34. Time-Dependent Perturbation Theory

Introduction¶

Time-dependent perturbation theory applies to Hamiltonians of the form

where $H_0$ is solvable and $H_1$ is treated as a perturbation. In bound state perturbation theory (see 23. Bound-State Perturbation Theory) we were interested in the the shifts in the energy levels and eigenfunctions of the unperturbed system induced by the perturbation $H_1$, which was assumed to be time-independent. In time-dependent perturbation theory, on the other hand, we are usually interested in time-dependent transitions between eigenstates of the unperturbed system induced by the perturbation $H_1$. In time-dependent perturbation theory the perturbation $H_1$ is allowed to depend on time, as indicated by (1), but it does not have to be time-dependent, and in fact in practice often it is not. Time-dependent perturbation theory is especially useful in scattering theory, problems involving the emission and absorption of radiation, and in field theoretic problems of various kinds. Such problems will occupy us for the rest of the course.

Time-dependent transitions are usually described by the transition amplitude, defined as the quantity

where $U(t)$ is the exact time evolution operator for the Hamiltonian (1), and where $\ket{i}$ and $\ket{f}$ are two eigenstates of the unperturbed Hamiltonian $H_0$ (the “initial” and “final” states). The transition amplitude can be regarded as simply a matrix element of the exact time evolution operator in the eigenbasis of the unperturbed Hamiltonian, but it is also the amplitude to find the system in state $\ket{f}$ when it was known to be in the state $\ket{i}$ at $t=0$. Thus, the absolute square of the transition amplitude is the transition probability, the probability to make the transition $i\to f$ in time $t$. Often we are interested in transitions to some collection of final states, in which case we must sum the transition probabilities over all these states.

In these notes we shall develop the basic formalism of time-dependent perturbation theory and study some simple examples. For the most part we shall simply follow the formulas to see where they lead, without examining the conditions of validity or the limitations of the results. We will address those questions in the context of some examples, both in these and in successive notes.

Time-Evolution Operators¶

Let us denote the unperturbed time-evolution operator by $U_0(t)$ and the exact one by $U(t)$. Since the full Hamiltonian may depend on time, the exact time-evolution operator actually depends on two times, $t$ and $t_0$, but we shall set $t_0=0$ and just write $U(t)$. See The Time-Evolution Operator. These operators satisfy the evolution equations,

which are versions of (13). Since $H_0$ is independent of time, (3) can be solved,

but if $H_1$ depends on time then there is no similarly simple expression for $U(t)$.

The Interaction Picture¶

The interaction picture is a picture that is particularly convenient for developing time-dependent perturbation theory. It is intermediate between the Schr"odinger and Heisenberg pictures that were discussed in The Heisenberg Picture. Recall that in the Schr"odinger picture, the kets evolve in time but the operators do not (at least if they have no explicit time dependence), while in the Heisenberg picture the kets do not evolve but the operators do. In the interaction picture, the time evolution of the kets in the Schr"odinger picture that is due to the unperturbed system $H_0$ is stripped off, leaving only the evolution due to the perturbation $H_1$. This is presumably slower than the evolution due to the whole Hamiltonian $H$, since $H_1$ is assumed small compared to $H_0$. We will not attempt to state precisely what “small” means in this context, rather we will develop the perturbation expansion as a power series in $H_1$ and then examine its limitations in various examples.

In the following we shall use an $S$ subscript on kets and operators in the Schr"odinger picture, and an $I$ on those in the interaction picture. We will not use the Heisenberg picture in these notes. If the subscript is omitted, the Schr"odinger picture will be assumed. The the relation between the kets in the Schr"odinger and interaction pictures is

Compare this to (17), which shows the relation between kets in the Schr"odinger and Heisenberg pictures. The difference is that in (6) we are only stripping off the evolution due to $H_0$, not the whole time evolution. Notice that at $t=0$ the Schr"odinger and interaction picture kets agree,

As for operators in the interaction picture, they are defined by

In practice $A_S$ is often time-independent, but $A_I$ is always time-dependent. Compare this to (19), which shows the relation between operators in the Schr"odinger and Heisenberg pictures.

