Matrix formulation of quantum mechanics

hi everyone this is Professor M the science and today I want to tell you about the matrix formulation of quantum mechanics also known as matrix mechanics this is another one of our videos on rigorous quantum mechanics the matrix formulation of quantum mechanics was the first consistent formulation of quantum theory and it was first proposed by Heisenberg Bourne and Jordan as early as 1925 it is equivalent to the later wave mechanics formulation of Schrodinger and are both unified in the state space formalism so why do we care about matrix mechanics the matrix formulation is most useful when we deal with finite discrete basis and the reason for that is because then it reduces to the simple rules of matrix multiplication in fact for most practical solutions of quantum mechanics we use the matrix formulation whether we want to solve quantum mechanics on pen and paper using analytical solutions or we want to use the largest supercomputers on earth to find numerical solutions in this video I will introduce how to write down Katz brass and operators in the matrix formulation of quantum mechanics and then I will work through a number of examples to see how the usual rules of matrix multiplication can be used to calculate quantum mechanical quantities so let's go to introduce the matrix formulation of quantum mechanics we start with cats we expand the cat toy in a basis you where the expansion coefficient C are given by the bracket between you and beside this C coefficient are what we call the representation of the cat's eye in the U basis and if you need a refresher check out the video representations to define the matrix formulation of cats we arrange this coefficient into a column vector and we can do it either in terms of the brackets europe's eye or directly in terms of the expansion coefficient c therefore in the matrix formulation of quantum mechanics cats are written as column vectors now that we know how to write cats as column vectors we next one to look at brows we expand the brows I in the basis you in terms of the same expansion coefficients see that we used to expand the corresponding ket but in this case the expansion coefficients are the complex conjugates C star again if you need to refresh it about this check the video on representations that is linked in the description to defy the matrix formulation of brass we arranged this coefficient into a row vector we can again write this out in terms of the brackets IU or in terms of the complex conjugates of the C coefficients therefore in the matrix formulation of quantum mechanics bras are written as row vectors after Keds and brass the next quantity we need our operators an operator a can be written in the U basis as the sum over the outer products of the basis States and the expansion coefficients AI J are given by the matrix elements of a with respect to the basis States the very name matrix element for this quantity comes from the matrix formulation of quantum mechanics that we will see in a moment the expansion coefficients for an operator are labeled by two indices so what we will do is to arrange them in the form of a square matrix with the first index denoting the row of the matrix and the second index the column of the matrix therefore in the matrix formulation of quantum mechanics operators are written as matrices as a summary in the matrix formulation of quantum mechanics a ket sy which is expanded in terms of the C coefficient in the U basis is written as a column vector of these C coefficients abrupt sigh expanded in terms of the sister coefficient is written as a row vector and an operator a expanded in terms of the matrix elements a IJ is written as a matrix at this point you will all be thinking okay we can arrange the representations of ket's bras and operators as vectors and matrices but what is the point of all this and here is the point of matrix mechanics when we write States and operators as vectors and matrices then all the manipulations of these objects that we need to perform in quantum mechanics can be performed in terms of the usual rules of matrix multiplication in the rest of the video I will look at the four operations listed here to see what they look like in the matrix formulation of quantum mechanics some of these expressions give scalars some give cats and some give operators so we will see how all this fits together in the language of matrix multiplication let's start with the bracket sci fi which we now give the scalar we first write both sigh and Phi in the Yuba and we do this in terms of expansion coefficient c and v calculating the bracket between Si and Phi we first write down the expansion of both these terms in the U basis by copying the expansions we wrote on the Left we can then rearrange this expression in the usual manner and we obtain this double sum over I and J as the basis is also normal we recognize this bracket as Delta IJ so the double sum becomes sum over i CI star di this last expression here is the expression for the scalar product of a row and the column vector to see this explicitly we write the props I as a row vector of sistar coefficients and we write the cat file as a column vector of D coefficients using the usual rules of matrix multiplication we first get c1 star times d1 we then add C to star times d2 and so on and combining this term into a sum we obtain this comparing this term with this term we see that indeed they