QM with one continuous degree of freedom

Consider the Hilbert space with vectors represented in the position basis . The momentum operator is given by

and thus the Hamiltonian operator by

and the Schrödinger equation becomes

Time independent Schrödinger equation

If is time-independent the stationary states are given by the time-independent Schrödinger equation

and thus general solutions are given by¹

Properties of solutions

If is an even function then is either odd or even

Proof of 1

Let be even, i.e. Let be a solution to the TISE so that

it follows immediately that

Let , so

hence is even and is odd. Then by linearity, and , so any solution is a linear combination of even and odd eigenstates. Hence may be chosen as eigenstates. Note in cases of energy degeneracy there is always a choice.

General properties

The canonical commutation relations is

The energy of a normalizable solution must exceed the the infimum of the potential.

Particular potentials

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2018. Introduction to Quantum Mechanics, §2.1, p. 26 ↩