Quantum Mechanics(14/2)Taehwang Son Functions as vectors  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.

Quantum Mechanics(14/2)Taehwang Son Functions as vectors  The function f(x) is approximated by its values at three points, x 1, x 2, and x 3, and is represented as a vector in a three-dimensional space.  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..

Quantum Mechanics(14/2)Taehwang Son Functions as vectors  Dirac bra-ket notation bra vector ket 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…).

Quantum Mechanics(14/2)Taehwang Son Functions as vectors  Expansion coefficients  Identity matrix  Hilbert space

Quantum Mechanics(14/2)Taehwang Son Linear operator  An example of operator  Bilinear expansion

Quantum Mechanics(14/2)Taehwang Son 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.