
<p>The S-Matrix<sup style="top: -0.3615em;">1 </sup></p><p>D. E. Soper<sup style="top: -0.3615em;">2 </sup><br>University of Oregon <br>Physics 634, Advanced Quantum Mechanics <br>November 2000 </p><p>1 Notation for states </p><p>In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization </p><p>3</p><p></p><ul style="display: flex;"><li style="flex:1">~</li><li style="flex:1">~</li></ul><p>hp~ |ki = (2π) δ(p~ − k). </p><p>(1) <br>Recall that in a relativistic theory there is an extra factor of 2E on the right </p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2 1/2 </li></ul><p></p><p>~</p><p></p><ul style="display: flex;"><li style="flex:1">hand side of this relation, where E = [k + m ] </li><li style="flex:1">.</li></ul><p>We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)i<sub style="top: 0.1495em;">S</sub>. The relation between these is </p><p>|ψ(t)i<sub style="top: 0.1495em;">S </sub>= e<sup style="top: -0.4113em;">−iHt </sup>|ψi. </p><p>(2) <br>Thus these are the same at time zero, and the Schrdinger states obey </p><p>di<br>|ψ(t)i<sub style="top: 0.1495em;">S </sub>= H |ψ(t)i<sub style="top: 0.1495em;">S </sub></p><p>(3) </p><p>dt </p><p>In the Heisenberg picture, the states do not depend on time but the operators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schro¨dinger operator O<sub style="top: 0.1495em;">S </sub>by </p><p>O(t) = e<sup style="top: -0.4113em;">iHt </sup>O<sub style="top: 0.1495em;">S </sub>e<sup style="top: -0.4113em;">−iHt </sup></p><p>(4) <br>Thus </p><p>hψ|O(t)|ψi = <sub style="top: 0.1494em;">S</sub>hψ(t)|O<sub style="top: 0.1494em;">S</sub>|ψ(t)i<sub style="top: 0.1494em;">S</sub>. </p><p>(5) <br>The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference </p><p><sup style="top: -0.3012em;">1</sup>Copyright, 2000, D. E. Soper <sup style="top: -0.3012em;">2</sup>[email protected] </p><p>1frame we use to define t in |ψ(t)i<sub style="top: 0.1495em;">S</sub>. For operators, we can deal with local operators like, for instance, the electric field F<sup style="top: -0.3615em;">µν</sup>(~x, t). The {~x, t} dependence is given by a covariant relation </p><p></p><ul style="display: flex;"><li style="flex:1"><sup style="top: -0.2344em;">µ</sup>x<sub style="top: 0.083em;">µ </sub>µν </li><li style="flex:1">−iP<sup style="top: -0.2344em;">µ</sup>x<sub style="top: 0.083em;">µ </sub></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">F<sup style="top: -0.4113em;">µν</sup>(~x, t) = e<sup style="top: -0.4113em;">iP </sup>F (0, 0)e </li><li style="flex:1">.</li></ul><p></p><p>(6) </p><p>~</p><p>where P<sup style="top: -0.3615em;">0 </sup>= H and x<sup style="top: -0.3615em;">0 </sup>= t. We use the Heisenberg picture here even though we discuss nonrelativistic quantum mechanics. </p><p>2 In and out states </p><p>We deal here with the simplest version of scattering theory. We imagine that there is a free particle hamiltonian H<sub style="top: 0.1494em;">0 </sub>that consists of the kinetic energy operators for all of the particles in the theory but does not contain any terms that cause the particles to interact. The full hamiltonian is </p><p>H = H<sub style="top: 0.1494em;">0 </sub>+ V </p><p>(7) where V contains interactions among the particles. For example, in a nonrelativistic description of two particles we might have </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p>2</p><p></p><ul style="display: flex;"><li style="flex:1">~</li><li style="flex:1">~</li></ul><p>h~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>|H<sub