A.28 WKB Theory of Nearly Classical Motion

WKB theory provides simple approximate solutions for the energy eigen­functions when the conditions are almost classical, like for the wave packets of chapter 7.11. The approxi­mation is named after Wentzel, Kramers, and Brillouin, who refined the ideas of Liouville and Green. The bandit scientist Jeffreys tried to rob WKB of their glory by doing the same thing two years earlier, and is justly denied all credit.

Figure A.16: Harmonic oscillator potential energy $V$, eigen­function $h_{50}$, and its energy $E_{50}$.

The WKB approxi­mation is based on the rapid spatial variation of energy eigen­functions with almost macroscopic energies. As an example, figure A.16 shows the harmonic oscillator energy eigen­function $h_{50}$. Its energy $E_{50}$ is hundred times the ground state energy. That makes the kinetic energy $E-V$ quite large over most of the range, and that in turn makes the linear momentum large. In fact, the classical Newtonian linear momentum $p_{\rm {c}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $mv$ is given by

$$\fbox{$\displaystyle p_{\rm{c}} \equiv \sqrt{2m(E-V)} $} %$$ (A.207)

In quantum mechanics, the large momentum implies rapid oscill­ation of the wave function: quantum mechanics associates the linear momentum with the operator $\hbar{\rm d}$$\raisebox{.5pt}{$/$}$${\rm i}{\rm d}{x}$ that denotes spatial variation.

The WKB approxi­mation is most appealing in terms of the classical momentum $p_{\rm {c}}$ as defined above. To find its form, in the Hamiltonian eigenvalue problem

$$-\frac{\hbar^2}{2m} \frac{{\rm d}^2\psi}{{\rm d}x^2} + V\psi = E\psi$$

take the $V\psi$ term to the other side and then rewrite $E-V$ in terms of the classical linear momentum. That produces

$$\frac{{\rm d}^2\psi}{{\rm d}x^2} = - \frac{p_{\rm {c}}^2}{\hbar^2}\psi %$$ (A.208)

Now under almost classical conditions, a single period of oscill­ation of the wave function is so short that normally $p_{\rm {c}}$ is almost constant over it. Then by approxi­mation the solution of the eigenvalue problem over a single period is simply an arbitrary combin­ation of two exponentials,

$$\psi \sim c_{\rm {f}} e^{{\rm i}p_{\rm {c}} x/\hbar} + c_{\rm {b}} e^{-{\rm i}p_{\rm {c}} x/\hbar}$$

where the constants $c_{\rm {f}}$ and $c_{\rm {b}}$ are arbitrary. (The subscripts denote whether the wave speed of the corre­sponding term is forward or backward.)

It turns out that to make the above expression work over more than one period, it is necessary to replace $p_{\rm {c}}x$ by the anti­derivative $\int{p}_{\rm {c}}{\,\rm d}{x}$. Furthermore, the “constants” $c_{\rm {f}}$ and $c_{\rm {b}}$ must be allowed to vary from period to period propor­tional to 1/$\sqrt{p_{\rm {c}}}$.

In short, the WKB approxi­mation of the wave function is, {D.46}:

$$\fbox{$\displaystyle \mbox{classical WKB:} \qquad \psi... ...eta \equiv \frac{1}{\hbar} \int p_{\rm{c}} {\,\rm d}x $} %$$ (A.209)

where $C_{\rm {f}}$ and $C_{\rm {b}}$ are now true constants.

If you ever glanced at notes such as {D.12}, {D.14}, and {D.15}, in which the eigen­functions for the harmonic oscillator and hydrogen atom were found, you recognize what a big simplifi­cation the WKB approxi­mation is. Just do the integral for $\theta$ and that is it. No elaborate transfor­mations and power series to grind down. And the WKB approxi­mation can often be used where no exact solutions exist at all.

