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Introduction

In order to capture the interference phenomena that occur during a scattering process that the classical theory fails to address properly, yet to avoid a fully quantum treatment that is complicated, a semiclassical approach may be employed (see Quantum Scattering Theory for a fully quantum treatment and Classical Scattering Theory for a classical treatment of a scattering process). Semiclassical scattering theory makes use of trajectories, impact parameters, and the like that are classical and the superposition of scattering waves that are quantum to approximate or model a scattering phenomenon fairly accurately under the condition where each classical trajectories carry unique phases.

Semiclassical Physics

Definition

Semiclassical physics is "a vast discipline of theoretical physics that attempts to compute spectra and wave functions of quantum (wave) systems on the basis of their classical trajectories (rays)" with the addition of quantum superposition principle and Planck's constant $h$. (see also Semiclassical physics)

Usefulness

In general, semiclassical approach is said to be useful since very few analytically solvable quantum systems are known to date. Usually, semiclassical methods are incorporated as they allow us to compute spectra, i.e. energy levels, and wave functions of quantum systems based on classical trajectories even in the absence of the exact quantum approach. Note that the method still involves quantum phases and the quantum superposition principle.

Background

This section largely follows section 5.2 of the reference .

In order to obtain approximate solutions to a quantum system with potential $V(x)$, we assume stationary wave function $\psi (x)$

$\psi (x)=A(x)e^{{\frac {i}{\hbar }}B(x)},$

where $A(x)$ and $B(x)$ are chosen such that the wave function satisfies Schrödinger equation. Inserting the form into Schrödinger equation and solving for the equation generally, we obtain a "classical" (absence of $\hbar$) equation

${\frac {1}{2m}}(B')^{2}+V-E=0$

and a "quantum" equation

$\hbar A''+2iA'B'+iAB''=0.$

Solving for $B(x)$ using the classical equation,

$B(x)=\pm \int _{a}^{x}p(x){\mathrm {d} x}=\pm S(x;E),$

where $a$ is an arbitrary location in the classically allowed region, $p(x)$ is the momentum ${\sqrt {2m[V(x)-E]}}$, and $S(x;E)$ is the classical action. The importance of the result lies in the fact that the quantum phase angle shift $B(x)$ of the wave function $\psi (x)$ is now directly related to the classical action $S(x;E)$, which serves as the backbone of semiclassical treatments.

Semiclassical Approximation

Among many semiclassical approximations (see Semiclassical physics), we focus on Bohr-Sommerfeld Quantization and JWKB Approximation that are most useful in dealing with scattering problems.

Bohr-Sommerfeld Quantization

Dating from the early 20th century, Bohr-Sommerfeld Quantization procedure forms the basis of the "old quantum mechanics''. The approximation taken in this procedure is to assume the amplitude $A(x)$ to be constant, namely,

$\psi (x)\approx Ne^{{\frac {i}{\hbar }}B(x)}.$

Computing the accumulated phase angle for the wave that takes off from the starting position, say $a$, then returns to its initial position, we obtain

$\Delta B(a)=\oint p(x){\mathrm {d} x}=S(E).$

Since the wave function $\psi (x)$ is stationary and unique,

$S(E_{n})=2\pi n\hbar ,$

where $n$ is a positive integer.

JWKB Approximation

In the limit $\hbar$ is small, which is justified by the fact that we are in semiclassical realm, the "quantum" equation (see Background section above) may be approximated to be

$2A'B'+A'B''=0,$

which has a solution $A(x)$ of the form

$A(x)={\frac {C}{\sqrt {|p(x)|}}},$

where $C$ is a complex number. With $B(x)=\pm S(x;E)$ found in Background section, the wave function $\psi (x)$ in the classically allowed region may be written as

$\psi (x)={\frac {Ce^{iS/\hbar }+De^{-iS/\hbar }}{\sqrt {|p(x)|}}},$

and the wave function $\psi (x)$ in the classically forbidden region may be written as

$\psi (x)={\frac {Fe^{S/\hbar }+Ge^{-S/\hbar }}{\sqrt {|p(x)|}}},$

where $D$, $F$, and $G$ are complex numbers. At this point, we notice that the obtained wave functions are not valid at the classical turning point due to their non-zero momenta. Using linear approximation of the shape of $V(x)$ at the turning point, with the help of Airy functions and the boundary conditions on $\psi (x)$ and $\psi '(x)$ (see WKB approximation, , or for details), we obtain the wave function in the classically allowed region to be

$\psi (x)\sim \sin \left({\frac {\pi }{4}}+{\frac {1}{\hbar }}\int _{x_{0}}^{x}|p(x)|{\mathrm {d} x}\right),$

where $x_{0}$ denotes the location of the classical turning point. For the sake of completeness, we note in passing that in the case where there are two classical turning points, say $x_{1}$ and $x_{2}$, which would not be of interest for scattering purpose, one may show

$\Delta B=S(E_{n})=2\pi \hbar \left(n-{\frac {1}{2}}\right),$

as an analogous expression to the classical action found in Bohr-Sommerfeld Quantization. In the case where there is only one classical turning point $x_{0}$, which is of interest for investigating a scattering system, a more involved approach is needed (see below).

Applications to scattering theory

Applying the semiclassical approximation techniques to a scattering system is relatively straightforward. The theoretical treatment of a scattering system states (i) mass $m$ $\rightarrow$ reduced mass $\mu$, (ii) potential $V$ $\rightarrow$ effective potential $V_{\rm {eff}}$, and (iii) starting position $a$ $\rightarrow$ far limit $\infty$. Applying these changes to Bohr-Sommerfeld Quantization, we have a situation where the phase of the wave function oscilates on a nearly constant background provided by the potential, with the fictitious particle with mass $\mu$ coming in from $\infty$, rebounding from the inner wall of the effective potential $V_{\rm {eff}}$, and returning to $\infty$. During this process, the particle or the wave function, then, picks up an extra phase angle $\Delta B$ whose 1-D analytical computation is shown above. When we apply the same changes to JWKB approximation, in analogy to the 1-D case shown above, we obtain the 3-D wave function at far limit to be

$\psi _{l}^{(>)}(r)\approx \sin \left({\frac {\pi }{4}}+\int _{r_{0}}^{r}|p_{l}(r')|{\mathrm {d} r'}\right),$

where

${\frac {p_{l}^{2}(r')}{2\mu }}=E-V_{\rm {eff}}(r')=E-V(r')-{\frac {\left(l+{\frac {1}{2}}\right)^{2}}{r'^{2}}}.$

Rewriting the far limit wave function as

$\psi _{l}^{(>)}(r)=\sin \left({\frac {\pi }{4}}+{\frac {1}{\hbar }}\left\{\int _{r_{0}}^{r}\left[|p_{l}(r')|-{\sqrt {2\mu E}}\right]{\mathrm {d} r'}+{\sqrt {2\mu E}}(r-r_{0})\right\}\right),$

and comparing it with the form of the far limit of the exact quantum solution

$\psi _{l}^{(>),({\rm {exact}})}(r)\approx {\frac {\sqrt {2\mu E}}{\hbar }}r\times j_{l}\left({\frac {\sqrt {2\mu E}}{\hbar }}r\right),$

where $j_{l}$ is a spherical Bessel function of the first kind, together with the large $r$ approximation of $j_{l}$

$j_{l}^{(>)}(kr)\approx {\frac {\sin \left(kr-{\frac {l\pi }{2}}\right)}{kr}},$

we see that the accumulated or extra phase picked up via scattering in the limit $r\rightarrow \infty$ is given by

$\delta _{l}^{\rm {JWKB}}=\left(l+{\frac {1}{2}}\right){\frac {\pi }{2}}-kr_{0}+\int _{r_{0}}^{\infty }[k(r')-k]{\mathrm {d} r'},$

where we used

${\frac {|p(r')|}{\hbar }}=k(r')$

and

${\frac {\sqrt {2\mu E}}{\hbar }}=k.$

The usefulness of the semiclassical $\delta _{l}$ expression above is apparent given the wide use of $\delta _{l}$ in finding the following scattering related quantities. The list includes but is not limited to the total cross section

$\sigma ={\frac {4\pi }{k^{2}}}\sum _{l=0}^{\infty }(2l+1)\sin ^{2}\delta _{l},$

the scattering length

$a_{l}(k)={\frac {\tan \delta _{l}(k)}{k}},$

etc.