style="top: 0.1494em;">0</sub>|ψi = − </p><p></p><ul style="display: flex;"><li style="flex:1">∇<sub style="top: 0.2462em;">1 </sub>− </li><li style="flex:1">∇</li></ul><p></p><p>h~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>|ψi </p><p>(8) </p><p></p><ul style="display: flex;"><li style="flex:1">2m<sub style="top: 0.1495em;">1 </sub></li><li style="flex:1">2m<sub style="top: 0.1495em;">2 </sub></li></ul><p></p><p>and </p><p>h~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>|V |ψi = V(|~x<sub style="top: 0.1494em;">1 </sub>− ~x<sub style="top: 0.1494em;">2</sub>|) h~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>|ψi </p><p>(9) for some potential function V. We consider potential functions with the property that V(|~x<sub style="top: 0.1495em;">1 </sub>− ~x<sub style="top: 0.1495em;">2</sub>|) → 0 as |~x<sub style="top: 0.1495em;">1 </sub>− ~x<sub style="top: 0.1495em;">2</sub>| → ∞. <br>In scattering theory, we consider shooting particles at each other and then seeing what happens. With the kind of nonrelativistic theory just mentioned, what can happen is that the particles emerging with different momenta in a fashion consistent with conservation of the total energy and momentum. The main idea is that when the particles scatter, after awhile they are far apart and V doesn’t act anymore. In addition, before they scatter they are also are far apart and V doesn’t do anything. <br>Consider a state |ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">f </sub>whose corresponding Schr¨odinger wave function ψ<sub style="top: 0.1494em;">F </sub>(~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>) consists of approximately plane waves in “wave packets”. If we were using the free theory, this state would consist of two particles that </p><p>2propagate freely into the far future, with their wave packets separating from each other. But we are using interacting theory rather than the free theory. We let |ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">out </sub>denote the state in the full theory that, in the far future, looks like |ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">f </sub>would look if we used the free hamiltonian. That is </p><p>e<sup style="top: -0.4113em;">−iHt </sup>|ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">out </sub>≈ e<sup style="top: -0.4113em;">−iH t </sup>|ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">f </sub></p><p>(10) </p><p>0</p><p>for very large positive t. The wave function for the out state is not at all simple. But based on this physical argument, we may hope to construct the out state as a limit: </p><p>|ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">out </sub>= lim e<sup style="top: -0.4113em;">iHt </sup>e<sup style="top: -0.4113em;">−iH t </sup>|ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">f </sub></p><p>(11) </p><p>0</p><p>t→∞ </p><p>Consider a different state |ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">f </sub>whose corresponding Schr¨odinger wave function ψ<sub style="top: 0.1494em;">I</sub>(~x<sub style="top: 0.1494em;">1</sub>, ~x<sub style="top: 0.1494em;">2</sub>) also consists of approximately plane waves in “wave packets”. This state represents a description of the wave packets for the particles entering the collision. We let |ψ<sub style="top: 0.1495em;">I</sub>i<sub style="top: 0.1495em;">in </sub>denote the state in the full theory that, in the far past, looks like |ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">f </sub>would look if we used the free hamiltonian. That is </p><p>e<sup style="top: -0.4113em;">−iHt </sup>|ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">in </sub>≈ e<sup style="top: -0.4113em;">−iH t </sup>|ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">f </sub></p><p>(12) </p><p>0</p><p>for very large negative t. Based on this physical argument, we may hope to construct the in state as a limit: </p><p>|ψ<sub style="top: 0.1495em;">I</sub>i<sub style="top: 0.1495em;">in </sub>= lim e<sup style="top: -0.4113em;">iHt </sup>e<sup style="top: -0.4113em;">−iH t </sup>|ψ<sub style="top: 0.1495em;">I</sub>i<sub style="top: 0.1495em;">f </sub></p><p>(13) </p><p>0</p><p>t→−∞ </p><p>The S-matrix is defined to be the amplitude that a state that looks like <br>|ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">f </sub>in the far past will look like |ψ<sub style="top: 0.1494em;">F </sub>i<sub style="top: 0.1494em;">f </sub>in the far future. That is </p><p>S<sub style="top: 0.1494em;">FI </sub></p><p>=</p><p><sub style="top: 0.1494em;">out</sub>hψ<sub style="top: 0.1494em;">F </sub>|ψ<sub style="top: 0.1494em;">I</sub>i<sub style="top: 0.1494em;">in</sub>. </p><p>(14) </p><p>3 Analysis of the S-matrix </p><p>With our definition, we have </p><p>S<sub style="top: 0.1495em;">FI </sub></p><p></p><ul style="display: flex;"><li style="flex:1">=</li><li style="flex:1"><sub style="top: 0.1495em;">out</sub>hψ<sub style="top: 0.1495em;">F </sub>|ψ<sub style="top: 0.1495em;">I</sub>i<sub style="top: 0.1495em;">in </sub>= lim hψ<sub style="top: 0.1495em;">F </sub>|U(T<sub style="top: 0.1495em;">F </sub>, T<sub style="top: 0.1495em;">I</sub>)|ψ<sub style="top: 0.1495em;">I</sub>i </li><li style="flex:1">(15) </li></ul><p>(16) </p><p>TT</p><p>→+∞ </p><p>FI</p><p>→−∞ </p><p>where </p><p>U(T<sub style="top: 0.1494em;">F </sub>, T<sub style="top: 0.1494em;">I</sub>) = e<sup style="top: -0.4113em;">iH T </sup>e<sup style="top: -0.4113em;">−iH(T −T ) </sup>e<sup style="top: -0.4113em;">−iH T </sup></p><p>.</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">F</li><li style="flex:1">F</li><li style="flex:1">I</li><li style="flex:1">I</li></ul><p></p><p>3<br>We can write this in a better form by writing a differential equation for <br>U: </p><p>d</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">I</li></ul><p></p><p>i</p><p>U(t, T<sub style="top: 0.1494em;">I</sub>) = e<sup style="top: -0.4113em;">iH t </sup>(H − H<sub style="top: 0.1494em;">0</sub>) e<sup style="top: -0.4113em;">−iH(t−T ) </sup>e<sup style="top: -0.4113em;">−iH T </sup></p><p>dt </p><p>= e<sup style="top: -0.4114em;">iH t </sup>V e<sup style="top: -0.4114em;">−iH t </sup>e<sup style="top: -0.4114em;">iH t </sup></p><p>e</p><p><sup style="top: -0.4114em;">−iH(t−T ) </sup>e<sup style="top: -0.4114em;">−iH T </sup></p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">I</li></ul><p></p><p>= V (t) U(t, T<sub style="top: 0.1494em;">I</sub>), </p><p>(17) (18) where </p><p>V (t) ≡ e<sup style="top: -0.4114em;">iH t </sup>V e<sup style="top: -0.4114em;">−iH t </sup></p><p>.