# Scattering matrix

Required math: calculus

Required physics: Schrödinger equation

References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.52.

In a system with a localized potential (that is, a potential that is non-zero only for some finite range, such as the delta function or finite square well) we can always analyze the scattering problem, in which particles come in from either right or left (or both) and get transmitted or reflected. In general, the wave function in the left hand region where ${V=0}$ is

$\displaystyle \psi_{l}=Ae^{ikx}+Be^{-ikx} \ \ \ \ \ (1)$

where

$\displaystyle k\equiv\frac{\sqrt{2mE}}{\hbar} \ \ \ \ \ (2)$

On the right,

$\displaystyle \psi_{r}=Fe^{ikx}+Ge^{-ikx} \ \ \ \ \ (3)$

In the middle, where ${V(x)\ne0}$, we can’t say what the wave function will be until ${V(x)}$ is specified. In the cases we’ve analyzed so far, the only incident particles have been from the left, so we’ve always taken ${G=0}$. However, it’s not too difficult to generalize the results we’ve obtained for the delta function and finite square well to the case where we have incident particles from both directions.

Since a particle coming in from the left will be either transmitted (continue to the right past the potential region) or reflected (travel back to the left), particles incident from the left cannot affect the particle stream travelling to the left on the right side of the potential region. By symmetry, particles incident from the right cannot affect the particle stream travelling to the right on the left side of the potential. That is, we can always specify ${A}$ and ${G}$ in the wave functions above, and express ${B}$ and ${F}$ in terms of them. We can write this dependence as a matrix equation

$\displaystyle \left[\begin{array}{c} B\\ F \end{array}\right]=\left[\begin{array}{cc} S_{11} & S_{12}\\ S_{21} & S_{22} \end{array}\right]\left[\begin{array}{c} A\\ G \end{array}\right] \ \ \ \ \ (4)$

where the matrix ${S}$ is called the scattering matrix.

For the delta function, with potential ${V(x)=-\alpha\delta(x)}$ we’ve seen that in the case where ${G=0}$,

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{i\beta}{1-i\beta}A\ \ \ \ \ (5)$ $\displaystyle F$ $\displaystyle =$ $\displaystyle \frac{1}{1-i\beta}A\ \ \ \ \ (6)$ $\displaystyle \beta$ $\displaystyle \equiv$ $\displaystyle \frac{m\alpha}{\hbar^{2}k} \ \ \ \ \ (7)$

By symmetry, if ${A=0}$ so that particles come in only from the right,

 $\displaystyle F$ $\displaystyle =$ $\displaystyle \frac{i\beta}{1-i\beta}G\ \ \ \ \ (8)$ $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{1}{1-i\beta}G \ \ \ \ \ (9)$

If both ${A\ne0}$ and ${G\ne0}$, we can just add up the contributions from the two cases, since they don’t interfere with each other, and we get

$\displaystyle \left[\begin{array}{c} B\\ F \end{array}\right]=\frac{1}{1-i\beta}\left[\begin{array}{cc} i\beta & 1\\ 1 & i\beta \end{array}\right]\left[\begin{array}{c} A\\ G \end{array}\right] \ \ \ \ \ (10)$

For the finite square well of depth ${V_{0}}$, with ${G=0}$ we had

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{e^{-2ika}\left(k^{2}-\mu^{2}\right)\sin\left(2\mu a\right)}{\sin\left(2\mu a\right)\left(k^{2}+\mu^{2}\right)+2i\mu k\cos\left(2\mu a\right)}A\ \ \ \ \ (11)$ $\displaystyle F$ $\displaystyle =$ $\displaystyle \frac{2i\mu ke^{-2ika}}{\sin\left(2\mu a\right)\left(k^{2}+\mu^{2}\right)+2i\mu k\cos\left(2\mu a\right)}A \ \ \ \ \ (12)$

where

$\displaystyle \mu=\frac{\sqrt{2m\left(E+V_{0}\right)}}{\hbar} \ \ \ \ \ (13)$

Here, the general scattering matrix is, by symmetry

$\displaystyle \left[\begin{array}{c} B\\ F \end{array}\right]=\frac{e^{-2ika}}{\sin\left(2\mu a\right)\left(k^{2}+\mu^{2}\right)+2i\mu k\cos\left(2\mu a\right)}\left[\begin{array}{cc} \left(k^{2}-\mu^{2}\right)\sin\left(2\mu a\right) & 2i\mu k\\ 2i\mu k & \left(k^{2}-\mu^{2}\right)\sin\left(2\mu a\right) \end{array}\right]\left[\begin{array}{c} A\\ G \end{array}\right] \ \ \ \ \ (14)$

# Reflectionless potential

Required math: calculus

Required physics: Schrödinger equation

References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.51.

An interesting potential is

$\displaystyle V(x)=-\frac{\hbar^{2}a^{2}}{m}\text{sech}^{2}(ax) \ \ \ \ \ (1)$

where ${a}$ is a constant and ${\mathrm{sech}(ax)}$ is the hyperbolic secant, which is defined as ${\mathrm{sech}(ax)\equiv1/\cosh(ax)}$. The general shape of this potential is as shown in the figure.

We can verify by direct substitution that the function

$\displaystyle \psi_{0}(x)=A\text{sech}(ax) \ \ \ \ \ (2)$

is a solution. We get

 $\displaystyle -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi_{0}}{dx^{2}}+V\psi_{0}$ $\displaystyle =$ $\displaystyle -\frac{\hbar^{2}}{2m}Aa^{2}\mathrm{sech}\left(ax\right)\ \ \ \ \ (3)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -\frac{\hbar^{2}a^{2}}{2m}\psi_{0} \ \ \ \ \ (4)$

Thus the energy of this state is

$\displaystyle E_{0}=-\frac{\hbar^{2}a^{2}}{2m} \ \ \ \ \ (5)$

We can normalize ${\psi_{0}}$ to find ${A}$:

 $\displaystyle \int_{-\infty}^{\infty}\psi_{0}^{2}dx$ $\displaystyle =$ $\displaystyle 1\ \ \ \ \ (6)$ $\displaystyle A$ $\displaystyle =$ $\displaystyle \sqrt{a/2} \ \ \ \ \ (7)$

A plot of ${\psi_{0}(x)}$ looks like this:

For positive energies, we can verify that

$\displaystyle \psi_{k}(x)=B\left(\frac{ik-a\tanh(ax)}{ik+a}\right)e^{ikx} \ \ \ \ \ (8)$

is a solution of the Schrodinger equation for any energy by direct substitution. Here, as usual, ${k\equiv\sqrt{2mE}/\hbar}$.

We get

 $\displaystyle -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi_{k}}{dx^{2}}+V\psi_{k}$ $\displaystyle =$ $\displaystyle \frac{\hbar^{2}k^{2}B}{2m}\left(\frac{ik-a\tanh\left(ax\right)}{ik+a}\right)\ \ \ \ \ (9)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\hbar^{2}k^{2}}{2m}\psi_{k}\ \ \ \ \ (10)$ $\displaystyle$ $\displaystyle =$ $\displaystyle E\psi_{k} \ \ \ \ \ (11)$

The asymptotic behaviour of ${\psi_{k}}$ can be found from the limit ${\lim_{x\rightarrow\infty}\tanh(ax)=1}$. We therefore get:

 $\displaystyle \lim_{x\rightarrow\infty}\psi_{k}(x)$ $\displaystyle =$ $\displaystyle B\frac{ik-a}{ik+a}e^{ikx}\ \ \ \ \ (12)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -B\frac{(ik-a)^{2}}{a^{2}+k^{2}}e^{ikx} \ \ \ \ \ (13)$

For large negative ${x}$ ${\lim_{x\rightarrow-\infty}\tanh(ax)=-1}$ so we get

 $\displaystyle \lim_{x\rightarrow-\infty}\psi_{k}(x)$ $\displaystyle =$ $\displaystyle B\frac{ik+a}{ik+a}e^{ikx}\ \ \ \ \ (14)$ $\displaystyle$ $\displaystyle =$ $\displaystyle Be^{ikx} \ \ \ \ \ (15)$

Thus in both cases, the wave function represents a wave travelling to the right, with no leftward component. That is, there is no reflected wave. The modulus of the wave for large ${x}$ is

 $\displaystyle \lim_{x\rightarrow\infty}\left|\psi_{k}(x)\right|^{2}$ $\displaystyle =$ $\displaystyle \left|B\right|^{2}\left|\frac{(ik-a)^{2}}{a^{2}+k^{2}}\right|^{2}\ \ \ \ \ (16)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left|B\right|^{2}\ \ \ \ \ (17)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \lim_{x\rightarrow-\infty}\left|\psi_{k}(x)\right|^{2} \ \ \ \ \ (18)$

Thus the transmission coefficient is 1 for all positive energies, which means that any particle coming in from the left passes straight through with no reflection. There is, however, a change of phase due to the factor of ${\frac{(ik-a)^{2}}{a^{2}+k^{2}}}$.

# Finite drop potential

Required math: calculus

Required physics: Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.35.

The problem of the finite step potential can be inverted to give a finite drop potential by replacing ${V_{0}}$ by ${-V_{0}}$, so the potential is given by

$\displaystyle V(x)=\begin{cases} 0 & x\le0\\ -V_{0} & x>0 \end{cases} \ \ \ \ \ (1)$

If we assume particles coming in from the left, then we must have ${E>0}$ (otherwise the wave function would decay exponentially inside the barrier and we couldn’t have particles coming in from infinity on the left). In this case the reflection coefficient is

$\displaystyle R=\left[\frac{E-\sqrt{E\left(E+V_{0}\right)}}{E+\sqrt{E\left(E+V_{0}\right)}}\right]^{2} \ \ \ \ \ (2)$

and the transmission coefficient is

$\displaystyle T=\frac{4E^{3/2}\sqrt{E+V_{0}}}{\left(E+\sqrt{E\left(E+V_{0}\right)}\right)^{2}} \ \ \ \ \ (3)$

For ${V_{0}=0}$ the problem reduces to that of the free particle, and ${R=0}$, ${T=1}$ as we’d expect. As ${V_{0}}$ gets very large, ${R\rightarrow1}$, ${T\rightarrow0}$.

If we take ${E=V_{0}/3}$, then ${R=1/9}$.

Although the graph of the potential looks like a cliff, it doesn’t represent the behaviour of an object, such as a car, falling over a cliff. Classically, the energy of a car is kinetic + potential, which in the absence of other forces, remains a constant. If a car had a speed ${v_{1}}$ in a region where ${V=0}$, then its total energy is kinetic: ${E=K_{1}=\frac{1}{2}mv_{1}^{2}}$. If it suddenly encounters a region where ${V=-V_{0}}$, then we’d have ${E=K_{2}-V_{0}}$, so ${K_{2}}$ is larger than ${K_{1}}$, meaning that the car would instantaneously increase its speed, which of course doesn’t happen. In reality, a car driving off a cliff encounters a potential energy of ${-mgy}$ where ${y}$ is the distance it has fallen, so its kinetic energy increases gradually. Besides, a car falling off a cliff is essentially a two-dimensional problem, so trying to analyze it in one dimension won’t work.

A slightly more realistic case is that of a neutron which is fired at an atomic nucleus. The neutron experiences a sudden drop in potential from ${V=0}$ outside the nucleus to ${V=-V_{0}=-12}$ MeV inside. If we give the neutron an initial kinetic energy of ${E=4}$ MeV, then the probability of transmission into the nucleus is

 $\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{4\times4^{3/2}\sqrt{4+12}}{\left(4+\sqrt{4(4+12)}\right)^{2}}\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{128}{144}\ \ \ \ \ (5)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{8}{9} \ \ \ \ \ (6)$

# Finite step potential – scattering

Required math: calculus

Required physics: Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.34.

A variant of the finite square well is the finite step, which has the potential

$\displaystyle V(x)=\begin{cases} 0 & x\le0\\ V_{0} & x>0 \end{cases} \ \ \ \ \ (1)$

where ${V_{0}}$ is a positive constant energy.

There are two distinct cases here:

1. Energy below the barrier: ${0\le E\le V_{0}}$
2. Energy greater than the barrier: ${E>V_{0}}$

We’ll consider first the case where ${0\le E\le V_{0}}$.

In this case, the Schrödinger equation for ${x>0}$ is

 $\displaystyle -\frac{\hbar^{2}}{2m}\psi''+V_{0}\psi$ $\displaystyle =$ $\displaystyle E\psi\ \ \ \ \ (2)$ $\displaystyle \psi''$ $\displaystyle =$ $\displaystyle \mu^{2}\psi \ \ \ \ \ (3)$

where

$\displaystyle \mu=\frac{\sqrt{2m(V_{0}-E)}}{\hbar} \ \ \ \ \ (4)$

This has solution

$\displaystyle \psi(x)=Ce^{\mu x}+De^{-\mu x} \ \ \ \ \ (5)$

To keep the solution finite as ${x\rightarrow\infty}$ we must have ${C=0}$ so the solution is an exponentially decaying wave function:

$\displaystyle \psi(x)=De^{-\mu x} \ \ \ \ \ (6)$

To the left of the barrier, the Schrödinger equation is

$\displaystyle \psi''=-\frac{2mE}{\hbar^{2}}\psi \ \ \ \ \ (7)$

Assuming particles coming in from the left, we have

$\displaystyle \psi(x)=Ae^{ikx}+Be^{-ikx} \ \ \ \ \ (8)$

where

$\displaystyle k=\frac{\sqrt{2mE}}{\hbar} \ \ \ \ \ (9)$

Since the potential is finite everywhere, both ${\psi}$ and ${\psi'}$ are continuous everywhere, which gives us two boundary conditions at ${x=0}$.

 $\displaystyle A+B$ $\displaystyle =$ $\displaystyle D\ \ \ \ \ (10)$ $\displaystyle ik\left(A-B\right)$ $\displaystyle =$ $\displaystyle -\mu D \ \ \ \ \ (11)$

This has solution

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{ik+\mu}{ik-\mu}A\ \ \ \ \ (12)$ $\displaystyle D$ $\displaystyle =$ $\displaystyle \frac{2ik}{ik-\mu}A \ \ \ \ \ (13)$

The reflection coefficient is

 $\displaystyle R$ $\displaystyle =$ $\displaystyle \frac{\left|B\right|^{2}}{\left|A\right|^{2}}\ \ \ \ \ (14)$ $\displaystyle$ $\displaystyle =$ $\displaystyle 1 \ \ \ \ \ (15)$

That is, the probability of an incoming particle being reflected is 1. This is because the wave function for ${x>0}$ is exponentially decaying, so the probability of a particle reaching infinity is zero, thus no particles can be transmitted.

For ${E>V_{0}}$ the Schrödinger equation for ${x>0}$ is

 $\displaystyle -\frac{\hbar^{2}}{2m}\psi''+V_{0}\psi$ $\displaystyle =$ $\displaystyle E\psi\ \ \ \ \ (16)$ $\displaystyle \psi''$ $\displaystyle =$ $\displaystyle -\kappa^{2}\psi \ \ \ \ \ (17)$

where

$\displaystyle \kappa=\frac{\sqrt{2m(E-V_{0})}}{\hbar} \ \ \ \ \ (18)$

We now get travelling wave solutions instead of exponentially decaying ones:

$\displaystyle \psi(x)=Ce^{i\kappa x}+De^{-i\kappa x} \ \ \ \ \ (19)$

Assuming incoming particles arrive only from the left, we can set ${D=0}$. Applying the boundary conditions, we get

 $\displaystyle A+B$ $\displaystyle =$ $\displaystyle C\ \ \ \ \ (20)$ $\displaystyle ik\left(A-B\right)$ $\displaystyle =$ $\displaystyle i\kappa C \ \ \ \ \ (21)$

with solutions

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{k-\kappa}{k+\kappa}A\ \ \ \ \ (22)$ $\displaystyle C$ $\displaystyle =$ $\displaystyle \frac{2k}{k+\kappa}A \ \ \ \ \ (23)$

In this case, the reflection coefficient is

 $\displaystyle R$ $\displaystyle =$ $\displaystyle \frac{\left|B\right|^{2}}{\left|A\right|^{2}}\ \ \ \ \ (24)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\frac{k-\kappa}{k+\kappa}\right)^{2} \ \ \ \ \ (25)$

Substituting the expressions for ${k}$ and ${\kappa}$ we get

$\displaystyle R=\left[\frac{E-\sqrt{E\left(E-V_{0}\right)}}{E+\sqrt{E\left(E-V_{0}\right)}}\right]^{2} \ \ \ \ \ (26)$

From this we can get the transmission coefficient

 $\displaystyle T$ $\displaystyle =$ $\displaystyle 1-R\ \ \ \ \ (27)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{4E^{3/2}\sqrt{E-V_{0}}}{\left(E+\sqrt{E\left(E-V_{0}\right)}\right)^{2}} \ \ \ \ \ (28)$

Note that this is not equal to ${\left|C\right|^{2}/\left|A\right|^{2}=4E^{2}/\left(E+\sqrt{E\left(E-V_{0}\right)}\right)^{2}}$. Have we done something wrong?

The answer lies in the fact that the wave for ${x>0}$ is not the same as the wave for ${x<0}$, since the net energy on the right is ${E-V_{0}}$ while on the left it is just ${E}$. One way of looking at it is in terms of the probability current for the free particle. The probability current must be conserved; this is just a way of saying that particles cannot vanish into, nor arise from, thin air. Since the probability current for a free particle with stationary state

$\displaystyle \Psi(x,t)=Ae^{ikx}e^{-i\hbar k^{2}t/2m} \ \ \ \ \ (29)$

is

$\displaystyle J=\frac{\hbar k}{m}\left|A\right|^{2} \ \ \ \ \ (30)$

the conservation law implies, for the case of the step potential

$\displaystyle \frac{\hbar k}{m}\left[\left|A\right|^{2}-\left|B\right|^{2}\right]=\frac{\hbar\kappa}{m}\left|C\right|^{2} \ \ \ \ \ (31)$

That is, the influx of particles from the left minus the reflected beam must equal the transmitted beam. Dividing through by ${\frac{\hbar k}{m}\left|A\right|^{2}}$ we get

$\displaystyle \frac{\left|B\right|^{2}}{\left|A\right|^{2}}+\frac{\kappa}{k}\frac{\left|C\right|^{2}}{\left|A\right|^{2}}=1 \ \ \ \ \ (32)$

The first term is the reflection coefficient we calculated in 26. The second term is the transmission coefficient, which works out to

 $\displaystyle T$ $\displaystyle =$ $\displaystyle \sqrt{\frac{E-V_{0}}{E}}\frac{\left|C\right|^{2}}{\left|A\right|^{2}}\ \ \ \ \ (33)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \sqrt{\frac{E-V_{0}}{E}}\frac{4E^{2}}{\left(E+\sqrt{E\left(E-V_{0}\right)}\right)^{2}}\ \ \ \ \ (34)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{4E^{3/2}\sqrt{E-V_{0}}}{\left(E+\sqrt{E\left(E-V_{0}\right)}\right)^{2}} \ \ \ \ \ (35)$

which is what we got earlier.

# Double delta function well – scattering states

Required math: calculus

Required physics: Schrödinger equation

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.28.

In the last post we had a look at the bound states of the double delta function potential

$\displaystyle V(x)=-\alpha\left[\delta(x+a)+\delta(x-a)\right] \ \ \ \ \ (1)$

where ${\alpha}$ gives the strength of the well. In this post, we’ll look at the scattering states of this potential.

We will use a similar approach to that for the single delta function potential. At first glance, you might think the problem is a trivial extension of the single delta function case. If a stream of particles enters from the left, then a fraction will get reflected at the first delta function, with the remainder being transmitted. Of those that are transmitted, another fraction will get reflected at the second delta function and those left over from that reflection will be transmitted to travel on to infinity on the right.

The flaw in this argument is that those that get reflected at the second delta function will travel back to the left, and some of them will be reflected back to the right again when they reach the first delta function. This process continues ad infinitum, with part of the particle stream being bounced back and forth between the two delta functions. Thus we are faced with an infinite series of reflections and transmissions.

Probably the easiest way to analyze the problem is just to confront the mathematics head on and see where it leads. We therefore follow the procedure for the single delta function to obtain the solutions in each of the three regions. Since we are proposing a particle stream entering from the left, there is no left-travelling stream from the right, so the solution is asymmetric, meaning we can’t propose even and odd solutions to the problem.

The most general solution of this equation is (with ${k\equiv\frac{\sqrt{2mE}}{\hbar}}$; remember ${E}$ is positive so ${k}$ is real)

$\displaystyle \psi(x)=\begin{cases} Ae^{ikx}+Be^{-ikx} & x<-a\\ Ce^{ikx}+De^{-ikx} & -aa \end{cases} \ \ \ \ \ (2)$

We can apply boundary conditions to eliminate some of the constants.

Continuity of the wave function at ${x=-a}$ gives

$\displaystyle Ae^{-ika}+Be^{ika}=Ce^{-ika}+De^{ika} \ \ \ \ \ (3)$

The same condition at ${x=a}$ gives

$\displaystyle Ce^{ika}+De^{-ika}=Fe^{ika} \ \ \ \ \ (4)$

The change in derivative of the wave function across the delta function boundary satisfies the following condition at ${x=\pm a}$ (this is the same condition that we applied to the single delta function at ${x=0}$):

$\displaystyle \Delta\psi'=-\frac{2m\alpha}{\hbar^{2}}\psi(\pm a) \ \ \ \ \ (5)$

At ${x=-a}$ we have

 $\displaystyle \Delta\psi'$ $\displaystyle =$ $\displaystyle ik\left[Ce^{-ika}-De^{ika}-Ae^{-ika}+Be^{ika}\right]\ \ \ \ \ (6)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -\frac{2m\alpha}{\hbar^{2}}\left(Ae^{-ika}+Be^{ika}\right) \ \ \ \ \ (7)$

Similarly at ${x=a}$ we have

 $\displaystyle \Delta\psi'$ $\displaystyle =$ $\displaystyle ik\left[Fe^{ika}-Ce^{ika}+De^{-ika}\right]\ \ \ \ \ (8)$ $\displaystyle$ $\displaystyle =$ $\displaystyle -\frac{2m\alpha}{\hbar^{2}}Fe^{ika} \ \ \ \ \ (9)$

We now have four equations in the five unknowns ${A,B,C,D}$ and ${F}$. To get the transmission and reflection coefficients, however, we need only express the last four constants in terms of ${A}$. The four equations constitute a system of linear equations in the constants, so it is a straightforward matter of algebra to solve them. Doing this by hand is fairly laborious, but we can use software such as Maple’s ‘solve’ command to do it for us.

The results are

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{iz\left[4k\cos\left(2ka\right)-2z\sin\left(2ka\right)\right]}{4k\left(k-iz\right)+z^{2}\left(e^{4ika}-1\right)}A\ \ \ \ \ (10)$ $\displaystyle C$ $\displaystyle =$ $\displaystyle -\frac{2ki(z+2ki)}{4k\left(k-iz\right)+z^{2}\left(e^{4ika}-1\right)}A\ \ \ \ \ (11)$ $\displaystyle D$ $\displaystyle =$ $\displaystyle \frac{2ikze^{2ika}}{4k\left(k-iz\right)+z^{2}\left(e^{4ika}-1\right)}A\ \ \ \ \ (12)$ $\displaystyle F$ $\displaystyle =$ $\displaystyle \frac{4k^{2}}{4k\left(k-iz\right)+z^{2}\left(e^{4ika}-1\right)}A \ \ \ \ \ (13)$

where

$\displaystyle z\equiv\frac{2m\alpha}{\hbar^{2}} \ \ \ \ \ (14)$

The transmission coefficient is then

 $\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{\left|F\right|^{2}}{\left|A\right|^{2}}\ \ \ \ \ (15)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{8k^{4}}{8k^{4}+4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]} \ \ \ \ \ (16)$

The reflection coefficient is

 $\displaystyle R$ $\displaystyle =$ $\displaystyle \frac{\left|B\right|^{2}}{\left|A\right|^{2}}\ \ \ \ \ (17)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{2z^{2}\left(2k\cos\left(2ka\right)-z\sin\left(2ka\right)\right)^{2}}{8k^{4}+4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]}\ \ \ \ \ (18)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]}{8k^{4}+4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]} \ \ \ \ \ (19)$

As a check, we note that ${R+T=1}$.

For reference, the two internal rates are

 $\displaystyle T_{i}=\frac{\left|C\right|^{2}}{\left|A\right|^{2}}$ $\displaystyle =$ $\displaystyle \frac{2k^{2}z^{2}+8k^{4}}{8k^{4}+4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]}\ \ \ \ \ (20)$ $\displaystyle R_{i}=\frac{\left|D\right|^{2}}{\left|A\right|^{2}}$ $\displaystyle =$ $\displaystyle \frac{2k^{2}z^{2}}{8k^{4}+4k^{2}z^{2}+z^{4}-4kz^{3}\sin\left(4ka\right)+z^{2}\cos\left(4ka\right)\left[4k^{2}-z^{2}\right]} \ \ \ \ \ (21)$

The first quantity represents the flow to the right after the first delta function, and we observe that it is larger than the second quantity, which represents the flow to the left. This makes sense, since we would expect that as the main particle stream enters from the left, and of that which gets transmitted past the first well, some will get transmitted past the second well and escape, while some will get reflected back towards the first well. In fact, we have ${T+R_{i}=T_{i}}$ which says that the probability of being transmitted past the first well is the sum of the probabilities of being reflected from the second well and being transmitted through the second well.

# Delta-function well – scattering

Required math: calculus, delta function

Required physics: energy, quantum mechanics basics

Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Sec 2.5.2.

Shankar, R. (1994), Principles of Quantum Mechanics, Plenum Press. Section 5.4, Exercise 5.4.2 (a).

The delta-function well always has exactly one bound state, where the energy of the particle is less than zero. If we consider states where the energy is greater than zero, we can investigate the phenomenon of scattering.

