Functions as vectors

1. Functions as vectors • If we know , it is possible to obtain all observable values. • In order to deal with in more complex problems, we need to introduce linear algebra. • Wave function → a list of numbers or a vector • Operators → matrices • Operation → the multiplication of the vector by the operator matrix.

2. Functions as vectors • We can imagine that the set of possible values of the argument is a list of numbers (x), and the corresponding set of values of the function (f(x)) is another list.. • The function f(x)is approximated by its values at three points, x1 , x2 , and x3 , and is represented as a vector in a three-dimensional space.

3. Functions as vectors • Dirac bra-ket notation cf) ket vector bra vector inner product • Now, let’s represent a function as an expansion of orthonormal basis set. • We have merely changed the axes, and hence the coordinates in our new representation of the vector have changed(now they are the numbers c1 , c2 , c3…).

4. Functions as vectors • Expansion coefficients → Projection • Identity matrix Ex) • Hilbert space The vector space formed by a set of orthogonal functions e.g.

5. Linear operator • An example of operator → Eigen value problem! • Bilinear expansion

6. Linear operator • Trace of an operator When calculating physical parameters, basis is not important. • Hermitian matrix • If A is Hermitian, eigenvalue is real and eigenvector is orthogonal each other. • All observables are Hermitian, so they are real value.