</p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>The solution of this is </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">Z</li></ul><p></p><p>t</p><p>U(t, T<sub style="top: 0.1495em;">I</sub>) = T exp −i dτ V (τ) . </p><p>(19) </p><p>T<sub style="top: 0.1116em;">I </sub></p><p>Here the T is a time ordering instruction that tells us what order the noncommuting operators V (τ) belong in. We should expand the exponential and then put the V (τ) operators with the later values of the time argument to the left. That is </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p>T exp −i </p><p>dτ V (τ) = 1 − i </p><p>dτ V (τ) </p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p>1</p><p>− T </p><p>2</p><p></p><ul style="display: flex;"><li style="flex:1">dτ<sub style="top: 0.1494em;">2 </sub></li><li style="flex:1">dτ<sub style="top: 0.1494em;">1 </sub>V (τ<sub style="top: 0.1494em;">2</sub>) V (τ<sub style="top: 0.1494em;">1</sub>) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">t</li><li style="flex:1">t</li><li style="flex:1">t</li></ul><p></p><p>i</p><p>+</p><p></p><ul style="display: flex;"><li style="flex:1">T</li><li style="flex:1">dτ<sub style="top: 0.1494em;">3 </sub></li><li style="flex:1">dτ<sub style="top: 0.1494em;">2 </sub></li><li style="flex:1">dτ<sub style="top: 0.1494em;">1 </sub>V (τ<sub style="top: 0.1494em;">3</sub>) V (τ<sub style="top: 0.1494em;">2</sub>) V (τ<sub style="top: 0.1494em;">1</sub>) </li></ul><p></p><p>3! </p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li></ul><p></p><p>+ · · · </p><p>Z</p><p>t</p><p>= 1 − i </p><p>dτ V (τ) </p><p>T<sub style="top: 0.1116em;">I </sub></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>t</p><p>τ<sub style="top: 0.0922em;">2 </sub></p><p>−</p><p></p><ul style="display: flex;"><li style="flex:1">dτ<sub style="top: 0.1494em;">2 </sub></li><li style="flex:1">dτ<sub style="top: 0.1494em;">1 </sub>V (τ<sub style="top: 0.1494em;">2</sub>) V (τ<sub style="top: 0.1494em;">1</sub>) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1116em;">I </sub></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>t</p><p></p><ul style="display: flex;"><li style="flex:1">τ<sub style="top: 0.0922em;">3 </sub></li><li style="flex:1">τ<sub style="top: 0.0922em;">2 </sub></li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">+i </li><li style="flex:1">dτ<sub style="top: 0.1495em;">3 </sub></li><li style="flex:1">dτ<sub style="top: 0.1495em;">2 </sub></li><li style="flex:1">dτ<sub style="top: 0.1495em;">1 </sub>V (τ<sub style="top: 0.1495em;">3</sub>) V (τ<sub style="top: 0.1495em;">2</sub>) V (τ<sub style="top: 0.1495em;">1</sub>) </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.1117em;">I </sub></li><li style="flex:1">T<sub style="top: 0.1117em;">I </sub></li></ul><p></p><p>+ · ·<sup style="top: -1.1909em;">T</sup>·<sup style="top: 0.1117em;">I</sup>. </p><p>(20) <br>Exercise. Show that this series actually solves the operator differential equation for U(t, T<sub style="top: 0.1494em;">I</sub>). </p><p>4<br>Thus the operator appearing in the S-matrix is </p><p></p><ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul><p>Z</p><p>T<sub style="top: 0.1116em;">F </sub></p><p></p><ul style="display: flex;"><li style="flex:1">U(T<sub style="top: 0.1495em;">F </sub>, T<sub style="top: 0.1495em;">I</sub>) = T exp −i </li><li style="flex:1">dτ V (τ) . </li></ul><p></p><p>(21) </p><p>T<sub style="top: 0.1117em;">I </sub></p><p>4 Perturbative expansion for the S-matrix </p><p>We started by thinking of |ψ<sub style="top: 0.1494em;">F </sub>i as a product wave-packet states for the two particles, so that we could be sure that after a long time the wave functions for the two final state particles would not overlap in space. That way, the potential energy operator V could be neglected in the far future. Similarly, we wanted wave-packet states for |ψ<sub style="top: 0.1494em;">I</sub>i. Now, however, let’s let the momentum spread of the wave functions approach zero, so that we have momentum eigenstates. That will make calculation feasible. <br>Assuming that we are dealing with plane wave states, the states |ψ<sub style="top: 0.1495em;">I</sub>i and <br>|ψ<sub style="top: 0.1495em;">F </sub>i are eigenstates of H<sub style="top: 0.1495em;">0</sub>, with eigenvalues E<sub style="top: 0.1495em;">I </sub>and E<sub style="top: 0.1495em;">F </sub>respectively. With this assumption, let’s expand the S-matrix in powers of V : </p><p>∞</p><p>X</p><p>(n) </p><p>S<sub style="top: 0.1495em;">FI </sub></p><p>=</p><p></p><ul style="display: flex;"><li style="flex:1">S</li><li style="flex:1">.</li></ul><p></p><p>(22) (23) </p><p>FI </p><p>n=0 </p><p>We have </p><p>(0) </p><p>FI </p><p>S<br>= hψ<sub style="top: 0.1495em;">F </sub>|ψ<sub style="top: 0.1495em;">I</sub>i. </p><p>This is the no-scattering term. The Born approximation is </p><p>Z</p><p>∞</p><p>(1) </p><p>FI </p><p>dτ hψ<sub style="top: 0.1494em;">F </sub>|e<sup style="top: -0.4113em;">iH τ </sup>V e<sup style="top: -0.4113em;">−iH τ </sup>|ψ<sub style="top: 0.1494em;">I</sub>i </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>S</p><p>= −i </p><p>−∞ <br>∞</p><p>Z</p><p>= −i <sub style="top: 0.7149em;">−∞ </sub>dτ e<sup style="top: -0.4114em;">i(E </sup></p><p>hψ<sub style="top: 0.1494em;">F </sub>|V |ψ<sub style="top: 0.1494em;">I</sub>i </p><p><sub style="top: 0.1117em;">F </sub>−E<sub style="top: 0.1117em;">I </sub>)τ </p><p>= −i 2π δ(E<sub style="top: 0.1495em;">F </sub>− E<sub style="top: 0.1495em;">I</sub>) hψ<sub style="top: 0.1495em;">F </sub>|V |ψ<sub style="top: 0.1495em;">I</sub>i. </p><p>(24) <br>That’s pretty simple. We just need to calculate the matrix element of V in plane wave states. <br>The order V <sup style="top: -0.3615em;">2 </sup>approximation is </p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p>τ<sub style="top: 0.0922em;">2 </sub></p><p>(2) </p><p>FI </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">2</li><li style="flex:1">0</li><li style="flex:1">2</li><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><p>S</p><p>= − <sub style="top: 0.7149em;">−∞ </sub>dτ<sub style="top: 0.1495em;">2 </sub></p><p>dτ<sub style="top: 0.1495em;">1 </sub>hψ<sub style="top: 0.1495em;">F </sub>|e<sup style="top: -0.4113em;">iH τ </sup>V e<sup style="top: -0.4113em;">−iH (τ −τ ) </sup>V e<sup style="top: -0.4113em;">−iH τ </sup>|ψ<sub style="top: 0.1494em;">I</sub>i </p><p>−∞ <br>∞</p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p>∞</p><p>= − <sub style="top: 0.7149em;">−∞ </sub>dτ<sub style="top: 0.1494em;">1 </sub></p><p>dτ hψ<sub style="top: 0.1494em;">F </sub>|e<sup style="top: -0.4113em;">iH (τ+τ ) </sup>V e<sup style="top: -0.4113em;">−iH τ </sup>V e<sup style="top: -0.4113em;">−iH τ </sup>|ψ<sub style="top: 0.1494em;">I</sub>i </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">1</li><li style="flex:1">0</li><li style="flex:1">0</li><li style="flex:1">1</li></ul><p></p><p>0</p><p>5</p><p></p><ul style="display: flex;"><li style="flex:1">Z</li><li style="flex:1">Z</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">∞</li><li style="flex:1">∞</li></ul><p></p><p><sub style="top: 0.1117em;">F </sub>−E<sub style="top: 0.1117em;">I </sub>)τ<sub style="top: 0.0922em;">1 </sub></p><p>dτ e<sup style="top: -0.4114em;">i(E </sup></p><p>hψ<sub style="top: 0.1495em;">F </sub>|V e<sup style="top: -0.4113em;">i(E</sup><sup style="top: 0.1117em;">F </sup><sup style="top: -0.4113em;">−H )τ </sup>V |ψ<sub style="top: 0.1495em;">I</sub>i </p><p>0</p><p>= − <sub style="top: 0.715em;">−∞ </sub>dτ<sub style="top: 0.1495em;">1 </sub></p><p>0</p><p>1</p><p></p><ul style="display: flex;"><li style="flex:1">= −i 2πδ(E<sub style="top: 0.1494em;">F </sub>− E<sub style="top: 0.1494em;">I</sub>) hψ<sub style="top: 0.1494em;">F </sub>|V </li><li style="flex:1">V |ψ<sub style="top: 0.1494em;">I</sub>i </li></ul><p></p><p>(25) </p><p>E<sub style="top: 0.1494em;">F </sub>− H<sub style="top: 0.1494em;">0 </sub>+ iꢀ </p><p>This is characteristic of the higher order terms. We have factors of V and energy denominator factors 1/(E<sub style="top: 0.1495em;">F </sub>− H<sub style="top: 0.1495em;">0 </sub>+ iꢀ). <br>Let us summarize this result. One defines the T-matrix as </p><p>S<sub style="top: 0.1494em;">FI </sub>= hψ<sub style="top: 0.1494em;">F </sub>|ψ<sub style="top: 0.1494em;">I</sub>i − i(2π)δ(E<sub style="top: 0.1494em;">F </sub>− E<sub style="top: 0.1494em;">I</sub>) T<sub style="top: 0.1494em;">FI</sub>. </p><p>Then we expand T<sub style="top: 0.1494em;">FI </sub>in powers of V : <br>(26) (27) </p><p>X</p><p>(n) </p><p>FI </p><p>T<sub style="top: 0.1494em;">FI </sub></p><p>=</p><p></p><ul style="display: flex;"><li style="flex:1">T</li><li style="flex:1">,</li></ul><p></p><p>(n) </p><p>FI </p><p></p><ul style="display: flex;"><li style="flex:1">where T </li><li style="flex:1">is the contribution to T<sub style="top: 0.1494em;">FI </sub>proportional to n powers of V . The </li></ul><p>result just derived is </p><p></p><ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">i</li></ul><p></p><p>(n) </p><p>−iT = hψ<sub style="top: 0.1494em;">F </sub>|(−iV ) </p><p>FI </p><p>(−iV ) </p><p>· · · (−iV )|ψ<sub style="top: 0.1494em;">I</sub>i. </p><p></p><ul style="display: flex;"><li style="flex:1">E<sub style="top: 0.1494em;">F </sub>− H<sub style="top: 0.1494em;">0 </sub>+ iꢀ </li><li style="flex:1">E<sub style="top: 0.1494em;">F </sub>− H<sub style="top: 0.1494em;">0 </sub>+ iꢀ </li></ul><p></p><p>(28) <br>Let us see what this is for plane wave initial and final states, </p><p>~</p><p>|ψ<sub style="top: 0.1494em;">I</sub>i = |p~<sub style="top: 0.1494em;">I</sub>, k<sub style="top: 0.1494em;">I</sub>i </p><p>~<br>|ψ<sub style="top: 0.1494em;">F </sub>i = |p~<sub style="top: 0.1494em;">F </sub>, k<sub style="top: 0.1494em;">F </sub>i. </p><p>(29) <br>Wherever we see an energy denominator factor we can insert a sum over a complete set of plane wave states. For the ith such sum over intermediate states, we can write </p><p>Z</p><p>~dp~<sub style="top: 0.1495em;">i </sub>dk<sub style="top: 0.1495em;">i </sub></p><p>(2π)<sup style="top: -0.2878em;">3 </sup>(2π)<sup style="top: -0.2878em;">3 </sup></p><p></p><ul style="display: flex;"><li style="flex:1">~</li><li style="flex:1">~</li></ul><p></p><p>1 = </p><p>|p~<sub style="top: 0.1494em;">i</sub>, k<sub style="top: 0.1494em;">i</sub>ihp~<sub style="top: 0.1494em;">i</sub>, k<sub style="top: 0.1494em;">i</sub>|. </p><p>(30) <br>When H<sub style="top: 0.1494em;">0 </sub>acts on one of the plane wave states it gives </p><p></p><ul style="display: flex;"><li style="flex:1">"</li><li style="flex:1">#</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">p<sub style="top: 0.2463em;">i</sub><sup style="top: -0.3615em;">2 </sup></li><li style="flex:1">k<sub style="top: 0.2463em;">i</sub><sup style="top: -0.3615em;">2 </sup></li></ul><p></p>
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