Evolution in the Interaction Picture¶

Let us define $W(t)$ as the operator that evolves kets in the interaction picture forward from time 0 to final time $t$:

The operator $W(t)$ is a time-evolution operator, but we use the symbol $W$ to avoid confusion with the two other time-evolution operators introduced so far, $U_0(t)$ and $U(t)$.

It is easy to find a relation among these three operators. Substituting (6) and (7) into (9), we have

or, since $\ket{\psi_I(0)}$ is arbitrary,

The operator $W(t)$ is equivalent to first evolving forward for time $t$ under the exact Hamiltonian, then evolving backwards for the same time under the unperturbed Hamiltonian.

The $S$-Matrix¶

For an application in which the interaction picture leads to a interesting perspective, consider a scattering experiment in which a wave packet is directed against a target, described by a potential $U(\xvec)$. We assume that the potential dies off rapidly with distance, and that at the initial time the wave packet is far enough away from the scatterer that it is essentially free. For simplicity we assume that the potential (or other perturbing Hamiltonian) is time-independent.

Initially the wave packet evolves according to the free particle Hamiltonian, since it does not overlap with the potential. That is, the wave packet moves with an average velocity and simultaneously spreads, as studied in Prob. Paragraph. After some time, however, the wave packet begins to interact with the potential, producing a scattered wave that radiates outward from the potential in various directions. Depending on the size of the wave packet and that of the scatterer, some of the wave packet may effectively miss the scatterer and proceed in the forward direction, largely unaffected by the scattering process. In any case, after some time the scattered wave and the remaining part of the incident wave packet move away from the scatterer, and once again evolve according to the free particle Hamiltonian. Thus, the exact time evolution is that of a free particle both at large negative times and large positive times.

Let us now view the evolution of the quantum state in the interaction picture. Initially and for large negative times the wave function in the Schr"odinger picture evolves according the free particle Hamiltonian, so in the interaction picture the wave function does not evolve at all. The wave function in the interaction picture does not begin to change until the wave packet in the Schr"odinger picture starts to interact with the potential. Then the wave packet in the interaction picture does evolve, and continues to do so as long as the Schr"odinger wave function has any overlap with the scatterer. As the Schr"odinger wave function leaves the region of the scatterer, however, the wave function in the interaction picture ceases to evolve, and at large positive times it approaches another, constant wave function.

Thus, the interaction picture kets $\ket{\psi_I(-T)}$ and $\ket{\psi_I(T)}$ approach constant kets as $T\to\infty$. The scattering process associates a given initial state $\lim_{T\to\infty}\ket{\psi_I(-T)}$ with a definite final state $\lim_{T\to\infty}\ket{\psi_I(T)}$. This association is linear, and defines the $S$-matrix. It is usually called a “matrix” but really it is an operator, whose matrix elements in some basis form a matrix. The usual basis is that of free particle states, that is, plane waves. In this form, the $S$-matrix bears a close relationship to the cross .

We can easily express the $S$-matrix (or operator) in terms of time-evolution operators. Since

we have

But

where we have used $U_0^\dagger(T)=U_0(-T)$ and $U^\dagger(T)=U(-T)$. But by the definition of the $S$ operator, this implies

The $S$-matrix is a central object in advanced treatments of scattering theory, but its mathematics involves noncommuting limits and is tricky. In these notes we take a simpler approach to scattering theory, one that involves a straightforward perturbation expansion of the operator $W(t)$ in powers of the perturbing Hamiltonian $H_1$.

The Dyson Series¶

We obtain a differential equation for $W(t)$ by differentiating (11) and using (3),

where we have used $H=H_0+H_1$ in the third equality, whereupon the terms in $H_0$ cancel. The result can be written,

an equation of the same form as (3), but one in which only the perturbing Hamiltonian $H_1$ appears, and that in the interaction picture. By applying both sides of (18) to the initial state $\ket{\psi_I(0)}$, we obtain a version of the Schr"odinger equation in the interaction picture,

It is now easy to obtain a solution for $W(t)$ as a power series in the perturbing Hamiltonian $H_1$. First we integrate (18) between time 0 and time $t$, obtaining

where we have used $W(0)=1$. This is an exact result, but it is not a solution for $W(t)$, since $W(t)$ appears on the right-hand side. But by substituting the left-hand side of (20) into the right-hand side, that is, iterating the equation, we obtain

another exact equation in which the term containing $W(t)$ on the right hand side has been pushed to second order. Substituting (20) into the right-hand side of (21) pushes the term in $W(t)$ to third order, etc. Continuing in this way, we obtain a formal power series for $W(t)$,

This series is called the Dyson series. It is obviously a kind of a series expansion of $W(t)$ in powers of $H_1$.

The Usual Problem¶

Let us assume for simplicity that $H_0$ has a discrete spectrum, and let us write

where $n$ is a discrete index. The usual problem in time-dependent perturbation theory is to assume that the system is initially in an eigenstate of the unperturbed system, what we will call the “initial” state $\ket{i}$ with energy $E_i$,

This is just a special case of (23) (with $n=i$). That is, we assume the initial conditions are

We will be interested in finding the state of the system at a later time,

By applying the Dyson series (22) to this equation, we obtain a perturbation expansion for $\ket{\psi_I(t)}$,

Transition Amplitudes¶

Let us expand the exact solution of the Schr"odinger equation in the interaction picture in the unperturbed eigenstates,

with coefficients $c_n(t)$ that are functions of time. We can solve for these coefficients by multiplying (28) on the left by the bra $\bra{n}$, which gives

showing that $c_n(t)$ is a transition amplitude in the interaction picture, that is, a matrix element of the time-evolution operator $W(t)$ in the interaction picture with respect to the unperturbed eigenbasis. These are not the same as the transition amplitudes (2), which are the matrix elements of $U(t)$ (the time-evolution operator for kets in the Schr"odinger picture) with respect to the unperturbed eigenbasis. There is, however, a simple relation between the two. That is, substituting (11) into (29), we have

where we have allowed $U_0(t)^\hc$ to act to the left on bra $\bra{n}$. We see that the transition amplitudes in the interaction picture and those in the Schr"odinger picture are related by a simple phase factor. The phase factor removes the rapid time evolution of the transition amplitudes in the Schr"odinger picture due to the unperturbed system, leaving behind the slower evolution due to $H_1$. The transition probabilities are the squares of the amplitudes and are the same in either case,

The perturbation expansion of the transition amplitude $c_n(t)$ is easily obtained by multiplying (27) by the bra $\bra{n}$. We write the result in the form,

where the parenthesized numbers indicate the order of perturbation theory, and where

etc.

We now switch back to the Schr"odinger picture in the expressions for the $c_n^{(r)}$. The interaction picture is useful for deriving the perturbation expansion, but the Schr"odinger picture is more convenient for subsequent calculations. For $c_n^{(1)}$, we have

where we have allowed $U_0(t')^\hc$ and $U_0(t')$ to act to the left and right, respectively, on bra $\bra{n}$ and ket $\ket{i}$, bringing out the phase factor $e^{i\omega_{ni}t'}$ where $\omega_{ni}$ is the Einstein frequency connecting unperturbed states $n$ and $i$,

As for $c_n^{(2)}$, it is convenient to introduce a resolution of the identity $\sum \ketbra{k}{k}$ between the two factors of $H_{1I}$ in (34) before switching to the Schr"odinger picture. This gives

From this the pattern at any order of perturbation theory should be clear. In (38), the states $\ket{k}$ are known as intermediate states, the idea being that there is a time sequence as we move from the right to the left in the product of matrix elements, starting with the initial state $\ket{i}$, moving through the intermediate state $\ket{k}$ and ending with the final state $\ket{n}$. Notice that the variables of integration satisfy $t\ge t' \ge t'' \ge 0$.

The Case that $H_1$ is Time-independent¶

This is about as far as we can go without making further assumptions about $H_1$, so that we can do the time integrals. Let us now assume that $H_1$ is time-independent, an important case in practice. Then the matrix elements are independent of time and can be taken out of the integrals, and the integrals that remain are elementary. For example, in (36) we have the integral

so that

The case of $c_n^{(2)}$ is similar but more complicated. We will return to it in 43. Scattering of Radiation by Matter{} when we apply second order time-dependent perturbation theory to the problem of the scattering of photons.

The transition probability $P_n(t)$ can also be expanded in the perturbation series,

Notice that on taking the square there are cross terms, that is, interference terms between the amplitudes at different orders.

For now we assume that the final state we are interested in is not the the initial state, that is, we take the case $n \ne i$ so the first term in (41) vanishes, and we work only to first order of perturbation theory. Then we have

The transition probability depends on time and on the final state $n$. As for the time dependence, it is contained in the first factor in the parentheses, while both this factor and the matrix element depend on the final state $\ket{n}$. However, the factor in the parentheses depends on the state $n$ only through its energy $E_n$, which is contained in the Einstein frequency $\omega_{ni}$, while the matrix element depends on all the properties of the state $\ket{n}$, for example, its momentum, spin, etc.

The case $n=i$ is interesting, too, as it gives the amplitude and probability for the system to remain in the initial state. This case will be analyzed in some detail in 44. Natural Line Width, Long-Time Behavior and the Lamb Shift.

The Case of Time-Periodic Perturbations¶

Another case that is important in practice is when $H_1$ has a periodic time dependence of the form

where $\omega_0$ is the frequency of the perturbation and $K$ is an operator (generally not Hermitian). We call the first and second terms in this expression the positive and negative frequency components of the perturbation, respectively. This case applies, for example, to the interaction of spins or atoms with a given, classical electromagnetic wave, a problem that is studied in 35. The Photoelectric Effect.

To find the transition amplitude in this case we substitute (43) into (36) and perform the integration, whereupon we obtain two terms,

For any given final state $n$, both these terms are present and contribute to the transition amplitude, and when we square the amplitude to get the transition probability, there are cross terms (interference terms) between these two contributions to the amplitude.

Often, however, we are most interested in those final states to which most of the probability goes, which are the states for which one or the other of the two denominators in (45) is small. For these states we have $\omega_{ni}\mp\omega_0\approx0$, or

We see that the first (positive frequency) term is resonant when the system has absorbed a quantum of energy $\hbar\omega_0$ from the perturbation, whereas the second (negative frequency) term is resonant when the system has given up a quantum of energy $\hbar\omega_0$ to the perturbation. We call these two cases absorption and stimulated emission, respectively.

Taking the case of absorption, and looking only at final states $\ket{n}$ that are near resonance ($E_n \approx E_i + \hbar\omega_0$), we can write the transition probability to first order of perturbation theory as

Similarly, for nearly resonant final states in stimulated emission, we have

These formulas may be compared to (42). In all cases, $P_n$ has a dependence on time that is described by functions of a similar form.

How $P_n$ Depends on Time¶

Let us fix the final state $\ket{n}$ and examine how the probability $P_n(t)$ develops as a function of time in first order time-dependent perturbation theory. To be specific we will take the case of a time-independent perturbation and work with (42), but with $\omega_{ni}$ replaced by $\omega_{ni}\pm\omega_0$ and $H_1$ replaced by $K$ or $K^\hc$, everything we say also applies to absorption or stimulated emission.

Obviously $P_n(0) = 0$ (because $n\ne i$ and all the probability lies in state $\ket{i}$ at $t=0$). At later times we see that $P_n(t)$ oscillates at frequency $\omega_{ni}$ between 0 and a maximum proportional to $1/\omega_{ni}^2$. The frequency $\omega_{ni}$ measures how far the final state is “off resonance,” that is, how much the final energy differs from the initial energy. In some applications this difference can be regarded as a failure to conserve energy, which is allowed over finite time intervals by the energy-time uncertainty relation $\Delta E \Delta t >\approx \hbar$. If $\omega_{ni}$ is large, the probability $P_n(t)$ oscillates rapidly between zero and a small maximum. But as we move the state $\ket{n}$ closer to the initial state $\ket{i}$ in energy, $\omega_{ni}$ gets smaller, the period of oscillations becomes longer, and the amplitude grows.

If there is a final state $\ket{n}$ degenerate in energy with the initial state $\ket{i}$ (not the same state since we are assuming $n\ne i$), then $\omega_{ni}=0$ and the time-dependent factor in (42) takes on its limiting value, which is $t^2/4$. In this case, first order perturbation theory predicts that the probability $P_n(t)$ grows without bound, obviously an absurdity after a while since we must have $P_n \le 1$. This is an indication of the fact that at sufficiently long times first order perturbation theory breaks down and we must take into account higher order terms in the perturbation expansion. In fact, to get sensible results at such long times, it is necessary to take into account an infinite number of terms (that is, to do some kind of summation of the series). But at short times it is correct that $P_n$ for a state on resonance grows as $t^2$.

