A Primer on Quantum Mechanics and Orbitals



Tunneling and the Harmonic Oscillator

Introduction

I have talked in class about the concept of tunneling. This is where, because of their quantum mechanical nature, atomic/molecular scale particles may penetrate an energy barrier (potential) that they would be forbidden to under a purely classical scenario. In other words, even though the particle cannot surmount the barrier because it has less energy, the particle still has a small but finite probability of being ‘outside’ the potential. Tunneling is very important for a variety of chemical and biological processes, mostly involving hydrogen or electron transport. In general, the small a particle is the more it is able to tunnel through a barrier because small particles with low masses (i.e. low momenta) have long wavelengths. We can actually see this illustrated fairly clearly in harmonic oscillator problems which are very good representations of molecular vibrations.

Where does tunneling come in to play for the harmonic oscillator? Well, just remember how we represent the plot of ψ0 ( the lowest vibrational state of the harmonic oscillator wavefunction). We draw function that looks like a Gaussian but that spills out beyond the graph of the potential, in other words the particle has a non-zero probability of being outside the potential.

We will use a real world example to illustrate the properties of the Harmonic oscillator and also the concept of tunneling. Recall that in this case the variable ‘x’ represents the deviation from the equilibrium bond length in the molecule. Hence, a positive x means that the bond is stretched compared to the equilibrium bond length and a negative x means that the bond is compressed.

The Problem

The Raman spectrum of H2 has a dominant peak at 1751 cm-1(wavenumbers). Raman spectroscopy is similar to infrared spectroscopy in that vibrational transitions are examined but it is sometimes applicable to problems that are not amenable to IR methods, such is the case for homonuclear diatomic molecules like H2. Given this spectrum, what is the probability that the H-H bond length will be longer or shorter than would be allowed by classical mechanics.

Step 1: Converting wavenumbers to a vibrational frequency.

A vibrational frequency is often reported in terms of wavenumbers instead of direct frequencies. The definition of a wavenumber is

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and so wavenumbers are proportional to energy just as the frequency is. Using the definition of wavenumbers, it is a simple matter so find the relationship between wavenumbers and frequency. Please find the frequency of the radiation involved in the H2 vibrational transition.

Step 2: Calculate the reduced mass, μ, of H2 (in kg).

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Step 3: What is the fundamental vibrational frequency, ν0, of H2?

Hint: recall what the energetic spacing is between any two levels of the harmonic oscillator.

Step 4: Find the force constant, k, for the H-H bond

Use the relationship between the force constant and the fundamental frequency

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Step 5: Write the wavefunction for the ground state wavefunction of the harmonic oscillator. You can find this in your textbook. Oh Hell, here it is.

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While you’re at it, you’ll need to find ‘a’ for this problem, it’s defined (again in you book) as

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Hint: When doing calculations, especially as you go on, just represent the quantity as ‘a’, until you get to the very end.

Step 6: Now, find the classical turning point.

Whoa, what is a turning point? Well, if you think of a pendulum, you can think of the turning point as that point in its arc when it reaches an extreme in the upswing and stops. Then it will start swinging back toward x=0, when it reaches that point all of the potential energy that had been contained at the ‘turning point’ arc has been converted to kinetic and the pendulum swings back to the other side only to reconvert the kinetic into potential energy at the other turning point. So, when you are at the turning point, classically speaking, you know that all of the energy of the swing is potential energy. We can use this fact to determine what the turning point of a harmonic oscillator would be if it behaved classically.

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Solve for xtp.

Step 7: Now you’re ready. You need to calculate the probability that the bond will be stretched or compressed beyond the classical turning point on either end. This is the tunneling probability. You need to integrate the probability density, ψ0*ψ0, over the area x>xtp and x ................
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