This chapter discusses the quantum mechanical formulation in the position and the momentum representations. The position representation provides a natural means of constructing the quantum analog of a classical mechanical problem. The momentum representation emerges as the Fourier transform of the position representation and often provides a convenient means of studying the time evolution of the states.

Position Representation

For the sake of simplicity we consider first a particle constrained to move on the real axis. Recall that, according to the second postulate of quantum mechanics, to every classical dynamical variable there corresponds a unique hermitian operator. Accordingly, let the hermitian operator corresponding to classical position x be denoted by. Let be an eigenstate of the position operator corresponding to the eigenvalue x, i.e.,

where the second equation above is due to the hermiticity of. A state

in the state space of the particle is then represented by the function

called the wave function. By virtue of the axioms of the scalar product,. The representation of states and operators in the basis of the eigenstates fjxig of O x is called the position representation. As is expected of a self-adjoint operator, the eigenstates fjxig of O x constitute complete orthonormal set. The completeness relation reads

The orthonormality relation reads Hence, in terms of its position representative x(x), any state can be represented as

Now, we know from the fourth postulate that is the probability density which, on invoking (8.2), implies that