These considerations are important when the system has a discrete spectrum, for example, when a spin is interacting with a time-periodic magnetic field or when we are looking at a few discrete states of an atom in the presence of laser light. These are important problems in practice. Recall that in 14. Spins in Magnetic Fields\ we solved the Schr"odinger equation exactly for a spin in a certain kind of time-periodic magnetic field, but in more general cases an exact solution is impossible and we may have to use time-dependent perturbation theory. It is interesting to compare the exact solution presented in 14. Spins in Magnetic Fields\ with the perturbative solutions presented here, to see the limitations of the perturbative solutions.

In such problems with a discrete spectrum, the resonance condition $\omega_{ni}=0$ may not be satisfied exactly for any final state $n\ne i$. In fact, in problems of emission and absorption, if the frequency $\omega_0$ of the perturbation is chosen randomly, then it is unlikely that the resonant energy $E_i \pm \hbar\omega_0$ will exactly coincide with any unperturbed energy eigenvalue. In that case, first-order theory predicts that the transition probability to all final states just oscillates in time.

On the other hand, if the final states are members of a continuum, then there are an infinite number of final states arbitrarily close to the initial state in energy. For those cases, we must examine how the transition probability depends on energy.

How $P_n$ Depends on Energy¶

Now let us fix the time $t$ and examine how the expression for $P_n(t)$ in first order perturbation theory, (42), depends on the energy $E_n$ of the final state $\ket{n}$ (working for simplicity with the case of a time-independent perturbation). We shall concentrate on the energy dependence of the time-dependent factor in the parentheses, remembering that the matrix element also depends on the energy (and other parameters) of the final state. To do this we plot the function $\sin^2(\omega t/2)/\omega^2$ as a function of $\omega$, as shown in Figs.~Figure 1 and Figure 2 for two different times. In the plot, $\omega$ is to be identified with $\omega_{ni}=(E_n-E_i)/\hbar$, so that $\omega$ specifies the energy of the final state and $\omega=0$ is the resonance (energy conserving) condition.

The curve consists of oscillations under the envelope $1/\omega^2$, with zeroes at $\omega=(2n\pi/t)$. The central lobe has height $t^2/4$ and width that is proportional to $1/t$, so the area of the central lobe is proportional to $t$. As $t$ increases, the central lobe grows in height and gets narrower, a behavior that reminds us of functions that approach a $\delta$-function, but in this case the limit is not a $\delta$-function because the area is not constant. In fact, the total area is given exactly by an integral that can be evaluated by contour integration,

showing that the area is indeed proportional to $t$. Thus if we divide by $t$ we do get a $\delta$-function as $t\to \infty$,

For fixed $\omega\ne0$ the function under the limit in this expression approaches 0 as $t\to\infty$, while exactly at $\omega=0$ it grows in proportion to $t$, with a constant total area. This is exactly the behavior that produces a $\delta$-function in the limit.

In physical applications we never really go to infinite time, rather we work with times long enough that there is negligible error in replacing the function on the left-hand side of (50) by its limit. To deal with the case of finite time, we introduce the notation,

which defines the function $\Delta_t(\omega)$. Then the limit (50) can be written,

As we shall see when we take up some applications, the $\delta$-function in (50) enforces energy conservation in the limit $t\to\infty$, that is, only transitions to final states of the same energy as the initial state are allowed in that limit. At finite times, transitions take place to states in a range of energies about the initial energy, given in frequency units by the width of the function $\Delta_t(\omega)$. But as we have seen this width is of order $1/t$, or, in energy units, $\hbar/t$. This is an example of the energy-time uncertainty relation, $\Delta E \Delta t >\approx \hbar$, indicating that a system that is isolated over a time interval $\Delta t$ has an energy that is uncertain by an amount $\Delta E >\approx \hbar/\Delta t$.

Now we can write the transition probability (42) as

This applies in first order perturbation theory, in the case $n \ne i$.

