Generators for Continuous Coordinate Transformations

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1 Page 636 Lecture 37: Coordinate Transformations: Continuous Passive Coordinate Transformations Active Coordinate Transformations Date Revised: 2009/01/28 Date Given: 2009/01/26

2 Generators for Continuous Coordinate Transformations Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 637 General Properties of Continuous Passive Coordinate Transformations Consider a passive coordinate transformation that can be parameterized by and differentiated with respect to a continuous variable ε, T P = T P (ε). This might be the translation vector for a translation, the rotation angle for a rotation (such as in the example we just did), etc. Are there any interesting properties of such transformations that arise from this additional differentiability property? The first point we may make is that the infinitesimal version of such a transformation (that is, T P (ε) for ε 0) may always be written in the form T P (ε) = I i ε G where G is some operator to be determined. This property arises simply because T P (ε) I as ε 0. The choice of the i coefficient is of course motivated by prior knowledge of what will happen below, but is completely general.

3 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 638 What may we say about G? Let s investigate the consequences of the unitarity of T P (ε). To do this, we need to know what T 1 1 P (ε) is. It is TP (ε) = I + i ε G, which we can prove by applying it to T P (ε): T 1 P (ε) T P(ε) = I + i «I ε G i «ε G = I + O(ε 2 ) = I where we drop O(ε 2 ) terms because we are assuming ε is infinitesimal. From the above, we can see that G must be Hermitian: T P 1 (ε) = TP (ε) I + i G = I + i ε G G = G

4 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 639 With some knowledge of the properties of G, we may construct the full transformation for arbitary ε by taking an infinite product of infinitesimal transformations: T P (ε) = lim N h ε i» N T P = lim I i N N ε N N G = exp i «ε G where ε/n is infinitesimal as N. We have used a generic property of the exponential function in the last step, one that can be verified to be equivalent to the standard Taylor expansion via inductive proof. That G is Hermitian is now not surprising, as we know from previous work that e iλ is unitary if Λ is Hermitian. The above formula leads us to an explicit formula for G: T 1 P»i (ε) d dε T P(ε) = T 1 P (ε) T P(ε) G = G That is, G can be obtained by differentiating T P (ε). Due to the relation between G and T P (ε), G is called the generator of the passive coordinate transformation T P (ε). Because G is Hermitian, G is allowed to be an observable, a property that will be interesting for symmetry transformations.

5 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 640 We note that the explicit form for T P (ε) for continuous transformations lets us write down a more explicit form for how position-basis elements, states, and operators transform: q = exp i «ε G q O = exp i ««i ε G O exp ε G That is, rather than specifying T P (ε) by the mapping from unprimed to primed basis elements, we can now specify it simply by the action of G. The effect of the transformation T P (ε) with generator G may be particularly simply written in terms of the eigenstates of G. Consider first the action of the transformation on an eigenstate g of G: g = e i ε G g = e i ε g g That is, the eigenstates of G transform very simply under T P (ε): they retain the same direction in Hilbert space, picking up only a unity-modulus factor.

6 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 641 The above suggests that the transformation of the { g }-basis wavefunction of the state is trivial. This wavefunction is ψ g (g) = g ψ. Then we have ψ g (g ) = g ψ = g e i ε G ψ = e i ε g g ψ = e i ε g ψ g (g) Note the sign in the argument of the expoential, which arises because we have g not g in the expression. Our result shows that the { g }-basis wavefunction simply picks up a g-dependent unity-modulus factor. Of course, the g dependence of that factor is what results in interesting behavior. But, clearly, things are much simplified in this form. Finally, the above encourages us to look at the transformation of O in the { g } basis. That is: g 1 O g 2 = g 1 e i ε G O e i ε G g 2 = e i ε (g 1 g 2 ) g 1 O g 2 Again, we get a very simple relation between the matrix elements of O and those of O in the { g } basis. (Note that the above is different from the transformed operator defining relation, Equation 12.1, which here takes the form g 1 O g 2 = g 1 O g 2.)

7 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 642 Explicitly Determining the Generator The above is very nice, but how do you figure out what the generator is, explicitly? We have not found an explicit form for the transformation operators, so we cannot just Taylor expand it to find the generator. The answer is that we must Taylor expand the transformed wavefunction. Let s begin by Taylor-expanding the transformation operator: ψ q (q = u) = q = u ψ = q = u T P ψ q = u I + i «ε G ψ = q = u ψ + i ε q = u G ψ where we have done the usual substitution trick to avoid confusion and where the sign is used because we have considered an infinitesimal transformation by parameter value ε.

8 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 643 Let s move things around: i ε q = u G ψ q = u ψ q = u ψ = ˆ q = u q = u ψ We know that, for any transformation, there is an equality relationship between the { q } basis and the { q } basis that will allow us to rewrite q = u in terms of { q } basis elements. Let s write this as q = u = q = q(q = u, ε) where ε indicates the value of the transformation parameter that determines the relationship between q and q. So we have i ε q = u G ψ ˆ q = q(q = u) q = u ψ = ψ q(q = q(q = u, ε)) ψ q(q = u) That is, we have now written everything on the right side in terms of the { q }-basis wavefunction.

9 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 644 Finally, because we are considering an infinitesimal transformation, we may calculate the right side by Taylor expansion in ε near ε = 0. Note that ε = 0 corresponds to q = q, so we will be able to evaluate the derivatives at q = q = u and ε = 0. We have i ε q = u G ψ ε ψq q q q=u ε q=q =u ε=0 where the derivatives in the last expression are schematic there may be more than one coordinate, so one has to generalize appropriately to multiple coordinates. Regardless, one see that one can obtain the action of G on an arbitrary state, projected onto the { q } basis, which should let us determine explicitly what G is. We ll see this work in examples.

10 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 645 Example 12.3: Generator for the Passive Rotation Transformation Recall from Example 12.2 that the position-basis wavefunctions of the untransformed state in the { q } and { q } bases are: s ««4 a π x b π y ψ ab (x, y) = x, y ψ ab = sin sin L 1 L 2 L 1 L 2 s ψ ab,q (x, y ) = x, y 4 a π (x c θ y «s θ ) b π (x s θ + y «c θ ) ψ ab = sin sin L 1 L 2 L 1 L 2 Let s calculate the above Taylor expansion. First, let s recall that x(x, y ) = x c θ y s θ y(x, y ) = x s θ + y c θ Taking the derivatives and allowing θ to be infinitesimal gives x θ x=x =u y=y =v θ=0 v y θ x=x =u y=y =v θ=0 u

11 Generators for Continuous Coordinate Transformations (cont.) Section 12.2 Symmetries: Generators for Continuous Coordinate Transformations Page 646 So, our expression becomes i θ x = u, y = v G ψ = θ ψ ab,q x = θ x=u y=v x θ y x + x y x=x =u y=y =v θ=0 + θ ψ ab,q y «ψ ab,q (x, y) x=u y=v = i θ x = u, y = v (X Py Y Px ) ψ ab x=u y=v y θ x=x =u y=y =v θ=0 where we used i x ψ ab,q(x, y) = x, y P x ψ ab i y ψ ab,q(x, y) = x, y P y ψ ab So we have G = X P y Y P x. This expression should be familiar: straightforward replacement of classical x, y, p x, p y in l z = x p y y p x with the corresponding quantum mechanical operators yields L z = X P y Y P x. Hence, G = L z, the z-axis angular momentum operator, and we see that L z generates rotations about the z axis, T (θ) = exp i θ Lz.

12 Active Coordinate Transformations Section 12.3 Symmetries: Active Coordinate Transformations Page 647 Active Coordinate Transformations We have defined passive coordinate transformations as a simple relabeling of space. However, one could have viewed the transformation as a movement of all the particles and potentials in the problem relative to the original axes. This is called an active transformation. In contrast to the passive transformation case, we transform the states and the Hamiltonian also. We call the new state ψ = T A ψ where T A is the operator that maps from the old states to the new states (the active transformation operator). That said, we will need a way to define explicitly what we mean by the above. The natural way to do that is to consider a set of transformed axes {q } with corresponding new position and momentum basis elements { q } and { p q }, and to define the transformation in terms of its action on the basis elements in the same way as we did for passive transformations, e i θ q = T A q where θ is again real, and the prefactor exists for the same reasons discussed in connection to passive transformations. We shall take θ = 0 in general. Though the definition has the same form, there is indeed a distinction between passive and active transformation operators that we will explain shortly.

13 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 648 The unitarity of T A follows by an argument similar to that used for T P. Since we are just moving the system relative to the underlying space, the dimensionality of the new basis must be the same as that of the old basis; that is, the mapping is one-to-one, or invertible. Assuming the new basis elements are normalized in the same way as the old ones, the transformation also preserves inner products between basis elements because there are no coefficients in front. Therefore, it preserves all inner products, and hence is unitary, T A = T 1 A. The unitarity of T A then lets us see in a more intuitive way how the state transforms. The transformed state satisfies q ψ = q T A T A ψ = q T 1 A T A ψ = q ψ That is, the projection of the transformed state onto the transformed basis elements is the same as the projection of the untransformed state onto the untransformed basis elements. Since this projection is the position-basis wavefunction, what we are saying is that the position-basis wavefunction for the transformed state depends on the new (primed) coordinate axes in the same way as the position-basis wavefunction for the untransformed state depended on the old (unprimed) coordinate axes.

14 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 649 The definition of transformed operators follows in the same way as it did for the passive transformation: because we have new coordinate axes and new (position- and momentum-) basis elements, we want to define new operators that act on the transformed basis elements in the same way as the old operators acted on the untransformed basis elements: q 1 Q q 2 q 1 Q q 2 (12.2) By the definition of the transformation operator s action on basis elements, q = T A q, we also have q 1 Q q 2 = q 1 T A Q T A q 2. Combining the two statements gives q 1 Q q 2 = q 1 T A Q T A q 2 Since this relation holds for all q 1 and q 2, it therefore holds that Q = T A Q T A = T A Q T 1 P The above proof carries through for any operator, including the {P q}, and thus we now have a means to transform the operators. Notice how the formula is identical to the one we calculated for the passive transformation case; the distinction between active and passive transformations will be explained below. Again, if there is any confusion, one only needs to write the above in terms of matrix elements.

15 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 650 In contrast to the passive transformation case, we will transform H to H because we want to move the particles and potentials. Unitarity assures us that the transformed Hamiltonian s eigenstates are the transformed eigenstates of the untransformed Hamiltonian. That is, if ψ E is an eigenstate of H with eigenvalue E, then we have H (T A ψ E ) = T A H T A (T A ψ E ) = T A H ψ E = T A E ψ E = E (T A ψ E ) For continuous transformations, we may write the transformation operator in terms of generators. Recall that we were able to write the action of the transformation operator on the basis elements and operators as q = exp i «ε G q O = exp i ««i ε G O exp ε G The above forms continue to hold for active transformations, though, as we will see, one may not assume that the sign of ε will be the same for passive and active transformations. In addition, we may now also write ψ = exp i «ε G ψ H = exp i ««i ε G H exp ε G

16 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 651 The next obvious question to ask is: are the untransformed eigenstates of the untransformed Hamiltonian, the { ψ E }, also eigenstates of the transformed Hamiltonian, and are the eigenstates of the transformed Hamiltonian, the { ψ E = T A ψ E }, also eigenstates of the untransformed Hamiltonian? The answer to both questions is, in general, no: H ψ E = H T A ψ E H ψ E = H T A ψ E (12.3) We see that we need [H, T A ] = 0 in order for the above to simplify in the necessary fashion for ψ E to be an eigenstate of H and for ψ E to be an eigenstate of H. We shall discuss later symmetry transformations, which do satisfy the above commutation property and for which the answer to the above questions is yes.

17 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 652 Relation between Passive and Active Transformations It is clear that passive and active transformations are very much alike. Why make any distinction at all? What is the distinction? In the passive case, we define a new set of coordinate axes and corresponding basis elements, leave the states and Hamiltonian unchanged (i.e., leave the particles and potentials fixed relative to the old coordinate axes), and ask what the position-basis wavefunctions looks like in terms of the new coordinate axes and what the Hamiltonian looks like in terms of the new operators. That is, we are interested in q ψ and H `Q, P q. In the active case, we define a new set of coordinate axes and corresponding basis elements and transform the states and Hamiltonians along with the basis elements, and then we will ask what the new position-basis wavefunction looks like in terms of the old coordinate axes and what the Hamiltonian looks like in terms of the old operators. That is, we will be interested in q ψ and H (Q, P q). Before writing any formulae, let s first think conceptually about what the difference is, with our example of a coordinate system rotation for a particle in a 2d box in mind, Example 12.2.

18 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 653 For the passive transformation, we rotate the axes CCW by an angle θ and leave the box and state in place. We calculate the position-basis wavefunction in terms of the new coordinates and the Hamiltonian in terms of the new operators. The wavefunction and Hamiltonian look ugly because the box is at an angle θ with respect to the new coordinates. For the active transformation, we rotate the axes and the box and state CCW by an angle θ. We will see that the new wavefunction in terms of the old coordinates and the Hamiltonian in terms of the old operators are ugly because the box and wavefunction are rotated by an angle +θ relative to the old coordinate axes. We see that there is a sign flip of the transformation parameter involved because the box is at an angle θ relative to the new coordinate axes for the passive transformation while the box is at an angle +θ relative to the old coordinate axes for the active transformation. This is in general what happens, that the passive and active transformations are related by a sign flip of the transformation parameter. Effectively, T P = T 1 A and T A = T 1 P. For discrete transformations like the mirror transformation in Example 12.1, the distinction vanishes because doing the transformation twice returns one to the original situation, so TP 2 = I and T A 2 = I. Since the transformation is its own inverse, T 1 P = T P and T 1 A = T A and thus our relation T P = T 1 A yields T P = T A. In cases of discrete transformations where TP 2 I or T A 2 I, there will be a distinction and one will require T P = T 1 A in order for the two transformations to yield the same effects.

19 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 654 Now, let s write generic formulae that state the above: Passive: ψ q (q ) = q ψ = q T P ψ H = H(Q, P q) = H T P Q T P, T P P q T P Active: ψ q(q) = q ψ = q T A ψ H = T A H T A = H(Q, P q ) = H T A QT A, T AP qt A where we use H to indicate the classical Hamiltonian function, now treated as a function whose arguments can be operators; introducing H is necessary in order to be completely clear about what we mean. We immediately see that the old wavefunction in the new basis, q ψ, and the new wavefunction in the old basis, q ψ, are the same function of their arguments if and only if T P = T A, which is equivalent to T 1 P = T A, the condition we stated above. Similarly, the untransformed Hamiltonian is a function of the new operators in the same way that the transformed Hamiltonian is a function of the old operators if and only if the same condition is met, T P = T A. An important implication of the above is that, if one focuses on the case T P = T A for the above reason, then the transformed operators will not be the same in the two cases, but will only have the same form when T P = T A.

20 Active Coordinate Transformations (cont.) Section 12.3 Symmetries: Active Coordinate Transformations Page 655 For continuous transformations, which can be written as complex exponentials of generators, our explicit forms are: T P (ε) = exp i «ε G «i T A (ε) = exp ε G where we define the sign of ε be positive when the new axes are to positive ε of the old axes. For rotations, positive ε thus corresponds to the new axes being CCW from the old ones, as one would expect. For translation, positive ε corresponds to the origin displaced to a positive value in the particular coordinate for which the translation is being considered. For such continuous transformations, the expressions given on the previous page for the transformation of the wavefunction can be simplified if the { g } basis is used: Passive: Active: ψ g (g ) = g ψ = g e i ε G ψ = e i ε g g ψ = e i ε g ψ g (g) ψ g (g) = g ψ = g e i ε G ψ = e i ε g g ψ = e i ε g ψ g (g)

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