Classical to Quantum Mechanics Oscillator

Classically, Hooke’s law gives the force as a function of the position,  that substituted in Newton’s second law,  results in

When calling    the solution of the previous equation is  where  and  are integration constants that depend on the initial conditions of the oscillation.

 On the other side, given the form of the  force, , the associated potential energy can be calculated,  Substituting the previous potential energy in the steady state Schrödinger equation, , it becomes, Or

 The Constant Barrier Case

 In the case of a constant barrier, the Schrödinger equation is with the energy of the particle smaller than the barrier’s energy . Here, the Schrödinger equation is written as with For large , the solution to the previous equation is . This solution vanishes for large  as expected. The potential energy for the simple harmonic oscillator can be visualized as a potential made out of several steps as the one presented on the graph of the left (see graph below). However, in this case, the barrier increases as  is increased. Thus, the solution for the simple barrier, , need to be modified in order to account for the variation of the potential along the  axis. To visualize the modification, let us write the Schrödinger equation in the following form with for which an approximated solution can be . For large ,  which implies that a possible solution of the quantum mechanic oscillator may be  where  (with units of length) can be obtained from  with . Thus, .

To prove our guess, let us substitute this solution, , on the equation,

and

Thus, substituting in the Schrödinger equation results in

Since this equation must be valid for all ,  and . The first of these relations is the definition of  and the second equation set the value of the energy for this solution,

 Since there is not a node on this solution, this solution must be the solution with the lowest possible energy.

 Higher Energy Solutions

In order to explore the higher energy solutions, the following variable change is applied . Now,  and . Thus, the Schrödinger equation becomes

Renaming  the Schrödinger equation is reduced to

For large , in this case, large , just as in the previous case, the solution should be modulated by a function like  so the solution can be written as . Substituting in the previous equation, the differential equation for  can be obtained

The previous equation can be solved by using a power series expansion,

Substituting the previous expressions in the differential equation,

For the previous expression to be independent of , each square bracket must be zero; thus, , , and from where the series recurrent relation can be deduced

If no restriction is placed on , it can be proved that the series represents an exponential function of the form,   implying that  is a diverging solution for large . However, if  for a given . The previous series will stop at the given and the function  becomes a polynomial. These polynomials are called Hermite polynomials.

Therefore, the energy of the oscillator is quantized accordingly with

for

The corresponding normalized wave functions are written as

Where

 Hermite Polynomial 0 1 2 3 4 5

Graph of the first few Hermite Polynomials

Corresponding Wave Functions

 by Luis F. Sáez, Ph. D. Comments and Suggestions: LSaez@dallaswinwin.com