221A Lecture Notes Notes on Tensor Product


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1 What is Tensor? A Lecture Notes Notes on Tensor Product After discussing the tensor product in the class, I received many questions what it means I ve also talked to Daniel, and he felt this is a subject he had learned on the way here and there, never in a course or a book I myself don t remember where and when I learned about it It may be worth solving this problem once and for all Apparently, Tensor is a manufacturer of fancy floor lamps See For us, the word tensor refers to objects that have multiple indices In comparison, a scalar does not have an index, and a vector one index It appears in many different contexts, but this point is always the same Direct Sum Before getting into the subject of tensor product, let me first discuss direct sum This is a way of getting a new big vector space from two (or more) smaller vector spaces in the simplest way one can imagine: you just line them up Space You start with two vector spaces, that is ndimensional, and W that is mdimensional The tensor product of these two vector spaces is n + m dimensional Here is how it works Let { e, e,, e n } be the basis system of, and similarly { f, f,, f m } that of W We now define n + m basis vectors { e,, e n, f,, f m } The vector space spanned by these basis vectors is W, and is called the direct sum of and W
2 ectors Any vector v can be written as a linear combination of the basis vectors, v = v e + v e + + v n e n We normally express this fact in the form of a column vector, v v v = v n n () Here, it is emphasized that this is an element of the vector space, and the column vector has n components In particular, the basis vectors themselves can be written as, e =, e =,, e n = () If the basis system is orthonormal, ie, e i e j = δ ij, then v i = e i v Similarly for any vector w, we can write The basis vectors are f = W w = w w w n, f = W W m (3),, fm = W (4) The vectors are naturally elements of the direct sum, just by filling zeros
3 to the unused entries, v = v v n n + m, w = w w m n + m (5) One can also define a direct sum of two nonzero vectors, v w = v v n w w m = ( v w )} n + m, (6) where the last expression is used to save space and it is understood that v and w are each column vector 3 Matrices A matrix is mathematically a linear map from a vector space to another vector space Here, we specialize to the maps from a vector space to the same one because of our interest in applications to quantum mechanics, A :, eg, v A v, (7) or v v v n a a a n a a a n a n a n a nn v v v n (8) Similarly, B : W W, and B : w B w On the direct sum space, the same matrices can still act on the vectors, so that v A v, and w B w This matrix is written as A B This can 3
4 be achieved by lining up all matrix elements in a blockdiagonal form, A B = For instance, if n = and m = 3, and A = ( a a a a ) ( A n m m n B, B = ) (9) b b b 3 b b b 3 b 3 b 3 b 33 W, () where it is emphasized that A and B act on different spaces Their direct sum is a a a a A B = b b b 3 () b b b 3 b 3 b 3 b 33 The reason why the blockdiagonal form is appropriate is clear once you act it on v w, (A B)( v w) = ( A n m m n B ) ( v w ) = ( A v B w ) = (A v) (B w) () If you have two matrices, their multiplications are done on each vector space separately, (A B )(A B ) = (A A ) (B B ) (3) Note that not every matrix on W can be written as a direct sum of a matrix on and another on W There are (n+m) independent matrices on W, while there are only n and m matrices on and W, respectively Remaining (n + m) n m = nm matrices cannot be written as a direct sum Other useful formulae are det(a B) = (deta)(detb), (4) Tr(A B) = (TrA) + (TrB) (5) 4
5 3 Tensor Product The word tensor product refers to another way of constructing a big vector space out of two (or more) smaller vector spaces You can see that the spirit of the word tensor is there It is also called Kronecker product or direct product 3 Space You start with two vector spaces, that is ndimensional, and W that is mdimensional The tensor product of these two vector spaces is nmdimensional Here is how it works Let { e, e,, e n } be the basis system of, and similarly { f, f,, f m } that of W We now define nm basis vectors e i f j, where i =,, n and j =,, m Never mind what means at this moment We will be more explicit later Just regard it a symbol that defines a new set of basis vectors This is why the word tensor is used for this: the basis vectors have two indices 3 ectors We use the same notation for the column vectors as in Section The tensor product is bilinear, namely linear in and also linear in W (If there are more than two vector spaces, it is multilinear) What it implies is that v w = ( n i v i e i ) ( m j w j fj ) = n i mj v i w j ( e i f j ) In other words, it is a vector in W, spanned by the basis vectors e i f j, and the coefficients are given by v i w j for each basis vector One way to make it very explicit is to literally write it out For example, let us take n = and m = 3 The tensor product space is nm = 6 dimensional The new basis vectors are e f, e f, e f 3, e f, e f, and e f 3 Let us write them as a sixcomponent column vector, e f =, e f = 5, e f 3 =,
6 e f =, e f =, e f 3 = (6) Here, I drew a horizontal lines to separate two sets of basis vectors, the first set that is made of e, and the second made of e For general vectors v and w, the tensor product is 33 Matrices v w = v w v w v w 3 v w v w v w 3 (7) A matrix is mathematically a linear map from a vector space to another vector space Here, we specialize to the maps from a vector space to the same one because of our interest in applications to quantum mechanics, A :, eg, v A v, (8) or v a a a n v v a a a n v (9) v n a n a n a nn v n On the tensor product space, the same matrix can still act on the vectors, so that v A v, but w w untouched This matrix is written as A I, where I is the identity matrix In the previous example of n = and m = 3, 6
7 the matrix A is twobytwo, while A I is sixbysix, A I = a a a a a a a a a a a a () The reason why this expression is appropriate is clear once you act it on v w, (A I)( v w) = = a a a a a a a a a a a a (a v + a v )w (a v + a v )w (a v + a v )w 3 (a v + a v )w (a v + a v )w (a v + a v )w 3 v w v w v w 3 v w v w v w 3 = (A v) w () Clearly the matrix A acts only on v and leaves w W untouched Similarly, the matrix B : W W maps w B w It can also act on W as I B, where I B = b b b 3 b b b 3 b 3 b 3 b 33 b b b 3 b b b 3 b 3 b 3 b 33 () 7
8 It acts on v w as (I B)( v w) = = b b b 3 b b b 3 b 3 b 3 b 33 b b b 3 b b b 3 b 3 b 3 b 33 v (b w + b w + b 3 w 3 ) v (b w + b w + b 3 w 3 ) v (b 3 w + b 3 w + b 33 w 3 ) v (b w + b w + b 3 w 3 ) v (b w + b w + b 3 w 3 ) v (b 3 w + b 3 w + b 33 w 3 ) v w v w v w 3 v w v w v w 3 = v (B w) (3) In general, (A I)( v w) = (A v) w, and (I B)( v w) = v (B w) If you have two matrices, their multiplications are done on each vector space separately, (A I)(A I) = (A A ) I, (I B )(I B ) = I (B B ), (A I)(I B) = (I B)(A I) = (A B) (4) The last expression allows us to write out A B explicitly, A B = a b a b a b 3 a b a b a b 3 a b a b a b 3 a b a b a b 3 a b 3 a b 3 a b 33 a b 3 a b 3 a b 33 a b a b a b 3 a b a b a b 3 a b a b a b 3 a b a b a b 3 a b 3 a b 3 a b 33 a b 3 a b 3 a b 33 (5) It is easy to verify that (A B)( v w) = (A v) (B w) Note that not every matrix on W can be written as a tensor product of a matrix on and another on W Other useful formulae are det(a B) = (deta) m (detb) n, (6) Tr(A B) = (TrA)(TrB) (7) 8
9 In Mathematica, you can define the tensor product of two square matrices by Needs["LinearAlgebra MatrixManipulation "]; KroneckerProduct[a_?SquareMatrixQ, b_?squarematrixq] := BlockMatrix[Outer[Times, a, b]] and then KroneckerProduct[A,B] 34 Tensor Product in Quantum Mechanics In quantum mechanics, we associate a Hilbert space for each dynamical degree of freedom For example, a free particle in three dimensions has three dynamical degrees of freedom, p x, p y, and p z Note that we can specify only p x or x, but not both, and hence each dimension gives only one degree of freedom, unlike in classical mechanics where you have two Therefore, momentum eigenkets P x p x = p x p x from a basis system for the Hilbert space H x, etc The eigenstate of the full Hamiltonian is obtained by the tensor product of momentum eigenstates in each direction, p x, p y, p z = p x p y p z (8) We normally don t bother writing it this way, but it is nothing but a tensor product The operator P x is then P x I I, and acts as (P x I I)( p x p y p z ) = (P x p x ) (I p y ) (I p z ) = p x p x p y p z (9) and similarly for P y and P z The Hamiltonian is actually H = ( (Px I I) + (I P y I) + (I I P z ) ) (3) m Clearly we don t want to write the tensor products explicitly each time! 9
10 35 Addition of Angular Momenta Using the example of n = and m = 3, one can discuss the addition of s = / and l = is twodimensional, representing the spin /, while W is threedimensional, representing the orbital angular momentum one As usual, the basis vectors are ) ) for s = /, and, =, = (,, =,, = (,, =, (3), (3) for l = The matrix representations of the angular momentum operators are ( ) ( ) S + = h, S = h, S z = h ( ), (33) while L + = h, L = h, L z = h (34) On the tensor product space W, they become S + I = h, I L + = h, S I = h, I L = h,
11 S z I = h, I L z = h The addition of angular momenta is done simply by adding these matrices, J ±,z = S ±,z I + I L ±,z, J + = h, J = h, 3 J z = h 3 (36) (35) It is clear that 3, 3 =, 3, 3 = (37)
12 They satisfy J + 3, 3 = J 3, 3 = By using the general formula J ± j, m = j(j + ) m(m ± ) j, m ±, we find J 3, 3 = = 3 3,, J + 3, 3 = We normally write them without using column vectors, J 3, 3 = J,, = 3 3, (38) =,, +,, = 3 3,, (39) J + 3, 3 = J +,, =,, +,, = 3 3, (4) Their orthogonal linear combinations belong to the j = / representation,, = 3,, = 3 (4) It is easy to verify J +, = J, =, and J, =, Now we can go to a new basis system by a unitary rotation U = (4)
13 This matrix puts transpose of vectors in successive rows so that the vectors of definite j and m are mapped to simply each component of the vector, U 3, 3 =, U 3, =, U 3, =, U 3, 3 =, U, =, U 3, 3 = (43) For the generators, we find UJ z U = h 3 3, (44) UJ + U = h 3 3, (45) UJ U = h 3 3 (46) Here, the matrices are written in a blockdiagonal form What we see here is that this space is actually a direct sum of two representations j = 3/ and 3
14 j = / In other words, / = 3/ / (47) This is what we do with Clebsch Gordan coefficients Namely that the Clebsch Gordan coefficients define a unitary transformation that allow us to decompose the tensor product space into a direct sum of irreducible representations of the angular momentum As a mnemonic, we often write it as 3 = 4 (48) Here, the numbers refer to the dimensionality of each representation j + If I replace by and by +, my eightyear old daughter can confirm that this equation is true The added meaning here is that the righthand side shows a specific way how the tensor product space decomposes to a direct sum 4
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