are the same therefore a bracket in the matrix formulation of quantum mechanics is the matrix product of a row vector with a column vector next let's look at the action of an operator a on a cat side giving us another cat side Prime we start in the same way by first writing both sigh Prime and sigh in the U basis and in this case the corresponding expansion coefficients are C Prime and C the cat side prime is represented by the C prime coefficient and the cat SCI is represented by the sequel officiants therefore to describe the action of the a operator we need to find what the C prime coefficients are in terms of the seeker officiants to do that we first write CI prime equals the bracket between UI and beside prime we then use the definition of Sai Prime in terms of a to write this we then insert the identity operator after a with an insert the resolution of the identity in the U basis as shown here and we then rearrange to obtain a sum over J of these two terms the first term is the matrix element a IJ and the second term is CJ so we can write the whole thing sum over J a jcj this last expression here is the expression for the action of a matrix on a column vector to see this explicitly we first write down the matrix a we then write down the column vector C and then we use the usual rules of matrix multiplication so we start with the first row of the matrix acting on the column vector to obtain the first entry here and we can then continue with the subsequent rows of the matrix obtain the subsequent entries here each of the rows corresponds to one of these sums here so we can collect all the rows into the column vector for C Prime as expected multiplying a matrix with a column vector gives us another column vector just as the action of an operator on a ket gives us another ket the next example I want to look at is the adjoint operator which describes the action of an operator in the dual space we start with the matrix element IJ of the adjoint of a this is equal to the matrix element of a dagger with respect to UI and UJ we can then use conjugation to write this as UJ a UI star and this simplifies to complex conjugate of the matrix element ji of a what this is showing is that we need to exchange IJ by ji and then take the complex conjugate therefore in order to go from the operator a to the adjoint operator a dagger we first write the operator a in matrix form as shown here and we then transform it into the adjoint operator by exchanging rows with columns and then taking the complex conjugate of each entry therefore the matrix formulation of the adjoint of an operator a is given by the transpose conjugate matrix of the original operator the final example I want to look at is how we write an operator as now the product of two states we again start by writing Tsai and Phi in the U basis in terms of the expansion coefficients C and D we then build the outer product sci fi and then insert the expansion of these two state in the U basis as shown here rearranging this expression as usual we obtain a double sum over I and J for some coefficients C ID J star and then the outer product of the basis States comparing this to the usual expression for an opera in a particular basis we identified the CID J star term as the matrix elements of this operator since it is explicitly in the matrix formulation we write the cat psy as the column vector of C and multiply by the broth I as the row vector of D using the standard rules of matrix multiplication we first combine c1 with D 1 star to get the first entry here then c1 again with D 2 star to get the second entry and so on to fill the first row we can then repeat all these multiplications but now with the c2 rather than c1 so we obtain the second row here and so on and that expected the result is a matrix whose entries are precisely the matrix elements of the operator up here the outer product is therefore given by the matrix multiplication of a column vector with a row vector and the result is a matrix overall we have looked at four examples to show that the usual rules of matrix multiplication allow us to do the usual operations of quantum mechanics in the matrix formulation of this theory a bracket is written as a row vector times the column vector which leaves a scalar the action of an operator a on a ket sy is written as the multiplication of a matrix by a column vector which gives another column vector the adjoint operator is written as the transpose conjugate matrix and the outer product of two state is written as the multiplication of column vector with row vector which gives a matrix you can play with these and similar expressions yourself and here I have two more you should show that the first one gives a scalar and the second one gives a matrix so what have we learned today we have introduced the formulation of quantum mechanics known as matrix mechanics this is the most useful formulation for the practitioner of quantum mechanics because it reduces all operations to matrix multiplications if you liked this video or you would like to send me suggestions for future videos please subscribe

Channel: Professor M does Science

Views: 8,592

Rating: 4.9870548 out of 5

Keywords: quantum mechanics, quantum theory, quantum, matrix mechanics, matrix formulation, matrix representation, column vector, outer product, row vector, operator, quantum state, ket, bra

Id: wIwnb1ldYTI

Length: 10min 53sec (653 seconds)

Published: Wed Jun 24 2020