The potential function we are using is

$\displaystyle V(x)=-\alpha\delta(x) \ \ \ \ \ (1)$

where ${\alpha}$ is a positive constant that gives the strength of the well. In this case, the Schrödinger equation is

$\displaystyle -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}-\alpha\delta(x)\psi=E\psi \ \ \ \ \ (2)$

At all points except ${x=0}$ this equation becomes

$\displaystyle -\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}=E\psi \ \ \ \ \ (3)$

With ${E>0}$, we can write this as

$\displaystyle \frac{d^{2}\psi}{dx^{2}}=-k^{2}\psi \ \ \ \ \ (4)$

where

$\displaystyle k\equiv\frac{\sqrt{2mE}}{\hbar} \ \ \ \ \ (5)$

Since ${E>0}$, ${k}$ is real and can be taken to be positive. As in the case of the bound state, because of the singularity at ${x=0}$ we need to consider solutions on either side of this point separately.

The general solution of 4 is

$\displaystyle \psi(x)=Ae^{ikx}+Be^{-ikx} \ \ \ \ \ (6)$

so we can take this as the solution for ${x<0}$ and call it ${\psi_{-}(x)}$:

$\displaystyle \psi_{-}(x)=Ae^{ikx}+Be^{-ikx} \ \ \ \ \ (7)$

For ${x>0}$, we can write

$\displaystyle \psi_{+}(x)=Ce^{ikx}+De^{-ikx} \ \ \ \ \ (8)$

Because the potential is zero in these two regions, the solutions are those of the free particle and, as we saw when we considered that case, these solutions are not normalizable so cannot, on their own, represent a physically realizable state. However, because any linear combination of solutions is also a solution, we found in the case of the free particle that we could create a wave packet and that such a packet, although not itself a stationary state, was normalizable and represented a particle travelling through space.

Unfortunately, we also saw that the mathematics rapidly becomes pretty horrible when we attempt to work with wave packets, so much of what is known about them is derived through computer simulations. When dealing with scattering problems, the same problems arise, and realistic problems (that is, ones involving wave packets rather than single, non-normalizable functions) can be solved only by simulation.

We will work through the problem using stationary states to see how scattering problems are handled, but it should always be kept in mind that this is not a physically realizable situation. The problem is that we will be doing the analysis for only a single value of ${k}$ (and hence, of ${E}$), whereas a wave packet consists of contributions from many different values of ${k}$, and thus from many different energies. As we will see, the probabilities of reflection from the potential and of transmission through it both depend on ${k}$, so each value of ${k}$ behaves differently. A proper wave packet therefore scatters in quite a complex manner, and it’s a non-trivial problem to work this out in detail.

With that warning, let’s see how we solve a scattering problem for a single value of ${k}$. The idea behind a scattering experiment is that we imagine a particle coming in from one direction (say, coming in from the left, so moving towards the positive ${x}$ direction). When this particle hits the potential well, it may bounce back towards ${-x}$, or it may go through the ${x=0}$ point and emerge on the other side, still travelling in the direction of ${+x}$.

Remember that the full solution of the Schrödinger equation in this case is

$\displaystyle \Psi(x,t)=\psi(x)e^{-iEt/\hbar} \ \ \ \ \ (9)$

For ${x<0}$, this becomes

$\displaystyle \Psi_{-}(x,t)=Ae^{i(kx-Et/\hbar)}+Be^{i(-kx-Et/\hbar)} \ \ \ \ \ (10)$

and for ${x>0}$:

$\displaystyle \Psi_{+}(x,t)=Ce^{i(kx-Et/\hbar)}+De^{i(-kx-Et/\hbar)} \ \ \ \ \ (11)$

In each of these functions, the first term represents a wave travelling to the right, and the second term a wave travelling to the left. (A reminder of how to see this: consider the motion of a fixed point on the wave, where the exponent is a constant, and consider how ${x}$ must change as ${t}$ increases to see which way the wave is moving. If ${kx-Et/\hbar=}$constant, then ${x}$ must increase as ${t}$ increases; just the opposite is true if ${-kx-Et/\hbar=}$constant.)

Therefore, to represent the experiment we described above, with a particle coming in from the left and possibly reflecting back from or passing through the potential well, we need to include terms with waves travelling in both directions for ${x<0}$ and only one direction (travelling to the right) for ${x>0}$. Therefore, we can say that ${D=0}$ since there are no waves travelling to the left when ${x>0}$.

