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.

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}}},}

k

2

=

2

m

(

E

−

V

0

)

/

ℏ

2

{\displaystyle 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

=

ℏ

k

{\displaystyle p=\hbar k}

.

Boundary conditions

The coefficients A, B have to be found from the boundary conditions of the wave function at x = 0.