Connection to Classical and Quantum Scattering

Previously, we have shown that the classical action in semiclassical physics corresponds to the integral of the momentum of a given system over displacement, which, in 3-D, is written as

$S({\vec {r}};E,b)=\int _{{\vec {r}}_{\rm {initial}}}^{\vec {r}}{\vec {p}}({\vec {r}}')\cdot {\mathrm {d} {\vec {r}}'},$

where $b$ is the impact parameter. Dividing the coordinates into the radial and the angular parts, namely,

${\vec {p}}\cdot {\mathrm {d} {\vec {r}}}=p_{r}{\mathrm {d} r}+p_{\varphi }r{\mathrm {d} \varphi },$

and using the fact that the difference between the classical action in the presence of the potential and the classical action in the absence of the potential is the scattering phase shift $S(E,b)$ when integrated along a closed path, we write in general, with $\chi '(E,b)$ the deflection angle and $L$ the angular momentum $p_{\varphi }r$ of the system,

$S(E,b)=\Delta (E,b)-L\chi '(E,b),$

where

$\Delta (E,b)=\oint p_{r}(r'){\mathrm {d} r'}-\oint p_{r}^{V=0}(r'){\mathrm {d} r'}={\frac {\delta _{l}^{\rm {JWKB}}}{2\hbar }},$

which has the name "classical phase", and

$L\chi '=\int _{0}^{\pi -\chi '}p_{\varphi }r{\mathrm {d} \varphi }-\int _{0}^{\pi }p_{\varphi }r{\mathrm {d} \varphi }.$

We note that, with

$L=\mu v_{\infty }b,$

where $v_{\infty }$ is the asymptotic relative velocity of the two bodies participating in a scattering process, and

$p_{r}(r')=2\mu v_{\infty }{\sqrt {1-{\frac {b^{2}}{r'^{2}}}-{\frac {V(r')}{E}}}},$

the first $L$ derivative of $\Delta$ becomes

${\frac {\partial \Delta }{\partial L}}{\bigg |}_{E}=\chi (E,b),$

so that

${\frac {\partial S}{\partial L}}{\bigg |}_{E}=\chi (E,b)-\chi '.$

Applying the principle of least action, the action derivative, then, must read zero, meaning $\chi (E,b)=\chi '$, hence it is the classically allowed path. Said differently, the waves that are on the paths near the classically allowed one constructively interfere with one another whereas those that are not interfere destructively.

Comparing the two momenta expressions $p_{r}(r')$ above and $p_{l}(r')$ in the previous section, we obtain

$b={\frac {\lambda }{2\pi }}\left(l+{\frac {1}{2}}\right),$

where

$\lambda \equiv {\frac {2\pi }{k}}$

with $k$ defined in the previous section. The expression bears an importance as it connects the specific impact parameters with corresponding angular momentum quantum numbers, which may be used in the following ways.

Recalling the quantum mechanical expression for the total cross section above, we may now rewrite $\sigma$ as

$\sigma =8\pi \int _{0}^{\infty }b\sin ^{2}\left({\frac {\Delta (E,b)}{2\hbar }}\right){\mathrm {d} b}.$

In order to see the difference that semiclassical methods make, suppose we have a hard sphere (radius $R$) scattering, for instance. The evaluation of the integral results in $\sigma =2\pi R^{2}$, which is twice the classical value. The existence of additional contribution to $\sigma$, then, demonstrates the ability of semiclassical method capturing the effect of diffraction of the incident plane wave around the target.

This time, we take a look at the JWKB phase shift $\delta _{l}^{\rm {JWKB}}$. Instead of comparing the semiclassical wave function with the form of the exact quantum solution as in previous section, we write directly;

$\delta _{l}^{\rm {JWKB}}=2\mu v_{\infty }\left[\int _{r_{0}}^{\infty }\left(1-{\frac {b^{2}}{r^{2}}}-{\frac {V(r)}{E}}\right)^{\frac {1}{2}}{\mathrm {d} r}-\int _{b}^{\infty }\left(1-{\frac {b^{2}}{r^{2}}}\right)^{\frac {1}{2}}{\mathrm {d} r}\right].$

In the limit of large angular momentum, since the contribution from the potential $V(r)$ would be small, the turning point $r_{0}$ would approximately be $b$. Combining the two integrals with the common factor $1-b^{2}/r^{2}$ and using Taylor expansion up to first order in ${\frac {V(r)/E}{(1-b^{2}/r^{2})^{1/2}}}$, one can show

$\delta _{l}^{\rm {JWKB}}\approx -{\frac {1}{2}}\int _{b}^{\infty }{\frac {V(r)/E}{\left(1-{\frac {b^{2}}{r^{2}}}\right)^{\frac {1}{2}}}}{\mathrm {d} r},$

which was obtained by Massey and Mohr in 1934 . The expression may then be used to compute the total cross section given the long-range potential of the form $\sim {\frac {1}{r^{n}}}$, which is known as the Massey-Mohr formula.