In many appli­cations, it is more convenient to write the WKB approxi­mation in terms of a sine and a cosine. That can be done by taking the exponentials apart using the Euler formula (2.5). It produces

$$\fbox{$\displaystyle \mbox{rephrased WKB:} \qquad \psi... ...eta \equiv \frac{1}{\hbar} \int p_{\rm{c}} {\,\rm d}x $} %$$ (A.210)

The constants $C_{\rm {c}}$ and $C_{\rm {s}}$ are related to the original constants $C_{\rm {f}}$ and $C_{\rm {b}}$ as

$$\fbox{$\displaystyle C_{\rm{c}} = C_{\rm{f}} + C_{\rm{b}... ...xtstyle\frac{1}{2}} (C_{\rm{c}} + {\rm i}C_{\rm{s}}) $} %$$ (A.211)

which allows you to convert back and forward between the two formul­ations as needed. Do note that either way, the constants depend on what you chose for the integr­ation constant in the $\theta$ integral.

As an appli­cation, consider a particle stuck between two impenetrable walls at positions $x_1$ and $x_2$. An example would be the particle in a pipe that was studied way back in chapter 3.5. The wave function $\psi$ must become zero at both $x_1$ and $x_2$, since there is zero possi­bility of finding the particle outside the impenetrable walls. It is now smart to chose the integr­ation constant in $\theta$ so that $\theta_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ 0. In that case, $C_{\rm {c}}$ must be zero for $\psi$ to be zero at $x_1$, (A.210). The wave function must be just the sine term. Next, for $\psi$ also to be zero at $x_2$, $\theta_2$ must be a whole multiple $n$ of $\pi$, because that are the only places where sines are zero. So $\theta_2-\theta_1$ $\vphantom0\raisebox{1.5pt}{$=$}$ $n\pi$, which means that

$$\fbox{$\displaystyle \mbox{particle between impenetrable... ...rm{c}}({\underline x}) {\,\rm d}{\underline x}= n \pi $} %$$ (A.212)

Recall that $p_{\rm {c}}$ was $\sqrt{2m(E-V)}$, so this is just an equation for the energy eigen­values. It is an equation involving just an integral; it does not even require you to find the corre­sponding eigen­functions!

It does get a bit more tricky for a case like the harmonic oscillator where the particle is not caught between impenetrable walls, but merely prevented to escape by a gradually increasing potential. Classi­cally, such a particle would still be rigorously constrained between the so called “turning points” where the potential energy $V$ becomes equal to the total energy $E$, like the points 1 and 2 in figure A.16. But as the figure shows, in quantum mechanics the wave function does not become zero at the turning points; there is some chance for the particle to be found somewhat beyond the turning points.

A further compli­cation arises since the WKB approxi­mation becomes inaccurate in the immediate vicinity of the turning points. The problem is the requirement that the classical momentum can be approximated as a non­zero constant on a small scale. At the turning points the momentum becomes zero and that approxi­mation fails.

However, it is possible to solve the Hamiltonian eigenvalue problem near the turning points assuming that the potential energy is not constant, but varies approximately linearly with position, {A.29}. Doing so and fixing up the WKB solution away from the turning points produces a simple result. The classical WKB approxi­mation remains a sine, but at the turning points, $\sin\theta$ stays an angular amount $\pi$$\raisebox{.5pt}{$/$}$4 short of becoming zero. (Or to be precise, it just seems to stay $\pi$$\raisebox{.5pt}{$/$}$4 short, because the classical WKB approxi­mation is no longer valid at the turning points.) Assuming that there are turning points with gradually increasing potential at both ends of the range, like for the harmonic oscillator, the total angular range will be short by an amount $\pi$$\raisebox{.5pt}{$/$}$2.

Therefore, the expression for the energy eigen­values becomes:

$$\fbox{$\displaystyle \mbox{particle trapped between turn... ... d}{\underline x} = (n-{\textstyle\frac{1}{2}}) \pi $} %$$ (A.213)

The WKB approxi­mation works fine in regions where the total energy $E$ is less than the potential energy $V$. The classical momentum $p_{\rm {c}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{2m(E-V)}$ is imaginary in such regions, reflecting the fact that classi­cally the particle does not have enough energy to enter them. But, as the non­zero wave function beyond the turning points in figure A.16 shows, quantum mechanics does allow some possi­bility for the particle to be found in regions where $E$ is less than $V$. It is loosely said that the particle can “tunnel” through, after a popular way for criminals to escape from jail. To use the WKB approxi­mation in these regions, just rewrite it in terms of the magnitude $\vert p_{\rm {c}}\vert$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{2m(V-E)}$ of the classical momentum:

$$\fbox{$\displaystyle \mbox{tunneling WKB:}\qquad \psi ... ...\frac{1}{\hbar} \int \vert p_{\rm{c}}\vert {\,\rm d}x $} %$$ (A.214)

Note that $\gamma$ is the equivalent of the angle $\theta$ in the classical approxi­mation.

Key Points

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The WKB approxi­mation applies to situations of almost macroscopic energy.

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The WKB solution is described in terms of the classical momentum $p_{\rm {c}}$ $\vphantom0\raisebox{1.5pt}{$\equiv$}$ $\sqrt{2m(E-V)}$ and in particular its anti­derivative $\theta$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\int{p}_{\rm {c}}{\,\rm d}{x}$$\raisebox{.5pt}{$/$}$$\hbar$.

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The wave function can be written as (A.209) or (A.210), whatever is more convenient.

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For a particle stuck between impenetrable walls, the energy eigen­values can be found from (A.212).

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For a particle stuck between a gradually increasing potential at both sides, the energy eigen­values can be found from (A.213).

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The “tunneling” wave function in regions that classi­cally the particle is forbidden to enter can be approximated as (A.214). It is in terms of the anti­derivative $\gamma$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\int\vert p_{\rm {c}}\vert{\,\rm d}{x}$$\raisebox{.5pt}{$/$}$$\hbar$.

A.28 Review Questions

1. Use the equation
   
   $$\frac{1}{\hbar} \int_{{\underline x}=x_1}^{x_2} p_{\rm {c}}({\underline x}) {\,\rm d}{\underline x}= n \pi$$
   
   
   to find the WKB approxi­mation for the energy levels of a particle stuck in a pipe of chapter 3.5.5. The potential $V$ is zero inside the pipe, given by 0 $\raisebox{-.3pt}{$\leqslant$}$ $x$ $\raisebox{-.3pt}{$\leqslant$}$ $\ell_x$

   In this case, the WKB approxi­mation produces the exact result, since the classical momentum really is constant. If there was a force field in the pipe, the solution would only be approximate.

   Solution wkb-a

2. Use the equation
   
   $$\frac{1}{\hbar} \int_{{\underline x}=x_1}^{x_2} p_{\rm {c}}(... ...e x}) {\,\rm d}{\underline x}= (n-{\textstyle\frac{1}{2}}) \pi$$
   
   
   to find the WKB approxi­mation for the energy levels of the harmonic oscillator. The potential energy is ${\textstyle\frac{1}{2}}m\omega{x}^2$ where the constant $\omega$ is the classical natural frequency. So the total energy, expressed in terms of the turning points $x_2$ $\vphantom0\raisebox{1.5pt}{$=$}$ $-x_1$ at which $E$ $\vphantom0\raisebox{1.5pt}{$=$}$ $V$, is $E$ $\vphantom0\raisebox{1.5pt}{$=$}$ ${\textstyle\frac{1}{2}}m\omega{x_2}^2$.

   In this case too, the WKB approxi­mation produces the exact energy eigen­values. That, however, is just a coincidence; the classical WKB wave functions are certainly not exact; they become infinite at the turning points. As the example $h_{50}$ above shows, the true wave functions most definitely do not.

   Solution wkb-b