Requiring the wave function to be continuous at ${x=0}$ gives us one more condition:

$\displaystyle A+B=C \ \ \ \ \ (12)$

We can get another condition by using the same integration technique that we applied in the case of the bound state. Integrating the Schrödinger equation across the origin we get:

 $\displaystyle -\frac{\hbar^{2}}{2m}\int_{-\epsilon}^{\epsilon}\frac{d^{2}\psi}{dx^{2}}dx-\alpha\int_{-\epsilon}^{\epsilon}\delta(x)\psi dx$ $\displaystyle =$ $\displaystyle E\int_{-\epsilon}^{\epsilon}\psi dx\ \ \ \ \ (13)$ $\displaystyle -\frac{\hbar^{2}}{2m}\frac{d\psi}{dx}\Big|_{-\epsilon}^{\epsilon}-\alpha\psi(0)$ $\displaystyle =$ $\displaystyle E\int_{-\epsilon}^{\epsilon}\psi dx \ \ \ \ \ (14)$

Taking the limit as ${\epsilon\rightarrow0}$ gives us

$\displaystyle -\frac{\hbar^{2}ik}{2m}(C-A+B)-\alpha C=0 \ \ \ \ \ (15)$

We can now use 12 and 15 to eliminate two of the three constants, and we can express ${C}$ and ${B}$ in terms of ${A}$:

 $\displaystyle B$ $\displaystyle =$ $\displaystyle \frac{i\beta}{1-i\beta}A\ \ \ \ \ (16)$ $\displaystyle C$ $\displaystyle =$ $\displaystyle \frac{1}{1-i\beta}A\ \ \ \ \ (17)$ $\displaystyle \beta$ $\displaystyle \equiv$ $\displaystyle \frac{m\alpha}{\hbar^{2}k} \ \ \ \ \ (18)$

It might look as though we’re stuck at this point, since we can’t normalize the wave function, so we can’t determine ${A}$. However, what we really want is the probability that the particle will be reflected or transmitted, and we can get that by comparing ${B}$ and ${C}$ with ${A}$. Remember that the term ${Ae^{i(kx-Et/\hbar)}}$ represents the incoming particle, ${Be^{i(-kx-Et/\hbar)}}$ the reflected particle and ${Ce^{i(kx-Et/\hbar)}}$ the transmitted particle. For the region ${x<0}$, the probability that the particle is travelling to the left relative to the probability that it is travelling to the right should give the probability that it has been reflected. Comparing the probability that the particle is found in the region ${x>0}$ to the probability that it is travelling to the right in the region ${x<0}$ should give the probability that it has been transmitted.

At this point, you might be thinking there is something wrong with the logic here. After all, the particle can’t be both travelling to the right and to the left at the same time in the region ${x<0}$, nor can it be on both sides of the origin at the same time. Not only that, but we are analyzing an explicitly time-dependent problem using only stationary states, and these states are just waves of constant amplitude that extend out to infinity, rather than wave packets describing real particles. There is no honest way around this problem; essentially we are fudging the answer since we are analyzing a non-physical system anyway. One way of thinking about it that might make you feel a bit better is, instead of imagining a single particle travelling in and either bouncing off or passing through the well, imagine a steady stream of particles being beamed at the well from the left. In that case, we would reach a steady state in which a certain fraction of particles would get reflected and the remainder would get transmitted. This situation at least gets rid of any explicit time dependence, although the problem of non-normalizability of the wave function is still there.

In the final analysis, the only honest way of analyzing this problem is by constructing a wave packet out of multiple values of ${k}$, doing the normalization properly and then working out the probabilities. However, as you might imagine, that is far from easy.

In the meantime, we can get expressions for our steady state reflection and transmission probabilities ${R}$ and ${T}$ by finding the appropriate ratios:

 $\displaystyle R$ $\displaystyle =$ $\displaystyle \frac{|B|^{2}}{|A|^{2}}\ \ \ \ \ (19)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\beta^{2}}{1+\beta^{2}}\ \ \ \ \ (20)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{1}{1+2\hbar^{2}E/m\alpha^{2}}\ \ \ \ \ (21)$ $\displaystyle T$ $\displaystyle =$ $\displaystyle \frac{|C|^{2}}{|A|^{2}}\ \ \ \ \ (22)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{1}{1+\beta^{2}}\ \ \ \ \ (23)$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{1}{1+m\alpha^{2}/2\hbar^{2}E} \ \ \ \ \ (24)$

after substituting for ${\beta}$ and then for ${k}$ in terms of ${E}$. Note that ${R+T=1}$ so the particle must be either reflected or transmitted.

The derivation of ${R}$ and ${T}$ is also valid for a delta function barrier if we set ${\alpha<0}$, since nothing in the derivation relied on ${\alpha}$ being positive, and the final values of ${R}$ and ${T}$ depend only on ${\alpha^{2}}$.