3.5 Finite Rotations in 3D Euclidean Space and Angular Momentum in QM

Transcription

1 3.5 Finite Rotations in 3D Euclidean Space and Angular Momentum in QM An active rotation in 3D position space is defined as the rotation of a vector about some point in a fixed coordinate system (a passive rotation being the rotation of the coordinate system while the vector stays fixed). We claim that the rotation of a vector about some axis in space is induced by a corresponding unitary and hermitian operator expressed as a 3 by 3 matrix. These rotations are in direct correspondence to the transformation of a state vector in an angular momentum basis (i.e. where the eigenvalues are angular momentum). That is, the rotation of a vector r, defining the value of a function Ψ(r) at a point in position space, corresponds to a change of state of the corresponding state vector Ψ. In order to develop the proper operators inducing finite rotations, one determines the changes on a function (usually a vector function, say, f(r ) under an infinitesimal rotation which is induced by what we call generators of infinitesimal rotations. Then one integrates the change in the function δf(r) to obtain the proper rotation operators U. If the function f(r) is changed at r by an infinitesimal vector ε then we obtain the function f(r ) = f(r-ε) (if you find this confusing, think of shifting, say, the sine function by a certain amount just in 1D ). Since ε is supposed to be a very small vector, we can expand the new function in a Taylor series to first order: And the corresponding change in the function is (to first order) In general, a finite rotation R by an angle φ around an axis pointing along the unit vector n results in the displacement of an entity or a change in property of some medium that pervades all space (think of the electric field varying with position), located by the vector r, is given by the vector (see figures below): (See next page.)

2 Proof: Assume we are rotating around the z-axis (we can always rotate the coordinate system to satisfy this) and that the vector we want to rotate is given by: The usual matrix rotation about z is: With this, we have:

3 If the rotation angle is infinitesimal (δφ ), then eqn becomes (via Taylor expansion again) Where is a vector, pointing in the direction n of the axis of rotation, defined by the right hand screw rule. So, to first order in nδφ the change of the function f(r) under an infinitesimal rotation ε by an angle δφ is, Then, using we find Where α = nδφ is now defined to be the infinitesimal angle of rotation. With this result, we see that the rotated function is given by From here, we can go to a finite rotation φ around n by performing N such infinitesimal rotations in succession, such that

4 If we decide to make all of these infinitesimal rotations the same magnitude, then the total rotation is just So, for N successive, infinitesimal rotations (it s now time to start calling them transformations), a finite rotation in eqn becomes However, in the limit as N, this becomes exact, and we find: Accordingly, we write the rotation operator for finite rotations about n as For any state Ψ = l Ψ(r) r, l 1 the rotation operator (eqn ) rotates all probability amplitudes Ψ(r) into new amplitudes Ψ (r ), describing a new state Ψ. Ψ (r ) = U R Ψ(r) For a proper rotation, if the initial wave functions (and the state itself) are properly normalized, then so are the transformed wave functions (i.e. a proper rotation can not change the length of a vector!). For this to be the case, we must have Ψ Ψ = Ψ U R t U R Ψ = 1 If the matrices (operators) L in the rotation operator are hermitian (which we will verify later), then this condition is satisfied, since then U R t = U R -1 or UR t U R = I, as you can easily verify from looking at the rotation operator. 1) This is a sum and integral over simultaneous discrete angular momentum eigenvectors and continuous position eigenvectors this is basically a state assembled from a complete set of solutions or eigenvectors of the Schrödinger equation. There is more detail in the state as is indicated in this notation, but that is irrelevant at this point. We will find the full state, for all simultaneously diagonalizable operators later on.

5 For any operator A, the definition of a rotationally transformed operator A is given by A Ψ (r ) = U R AΨ(r) and inserting the identity operator expressed in terms of the rotation operators gives A Ψ (r ) = U R A U t R U R Ψ(r) t (A - U R A U R ) Ψ (r ) = 0 Which mean, that, since in general Ψ (r ) 0, we must have A = U R A U R t = e -iφn L/ћ A e iφn L/ћ The so called generator of rotation L used here (corresponding to the symbol commonly used to indicate orbital angular momentum) is just one example, serving to illustrate the concept for all instances of angular momenta. In addition to angular momentum, we also have spin angular momentum, denoted by S, which is just as special case of angular momentum in general, as described in the introduction to this chapter. There is then also total angular momentum J which is used to identify the total, added angular momentum of a system and may include the addition of several orbital angular momenta, several spin angular momenta, or the addition of spin and orbital angular momenta. All of these represent some form of rotation in Euclidean space and the corresponding transformation of a state, so that, in general we write 3.6 Angular Momentum Commutation Relations We want to establish the quantum mechanical behavior of angular momentum, and the first information we need for that is how the angular momentum operators commute with each other as well as with other operators. To this end, consider calculating the expectation value of some operator A. Assume that A is in fact a three vector, such as another angular momentum operator. Then, A = Ψ A Ψ A = Ψ A Ψ = Ψ e -iφn L/ћ A e iφn L/ћ Ψ

6 This expectation value is a sum of vectors of eigenvalues 1 (of actual 3D vectors, as we are used to, by virtue of the operator A being a vector operator). But for any individual eigenstate of the hamiltonian with probability amplitude Ψ(r) the expectation value corresponds to a single vector as written in eqn Where the last equality takes the infinitesimal rotation approximation with angle α. Then, taking the form of the infinitesimal rotation operator from eqn , we get Where we only take terms on the left hand side to order α ( throwing away the term with α 2 ), so as to keep consistency with the first order expansion that lead to eqn Therefore, we find that Since J itself is a vector operator and this derivation made no reference to or any restriction on any particular type, we can just plug in J itself again and see that this leads to Or, as we are more used to seeing These are the commutation relations of the angular momentum components with themselves. 1) It is a sum, because the general state is a superposition of eigenstates which all have (in general) different eigenvalues.

7 Properties of the Levi-Civita symbol: