Harmonic Oscillator Model:

The Harmonic Oscillator Model:

The harmonic oscillator model is important because not only does it demonstrate quantization of energy, but it also shows the phenomenon called quantum tunnelling, in which an electron can pass into a finite potential barrier, which is not permitted by classical mechanics.

This model is similar to the one-dimensional box situation, the electron is confined to a one-dimensional region, the x-axis. This case is different however, in that the electrostatic potential is different. Instead of being zero between two points and approaching infinity in the regions beyond, the potential increases with the power of x². The potential is centered such that its value at x=0 is zero. The electron can be thought of as being attached to one end of an imaginary spring, with the other end secured at the origin. For this system, the energy expression will be derived first, then the wave functions will be derived second (this makes it easier to understand what is going on). To begin, one first writes the Schrödinger Equation, in this case the time-independent form.

eq.1

Notice in this case, the potential energy term is everywhere nonzero, unlike the one-dimensional box model in which it was zero for the region of interest. The potential energy term is identical to the energy representation for a spring in classical mechanics, where k is the spring constant of the imaginary spring to which the electron is attached. To begin simplifying, in order that this expression may later be broken down, the second-order term on the left is isolated from any constants. One multiplies by 2m / -(h-bar)²:

eq.2

Eq. 2 is a non-homogeneous differential equation with variable coefficients. Even the simplest of systems often produce complicated expressions which take time to solve, mostly by relating them with differential equations already solved, or by using approximation schemes to find a reliable answer. This equation is similar to one solved by Charles Hermite, a French mathematician. To begin solving, the asymptotic solution to this expression is found by noting which terms disappear for large values of |x|. Here, the term on the right-side goes to zero, relative to the terms on the left-side. Rearranging:

eq.3

The asymptotic form of the wave function is labeled y _Inf. The general solution to y _Inf is:

eq.4

The solution to the actual wave function is the product of the asymptotic solution and another function, which is labeled here as S:

eq.5

Inserting this expression into eq.2 yields:

eq.6

Rewritting explicitly:

eq.7

Canceling the first and fifth terms in brackets, dividing both sides by exp(x²(mk)^1/2 / (2*h-bar)), and rearranging:

eq.8

To determine the values of E, one treats S as a power series in order to derive a recursion relation:

eq.9

Implementing this:

eq.10

Re-indexing the first term on the left and rewritting as one summation:

eq.11

In order for this to be true, the quantity in brackets must be zero for every value of x:

eq.12

Rearranging:

eq.13

In order for the wave function to be finite, one of the three conditions of a wave function, this recursion relation must terminate, otherwise the summation occurs without end. This relation terminates for when n equals a specific value, denoted as u:

eq.14

Rearranging:

eq.15

Isolating energy, produces the sought after eigenvalue expression :

eq.16

This can also be written in terms of the harmonic frequency of the electron's motion, denoted n:

eq.17

u is the quantum number for this system; it determines the specific state which the harmonic oscillator electron occupies. Now returning to the function S to derive the wave functions. S is any number of functions belonging to the set known as the Hermite polynomials. These certain polynomials when multiplied by the asymptotic solution, produce the actual solutions for the wave function. A partial list of the Hermite polynomials is presented below (Atkins 62):

These polynomials are generated by the expression (Mortimer A-45):

eq.18

The set of full wave functions for this system is generated by:

eq.19

Graphically demonstrating the wave functions for u=0,1,2,3,4,5:

Again, just as with the one-dimensional box system, the state, and therefore energy, of the electron is quantized, determined by the quantum number u which is any integer greater than or equal to zero. A surprising result is discovered by analyzing the positions of the wave functions relative to the classical potential energy curve in the above graph. Each wave function extends beyond the potential energy curve, representing the classically allowed maximum displacement of the oscillator. This effect, where the electron occupies space which it is not allowed by classicsal mechanics, is called quantum tunnelling. This effect does not mean that classical mechanics is necessarily incorrect, rather it helps one to understand that classical mechanics concerns itself with average values pertaining to objects which possess large amounts of energy, relative to electrons.