{0},&x\geq 0\end{cases}}}
The barrier is positioned at x = 0, though any position x0 may be chosen without changing the results, simply by shifting position of the step by −x0. The first term in the Hamiltonian, − ℏ 2 2 m d 2 d x 2 ψ {\textstyle -{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}\psi } is the kinetic energy of the particle.
Solution The step divides space in two parts: x < 0 and x > 0. In any of these parts the potential is constant, meaning the particle is quasi-free, and the solution of the Schrödinger equation can be written as a superposition of left and right moving waves (see free particle)
ψ 1 ( x ) = ( A → e i k 1 x + A ← e − i k 1 x ) x < 0 , {\displaystyle \psi _{1}(x)=\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow }e^{-ik_{1}x}\right)\quad x<0,}
ψ 2 ( x ) = ( B → e i k 2 x + B ← e − i k 2 x ) x > 0 {\displaystyle \psi _{2}(x)=\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow }e^{-ik_{2}x}\right)\quad x>0}
where subscripts 1 and 2 denote the regions x < 0 and 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 = 2 m E / ℏ 2 , {\displaystyle k_{1}={\sqrt {2mE/\hbar ^{2}}},}
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