Chapter 16 Variational and WKB Methods

16.1 The Variatonal Method

Exercise 16.1.1 Try $\psi=\exp \left(-\alpha x^2\right)$ for $V=\dfrac{1}{2} m \omega^2 x^2$ and find $\alpha_0$ and $E\left(\alpha_0\right)$.

Exercise 16.1.2 For a particle in a box that extends from $-a$ to $a$, try (within the box) $\psi=(x-a)(x+a)$ and calculate $E$. There is no parameter to vary, but you still get an upper bound. Compare it to the true energy, $E_0$. (Convince yourself that the singularities in $\psi^{\prime \prime}$ at $x= \pm a$ do not contribute to the energy.)

Exercise 16.1.3 For the attractive delta function potential $V=-a V_0 \delta(x)$ use a Gaussian trial function. Calculate the upper bound on $E_0$ and compare it to the exact answer ($-m a^2 V_0^2/2\hbar^2$) (Exercise 5.2.3).

Exercise 16.1.4 For the oscillator choose

Calculate $E(a)$, minimize it and compare to $\hbar \omega / 2$.

Exercise 16.1.5 Solve the variational problem for the $l=1$ states of the electron in a potential $V=-e^2/r$. In your trial function incorporate (i) correct behavior as $r \rightarrow 0$, appropriate to $l=1$, (ii) correct number of nodes to minimize energy, (iii) correct behavior of wave function as $r \rightarrow \infty$ in a Coulomb potential (i.e., exponential instead of Gaussian damping). Does it matter what $m$ you choose for $Y_l^m$ ? Comment on the relation of the energy bound you obtain to the exact answer.

16.2 The Wentzel-Kramers-Brillouin Method

Exercise 16.2.1 Consider the function $W(E)$ introduced in Eq. (16.2.21a). Since $-E$ is the $t$ derivative of $S(t)$, it follows that $W(E)$, must be the Legendre transform of $S$. In this case $t$ must emerge as the derivative of $W(E)$ with respect to $E$. Verify that differentiation of the formula

gives the time $t$ taken to go from start to finish at this preassigned energy.

Exercise 16.2.2 Consider the free particle problem using the approach given above. Now that you know the wave functions explicitly, evaluate the integral in Eq. (16.2.18b) by contour integration to obtain

In doing the contour integrals, ask in which half-plane you can close the contour for a given sign of $x-x^{\prime}$. Compare the above result to the semiclassical result and make sure it all works out. Note that there is no need to form the combination $U\left(x, x^{\prime}, t\right)+U^*\left(x, x^{\prime}, t\right)$; we can calculate both the principal part and the delta function contributions explicitly because we know the $p$-dependence of the integrand explicitly. To see this more clearly, avoid the contour integral approach and use instead the formula for $(x \pm i \varepsilon)^{-1}$ given above. Evaluate the principal value integral and the contribution from the delta function and see how they add up to the result of contour integration. Although both routes are possible here, in the problem with $V \neq 0$ contour integration is not possible since the $p$-dependence of the wave function in the complex $p$ plane is not known. The advantage of the $U+U^*$ approach is that it only refers to quantities on the real axis and at just one energy.

Exercise 16.2.3 Let us take a second look at our derivation. We worked quite hard to isolate the eigenfunctions at one energy: we formed the combination $U\left(x, x^{\prime}, z\right)+U^*\left(x, x^{\prime}, z\right)$ to get rid of the principal part and filter out the delta function. Now it is clear that if in Eq. (16.2.18a) we could integrate in the range $-\infty \leqslant t \leqslant \infty$, we would get the $\delta$ function we want. What kept us from doing that was the fact the $U(t)$ was constructed to be used for $t>0$. However, if we use the time evolution operator $\mathrm{e}^{-(\mathrm{i}/\hbar)Ht}$ for negative times, it will simply tell us what the system was doing at earlier times, assuming the same Hamiltonian. Thus we can make sense of the operator for negative times as well and define a transform that extends over all times. This is true in classical mechanics as well. For instance, if a stone is thrown straight up from a tall building and we ask when it will be at ground level, we get two answers, one with negative $t$ corresponding to the extrapolation of the given initial conditions to earlier times. Verify that $U\left(x, x^{\prime}, z\right)+U^*\left(x, x^{\prime}, z\right)$ is indeed the transform of $U(t)$ for all times by seeing what happens to $\langle x\mid \mathrm{e}^{-(\mathrm{i}/\hbar)Hf}\mid x^{\prime}\rangle$ under complex conjugation and the exchange $x \leftrightarrow x^{\prime}$. Likewise, ask what happens to $U_{\mathrm{cl}}$ when we transform it for all times. Now you will find that no matter what the sign of $x-x^{\prime}$ a single right moving trajectory can contribute-it is just that the time of the stationary point, $t^*$, will change sign with the sign of $x-x^{\prime}$. (The same goes for a left moving trajectory.)

Exercise 16.2.4 Alpha particles of kinetic energy $4.2~\mathrm{MeV}$ come tunneling out of a nucleus of charge $Z=90$ (after emission). Assume that $V_0=0$, $x_0=10^{-12}~\mathrm{cm}$, and $V(x)$ is just Coulombic for $x \geqslant x_0$. (See Fig. 16.1) Estimate the mean lifetime. [Hint: Show $\gamma=\left(8 Z e^2 / \hbar v\right)\left[\cos ^{-1} y^{1 / 2}-y^{1 / 2}(1-y)^{1 / 2}\right]$, where $y=x_0 \mid x_e$. Show that $y \ll 1$ and use $\cos^{-1} y^{1/2} \simeq \dfrac{1}{2}\pi -y^{1/2}$ before calculating numbers.]

Exercise 16.2.5 In 1974 two new particles called the $\psi$ and $\psi^{\prime}$ were discovered, with rest energies $3.1$ and $3.7~\mathrm{GeV}$, respectively ($1~\mathrm{GeV}=10^9~\mathrm{eV}$). These are believed to be nonrelativistic bound states of a “charmed” quark of mass $m=1.5~\mathrm{GeV}/c^2$ (i.e., $mc^2=1.5~\mathrm{GeV}$) and an antiquark of the same mass, in a linear potential $V(r)=V_0+kr$. By assuming that these are the $n=0$ and $n=1$ bound states of zero orbital angular momentum, calculate $V_0$ using the WKB formula. What do you predict for the rest mass of $\psi^{\prime \prime}$, the $n=2$ state? (The measured value is $\simeq 4.2~\mathrm{GeV}/c^2$.) [Hints: (1) Work with $\mathrm{GeV}$ instead of $\mathrm{eV}$. (2) There is no need to determine $k$ explicitly.]

Exercise 16.2.6 Obtain Eq. (16.2.39) for the $\lambda x^4$ potential by the scaling trick.

Exercise 16.2.7 Find the allowed levels of the harmonic oscillator by the WKB method.

Exercise 16.2.8 Consider the $l=0$ radial equation for the Coulomb problem. Since $V(r)$ is singular at the turning point $r=0$, we can't use ($n+3/4$).

(1) Will the additive constant be more or less than $3/4$?

(2) By analyzing the exact equation near $r=0$, it can be shown that the constant equals $1$. Using this constant show that the WKB energy levels agree with the exact results.