Delta potential well (QM)

From Wikipedia, the free encyclopedia

It has been suggested that this article or section be merged into Delta function potential. (Discuss)
Please help improve this article or section by expanding it.
Further information might be found on the talk page or at requests for expansion.
This article has been tagged since January 2007.
This article or section needs copy editing for grammar, style, cohesion, tone and/or spelling.
You can assist by editing it now. A how-to guide is available, as is general documentation.
This article has been tagged since January 2007.

The Delta potential well is a common theoretical problem of quantum mechanics. It consists of a time-independent Schrödinger equation for a particle in a potential well defined by a delta function in one dimension.

Contents

[edit] Calculation

The time-independent Schrödinger equation for the wave function ψ(x) is

H\psi(x)=\left[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)\right]\psi(x)=E\psi(x),

where H is the Hamiltonian, \hbar is the (reduced) Planck constant, m is the mass, E the energy of the particle, and

V(x)=\frac{\lambda}{m}\delta(x)

is the delta function well with strength λ < 0. The potential is located at the origin. Without changing the results, any other shifted position was possible.

The potential well splits the space in two parts (x < 0,x > 0). In any of these parts the potential energy is constant, and the solution of the Schrödinger equation can be written as a superposition of exponentials:

\psi_L(x)= A_r e^{i k x} + A_l e^{-ikx}\quad x<0, and
\psi_R(x)= B_r e^{i k x} + B_l e^{-ikx}\quad x>0

where the wave vector is related to the energy via

k=\sqrt{2m E}/\hbar .

The index r/l on the coefficients A and B denotes the direction of the velocity vector (for E > 0). Even though the association with propagating waves only holds for positive energies (real wave vectors), we keep this notation also for E < 0. The coefficients A,B have to be found from the boundary conditions of the wave function at x = 0:

Scattering at a delta function well of strength λ. The amplitudes and direction of left and right moving waves are indicated. In red, those waves used for the derivation of the reflection and transmission amplitude. E>0

ψL = ψR,
\frac{d}{dx}\psi_L = \frac{d}{dx}\psi_R - \frac{2\lambda}{\hbar^2} \psi_R.

The second of these equations follows from integrating the Schrödinger equation with respect to x in the vicinity of x=0. The boundary conditions thus give the following restrictions on the coefficients

Ar + Al = Br + Bl
ik(A_r-A_l-B_r+B_l)=-\frac{2\lambda}{\hbar^2}(B_r+B_l).

[edit] Transmission and Reflection

[edit] E > 0

For positive energies, the particle is free to move in either half-space: x < 0,x > 0. It may be scattered at the delta-function well. The calculation is identical to the one in Delta potential barrier (QM) the only difference being that λ is now negative.

To study the quantum case, let us consider the following situation: a particle incident on the barrier from the left side (Ar). It may be reflected (Al) or transmitted (Br). To find the amplitudes for reflection and transmission for incidence from the left, we put in the above equations Ar = 1 (incoming particle), Al = r (reflection), Bl=0 (no incoming particle from the right) and Br = t (transmission) and solve for r,t. The result is:

t=\frac{1}{\frac{i\lambda}{\hbar^2k}+1}
r=\frac{1}{\frac{i\hbar^2 k}{\lambda}-1}.

Due to the mirror symmetry of the model, the amplitudes for incidence from the right are the same as those from the left. The surprising result is that there is a non-zero probability

R=|r|^2\frac{1}{1+\frac{\hbar^4k^2}{\lambda^2}}= \frac{1}{1+\frac{2m\hbar^2 E}{\lambda^2}}.

for the particle to be reflected from the barrier. This is a purely quantum effect which does not appear in the classical case.

For completeness, the probability for transmission is:

T=|t|^2=1-R=\frac{1}{1+\frac{\lambda^2}{\hbar^4k^2}}= \frac{1}{1+\frac{\lambda^2}{2m\hbar^2 E}}.
Transmission (T) and reflection (R) probability of a delta potential well. The energy E > 0 is in units of . Dashed: classical result. Solid line: quantum mechanics.
Transmission (T) and reflection (R) probability of a delta potential well. The energy E > 0 is in units of \frac{\lambda^2}{2m\hbar^2}. Dashed: classical result. Solid line: quantum mechanics.

[edit] Bound state E < 0

As with any one-dimensional attractive potential there will be a bound state. To find its energy, note that for E<0, k=i\sqrt{2m |E|}/\hbar is complex and the wave functions which were oscillating for positive energies in the calculation above, are now exponentially increasing or decreasing functions of x (see above). Requiring that the wave functions do not diverge at x\to\pm \infty eliminates half of the terms: Ar = Bl = 0. The wave function is then

\psi_L(x)= A_l e^{|k|x}\quad x<0, and
\psi_R(x)= B_r e^{- |k| x} \quad x>0.

From the first of the above boundary conditions, we have Al = Br and from the second we obtain a relation between k and the well strength λ

k=i\frac{\lambda}{\hbar^2}.

The energy of the bound state is then

E=\frac{\hbar^2k^2}{2m}=-\frac{\lambda^2}{2\hbar^2m}.

[edit] Remarks

The delta function potential well is a special case of the finite potential well and follows as a limit of infinite depth and zero width of the well, keeping the product of width and depth constant equal to λ2 / m2.

[edit] See also

Retrieved from "http://en.wikipedia.org../../../d/e/l/Delta_potential_well_%28QM%29_2ce6.html"