Finally, consider the partial wave expansion of the (axially-symmetric) scattering amplitude, the amplitude of the scattered wave function whose radial dependence is $\sim {\frac {e^{ikr}}{r}}$,

$f(\theta )={\frac {1}{2ik}}\sum _{l=0}^{\infty }(2l+1)(e^{2i\delta _{l}}-1)P_{l}(\cos \theta ),$

where $\delta _{l}$ is the extra phase picked up by the presence of potential in the scattering process associated with the angular quantum number $l$ and $P_{l}$ is the Legendre polynomials. Assuming those terms in the sum with $l\theta \leq 1$ are small, replacing $P_{l}(\cos \theta )$ in the sum with an approximate formula that holds for $l\theta \geq 1$

$P_{l}(\cos \theta )\approx {\frac {\sin \left[\left(l+{\frac {1}{2}}\right)\theta +{\frac {\pi }{4}}\right]}{\sqrt {{\frac {\pi }{2}}\left(l+{\frac {1}{2}}\right)\sin \theta }}}$

results in

$f(\theta )={\frac {-1}{\sqrt {\lambda \sin \theta }}}\int _{0}^{\infty }{\sqrt {b}}(e^{i\varphi _{+}}-e^{i\varphi _{-}}){\mathrm {d} b},$

where we first replaced $l$ and $k$ with the correspondent $b$ and $\lambda$ expressions respectively (see above) then used integral approximation on $b$ and

$\varphi _{\pm }=2\delta (b)\pm \left({\frac {2\pi b\theta }{\lambda }}+{\frac {\pi }{4}}\right).$

Now relaxing the positivity condition of $b$, with $\theta =|\chi '|$, we obtain

$f(\theta )={\frac {e^{-i{\frac {\pi }{4}}}}{\sqrt {\lambda \sin \theta }}}\int _{-\infty }^{+\infty }{\sqrt {b}}e^{i{\frac {S(\theta ,b)}{\hbar }}}{\mathrm {d} b},$

where we have $\theta =\chi '>0$ for the repulsive trajectory with $b>0$ and $-\theta =\chi '<0$ for the attractive trajectory with $b<0$. We note that we have replaced the argument $E$ of $S$ with $\theta$ since $\theta (E,b)=|\chi '(E,b)|$ is a function of $E$ and $b$. Incorporating the stationary phase or the steepest descent method, justified by the fact that the action must not change rapidly in $b$ if the integral were not to be 0 in consistency with the least action principle, $f(\theta )$ now evaluates to

$f(\theta )=\sum _{b_{i}(\theta )}{\sqrt {\frac {2\pi b_{i}(\theta )}{\lambda \sin \theta \Delta ''(b_{i})/\hbar }}}e^{iS(\theta ,b_{i})/\hbar },$

where we used

${\frac {\partial ^{2}S}{\partial b^{2}}}={\frac {\partial ^{2}\Delta }{\partial b^{2}}}$

and $b_{i}$ denotes the stationary point. Obviously, if there are more than one $b_{i}$, the different paths will cause interference phenomena. Solving for the spacing of the maxima given two different $b_{i}$'s, namely $b_{1}(\theta )$ and $b_{2}(\theta )$, we obtain

$\Delta \theta ={\frac {\lambda }{|b_{1}(\theta )-b_{2}(\theta )|}},$

which is nothing but the spacing of interference maxima in Young's double-slit experiment, for which $\Delta \theta =\lambda /2\pi d$; multiple amplitudes will lead to multiple sets of interference minima and maxima. In analogy to the double-slit experiment, we conclude that the greater the spacing between the $b_{i}$'s are, the closer the spacing of the fringes, a.k.a. diffraction oscillations.

Limitations

Semiclassical approximations taken here do have their own limitations. Among many other possible limitations, the notable one includes the breakdown of $f(\theta )$ expression at $\theta =\theta _{R}$, the rainbow angle. This is so, because the stationary phase approximation does not hold at the rainbow angle as the second derivative of action disappears. This failure of the primitive semiclassical approximation may be saved by

$S(\theta ,b)\approx S(\theta _{R},b_{R})+{\frac {1}{6}}{\frac {\partial ^{3}S}{\partial b^{3}}}{\bigg |}_{b=b_{R}}(b-b_{R})^{3},$

where $b_{R}$ is the rainbow impact parameter. Because now the integral that needs to be done includes an exponential in a cubic power of $b$, the final form of the integral is an Airy function.