Cross and Differential Cross In preparation for applications to scattering, we make a digression to define and discuss cross s and differential cross s. These concepts are best understood in a classical context, but most of the ideas carry over without trouble into quantum mechanics.¶

We first discuss classical scattering from a fixed target, which is illustrated in Figure 3. An incident beam of particles of momentum $\pvec$ and uniform density is directed against a target, illustrated by the shaded region in the figure. The target is described by a potential $U(\xvec)$. The origin of a coordinate system is located in or near the target, and in the figure the beam is directed in the $z$-direction. A transverse plane $T$ is erected perpendicular to the beam at a large, negative value of $z$, where the potential $U(\xvec)$ is negligible. The plane $T$ is parallel to the $x$-$y$ plane, and when the particles cross it their momentum is purely in the $z$-direction, since no interaction with the potential has occurred yet. The negative $z$-axis is extended back to the plane $T$ along a center line $C$, intersecting it at point $O$, which serves as an origin in the plane.

The trajectory of one particle is illustrated in the figure. It crosses the plane $T$ at a position described by the impact parameter, a vector $\bvec$, which goes from the origin $O$ in the plane to the interpoint. The impact parameter only has $x$- and $y$-components. The particle continues forward and interacts with the potential, going out in some direction ${\hat{\mathbf{n}}}=(\theta,\phi)$. This direction is defined asymptotically, that is, it is the direction of the particle’s momentum when it is once again at a large distance from the target. The outgoing direction is a function of the impact parameter,

which can be determined by integrating the equations of motion from given initial conditions on the plane $T$.

(54) defines a mapping from the transverse plane $T$ to the unit sphere of outgoing directions, and the differential cross is closely related to the Jacobian of this map. That is, $d\sigma/d\Omega$ is the ratio of an infinitesimal area element in $T$ to a corresponding infinitesimal solid angle on the sphere.

To visualize this in more detail, we construct a small cone of solid angle $\Delta\Omega$, centered on the outgoing direction ${\hat{\mathbf{n}}}=(\theta,\phi)$, as illustrated in Figure 4. The particles whose asymptotic, outgoing directions lie inside this cone cross the plane $T$ inside an area indicated by the shaded area in plane $T$ in the figure. This area represents the portion of the incident flux of particles that is directed into the small cone by the scattering process. It defines the *differential cross * $d\sigma/d\Omega$ by the formula,

The differential cross $d\sigma/d\Omega$ is a function of $(\theta,\phi)$.

It is possible that more than one impact parameter $\bvec$ will lead to the same final outgoing direction ${\hat{\mathbf{n}}}$, for example, depending on $\bvec$ a particle may orbit the target one or more times before emerging in a definite final direction. In that case the area in (55) should be replaced by a sum of areas, each corresponding to the same small solid angle $\Delta\Omega$ in the outgoing direction.

Another point of view deals with counting rates. We use the symbol $w$ to stand for a rate, with dimensions of $time^{-1}$, for example, number of particles per unit time or probability per unit time. A detector situated in Figure 4 so as to intercept all particles coming out in the cone will have a counting rate given by the rate at which particles cross the shaded area in plane $T$. We denote this counting rate by $(dw/d\Omega)\Delta\Omega$. But this is just the flux of incident particles times the shaded area, that is,

As for the incident flux, it is given by

where $n$ is the number of particles per unit volume in the incident beam and $\vvec=\pvec/m$ is the incident velocity. The magnitude $J_inc = |\Jvec_inc|$ appears in (56), since the transverse plane is orthogonal to the velocity $\vvec$. It is obvious that the counting rate is proportional to the incident flux, so (56) gives another way of thinking about the differential cross : It is the counting rate, normalized by the incident flux.

The total scattering rate $w$ is the rate at which particles are scattered at any nonzero angle. It is the integral of the differential scattering rate,

It is related to the total cross $\sigma$ by

where

In classical scattering, the total cross is often infinite, due to a large number of particles that are scattered by only a small angle.

The relation (56) applies in the case of a single scatterer located inside the incident beam. In many practical circumstances there are multiple, identical scatterers. An example is Rutherford’s scattering experiment, in which a beam of $\alpha$-particles is directed against a gold foil. The individual gold nuclei are the scatterers, of which there are a large number in the region of the foil crossed by the beam.

In this case we can speak of the scattering rate (differential or total) per unit volume of the scattering material, which is $n_inc n_targ v$ times the cross (differential or total), where $n_inc$ and $n_targ$ are the number of incident particles and scatterers per unit volume, respectively. Then integrating over the interaction region, we find that (56) is replaced by

where $v$ is again the velocity of the beam. In this formula, both $n_inc$ and $n_targ$ can be functions of position, as they often are in practice. We are assuming that $v$ is constant, so that it can be taken out of the integral (the beam consists of particles of a given momentum).

Another case that is common in practice is when there are two beams intersecting one another. In this case it is easiest to work in the center-of-mass frame, in which the momenta of the particles in the two beams are equal and opposite. As we have seen in 16. Central Force Motion, when two particles interact by means of a central force potential, their relative motion is described by a pseudo-one-body problem. Thus, the results for scattering from a fixed target with central force potential $U$ can be transcribed into those for scattering of two particles in the center of mass frame, in which the position vector $\xvec$ of the beam particle relative to the scatterer is replaced by $\rvec=\xvec_2 -\xvec_1$, the relative position vector between two particles in the two beams, and where the mass $m$ of the beam particle is replaced by the reduced mass $\mu$ of the two particle system. Then the transition rate is given by a modified version of (61),

where $n_1$ and $n_2$ are the densities of the two beams, which may be functions of position, and where $v$ is now the relative velocity of the two beams. The integral is taken over the region where the two beams overlap.

The transformation between the lab positions and momenta of the two particles, $(\xvec_1,\pvec_1)$ and $(\xvec_2,\pvec_2)$, and the center of mass position and momentum $(\Rvec,\Pvec)$ and the relative position and momentum $(\rvec,\pvec)$, is given by Eqs.~((44){}--(45)) and ((50){}--(53)). The definitions of $\Rvec$, the center of mass position, and $\Pvec=\pvec_1+\pvec_2$, the total momentum of the two particle system as seen in the lab frame, are clear physically. So also is the definition of $\rvec =\xvec_2 -\xvec_1$, the relative separation between the two particles. But the definition of $\pvec$, the momentum conjugate to $\rvec$,

requires some comment. (This is essentially (53).

We offer two interpretations of this equation. First, we compute $\pvec/\mu$, where $\mu$ is the reduced mass, given by (58), or, equivalently,

Dividing this into (63) gives

or,

where

is the relative velocity. In other words, we have a version of the usual formula $\pvec=m\vvec$, where $\pvec$ is the momentum conjugate to the relative position vector, $\vvec$ is the relative velocity, and $m$ is replaced by the reduced mass.

Another interpretation is to imagine the two particle system as seen in the center-of-mass frame, in which $\Pvec=0$. If we let $\pvec_2=\qvec$, then $\pvec_1 = -\qvec$, which when substituted into (63) gives $\pvec=\qvec$. Thus, the momentum $\pvec$, defined as the conjugate to the relative position $\rvec$, or, equivalently, by (63), is in fact the momentum of one or the other of the particles (to within a sign) as seen in the center-of-mass frame.

Application: Potential Scattering¶

We return now to time-dependent perturbation theory, and examine an application, namely, potential scattering of a spinless particle from a fixed target, described by a potential $U(\xvec)$. We let the unperturbed Hamiltonian be $H_0=\pvec^2/2m$ and we take the perturbation to be $H_1=U(\xvec)$, where $U$ is some potential. We do not assume the potential is rotationally invariant, but it should approach zero sufficiently fast as $r=|\xvec|\to\infty$. We will examine this condition more carefully in Asymptotic Form of the Radial Wave Functions.

The unperturbed eigenstates are free particle solutions, which we take to be plane waves. In order to deal with discrete final states, we place our system in a large box of side $L$ and volume $V=L^3$, and we adopt periodic boundary conditions. This is equivalent to dividing the universe up into boxes and demanding that all the physics be periodic, that is, the same in all the boxes. We shall assume that the size of the box is much larger than the range of the potential $U(\xvec)$. When we are done we take $V\to\infty$ to get physical results. We denote the unperturbed eigenstates by $\ket{\kvec}$, with wave functions

so that

Here we are normalizing the eigenfunctions to the volume of the box, and integrating over the volume of the box when forming scalar products as in (69). The quantized values of $\kvec$ are given by

where $\nvec=(n_x,n_y,n_z)$ is a vector of integers, each of which ranges from $-\infty$ to $+\infty$. The unperturbed eigenstates can be represented as a lattice of points in $\kvec$-space, in which the lattice spacing is $2\pi/L$ and the density is $(L/2\pi)^3 = V/(2\pi)^3$. We let $\ket{\kvec_i}$ be an incident plane wave (the initial state), and $\ket{\kvec}$ be some final state.

Notice that the initial state is somewhat unrealistic from a physical standpoint. The initial state is a plane wave $\exp(i\kvec_i\cdot\xvec)$ that fills up all of space, including the region where $U(\xvec)$ is appreciably nonzero. Let us suppose it is a potential well. Thinking in classical terms, it is as if all of space is filled with particles with exactly the same momentum and energy, even the particles in the middle of the well. Of course a particle coming in from infinity and entering a potential well will gain kinetic energy, and the direction of its momentum will change (this is the scattering process in action). But the particles of our initial state in the middle of the potential have the same kinetic energy and momentum as the particles that are coming in from infinity. Obviously this initial condition would be difficult to establish in practice. Nevertheless, it turns out that these particles with the wrong energy and momentum in the initial state do not affect the transition probabilities after sufficiently long times, and so they do not affect the cross that we shall compute. All they do is give rise to short-time transients that can be regarded as nonphysical since they arise from the artificialities of the initial conditions. We shall say more about these transients below, but for now we shall just continue to follow the formulas of time-dependent perturbation theory.

In this application the perturbing Hamiltonian is time-independent, so the transition amplitude in first order perturbation theory is given by (40), with the change of notation $\ket{i} \to \ket{\kvec_i}$, $\ket{n} \to \ket{\kvec}$, etc. The transition amplitude is

where

and the transition probability is

where we sum over some set of final states for which $\kvec\ne\kvec_i$. Note that (72) implies

where $v$ is the classical velocity of the beam particles. It is also the group velocity of free particle wave packets.

Which final states $\kvec$ do we sum over? This depends on what question we wish to ask. If we are interested in the transition rate to any final state, then we sum over all of them (every lattice point in $\kvec$-space except $\kvec_i$). Often, however, we are interested in more refined information. In the present case, let us sum over all final states (lattice points) that lie in a cone of small solid angle $\Delta\Omega\ll 1$ in $\kvec$-space, as illustrated in Figure 5. Let the cone be centered on a direction ${\hat{\mathbf{n}}}_f$, a given unit vector pointing toward some counting device in a scattering experiment. Then we define a “final wave vector” $\kvec_f$ by requiring that $\kvec_f$ have the same direction as ${\hat{\mathbf{n}}}_f$, and that it satisfy conservation of energy,

that is, $k_f = k_i$. Then

This is only a definition, and although $\kvec_f$ satisfies energy conservation, notice that the states in the cone that we sum over include states of all energies, from 0 to $\infty$.

With this understanding of the states we sum over in the expression (73), we see that we are computing the probability as a function of time that the system will occupy a momentum state lying in the cone. Under some circumstances transition probabilities are proportional to time, and then we can refer to a transition rate as the probability per unit time for the process in question. We will generally use the symbol $w$ for transition rates. In the present case, we divide the probability (73) by $t$ and write

where $dw/d\Omega$ is the transition rate per unit solid angle, a quantity that is generally a function of direction, in this case, the direction ${\hat{\mathbf{n}}}_f$.

Notice that the factors of $t$ have canceled in (77), but the right-hand side still depends on $t$ through $\Delta_t(\omega)$. If, however, $t$ is large enough that $\Delta_t(\omega)$ can be replaced by $\delta(\omega)$, then the right hand side does become independent of $t$, and the transition rate is meaningful. We see that at short times we do not have a transition rate, but that at longer times we do.

For now let us assume that $t$ is large enough that $\Delta_t(\omega)$ can be replaced by $\delta(\omega)$, since this gives the simplest answer. This condition will be examined quantitatively in Short-Time Behavior. Then using (72), we can transform $\delta(\omega)$ by the rules for $\delta$-functions,

where we use (74).

We must also take the limit $V\to\infty$ to obtain physical results. In this limit, the initial wave function $\psi_\kvec(\xvec)$ loses meaning (it goes to zero everywhere, since it is normalized to unity over the volume of the box), as does the differential transition rate $dw/d\Omega$. However, the differential cross $d\sigma/d\Omega$, which is the differential transition rate normalized by the incident flux, is well defined in the limit. The incident flux is