The Harmonic Oscillator

Quantum Mechanical Oscillator

The solution for the quantum mechanical harmonic oscillator requires the application of the Schrödinger wave equation, since the moving particle in such a situation is described by its “material” wave function Ψ(x,t), not being possible to establish its position. the way it is done for the particle in the classical harmonic oscillator.The proper way is to consider the probability that the particle is in a certain position, a value that is provided through the wave intensity formula.The wave intensity is a quantity proportional to the square of its amplitude. The amplitude of the wave corresponding to a particle will be given by the complex product:

Compared to the classical harmonic oscillator, here we cannot apply Newton’s laws. However, similarly, the application, on the mechanical system, of a potential energy given by a function, in the particular one-dimensional case, is given as a primary initial condition:

Due to the action of the potential V(x), the wave corresponding to the wave behavior of the particle will be confined in a space called a potential well. And it is considered that the phenomena of superposition of waves with the formation of standing waves will take shape there. So vibration patterns, or modes of vibration, are to be expected. These vibration modes will be harmonic modes with vibration amplitude, nodes and antinodes.

The problem now resides in finding the function that represents the amplitude of such vibration modes, that is:

so that the wave function of the particle of the system can be written as:

where n corresponds to the index relative to each vibration mode.

And so, the probability wave motion equation to be developed will be the Schrödinger equation, written in time-independent form, also called the Schrödinger amplitude equation (one-dimensional):

The solution of this equation is done through the techniques of asymptotic mathematics. Using variable substitution it is possible to rewrite it in the form:

where,

and

with

These coefficients are finite values, so that, in asymptotic terms, if we have

and

f and g will be asymptotically equivalent for large values ​​of x, so we can rewrite

whose analytical solution methodology, taking into account the postulates of quantum mechanics, results in the function

This result is imposed for asymptotic reasons, and, in order to have a solution compatible with small values ​​of x, another technique is used, assuming there is another function, modulated by this last result:

Retro substituting it into the rewritten differential equation gives:

In order to leave it in a specific pattern, this expression undergoes another variable and notation transformation, doing:

and

which results in

This pattern of differential equation is known as Hermite’s differential equation and its solution corresponds to a polynomial series, in this case the Hermite polynomial: H(ξ). With the proper algebraic manipulation it is possible to establish the recurrence relation for the coefficients b of the solution polynomial, given by the expression:

This analysis leads to the conclusion that the solution polynomial series of the equation must be interrupted when reaching a certain power, in such a way that, for this, a necessary condition is that

The use of the recurrence formula to determine the coefficients of the Hermite polynomial will require working with two constants, which would be the terms b₀ and b₁, to be manipulated using boundary conditions, that is, convergence of power series , and meeting the requirements of the Schrödinger wave function.

Examples:

On the other hand, substituting in the recurrence formula the respective expressions defined above, we have:

simplifying this, we arrive at the expression of the mechanical energy of the quantum-mechanical oscillator:

and, alternatively, one can write in the recurrent form:

where n is steady state, or characteristic energy values.

As for the wave amplitude function, it can be written, in recurrent form, as: