The Harmonic Oscillator: Zero Point Energy and Tunneling

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1 The Harmonic Oscillator: Zero Point Energy and Tunneling Lecture Objectives: 1. To introduce simple harmonic oscillator model using elementary classical mechanics.. To write down the Schrodinger equation for harmonic oscillator and describe the solutions. This module will not provide the method of solving the Schrodinger equation. 3. To express the energy levels of the harmonic oscillator in terms of the mass of the oscillator, the spring (force) constant and quantum numbers. 4. To plot the harmonic oscillator wave functions and their squares and identify the different sections of area under the square of the functions as probability of finding out the oscillator in that region. 5. To identify regions of the squares of wave functions beyond the classical potential energy (negative kinetic energy) region and associate with quantum mechanical tunneling. 6. To present an elementary summary of two dimensional and three dimensional isotropic harmonic oscillators using quantum mechanics. Lecture outcomes: 1. You will be able to solve for the vibrational frequencies of a simple diatomic molecule such as HCl, CO, NO using harmonic oscillator approximation.. You will be able to calculate bond force constants for different molecules using the harmonic oscillator approximation and arrange the molecule in terms of increasing stability. 3. You will be able to calculate the most probable vibrational coordinate values in which the molecule is likely to exist at very low temperatures. 4. By reading additional materials provided or pointed out in the last section, you will observe that harmonic model does not predict molecular dissociation at high temperatures or with the help of radiation, through vibrational motion, and therefore this model is only a preliminary account of vibrational spectroscopic concepts. Molecules do dissociate and through vibrational motion of the atoms around a bond. This is a deficiency which will hopefully prompt you to read more.

2 Lecture summary 1. The classical mechanical energy of the harmonic oscillator is given.. The Hamiltonian for the harmonic oscillator is given from the classical expression and the Schrödinger equation for the oscillator is written down. The equation is not solved in detail here. It is given in a supplement to this lecture. 3. The solutions, namely the eigenfunctions and eigenvalues of the Schrödinger equation, are described. 4. Plots of eigenvalues, eigenfunctions and the probabilities are included. A brief summary of why the study of harmonic oscillator is important for chemists and what new results that one gets from the analysis are indicated. 1. Classical Mechanics of the Oscillator: A simple harmonic oscillator is an example for small-amplitude motion of a mass point about an equilibrium point. Two flash movies of a grandfather clock and the diatomic molecule vibrating with small amplitudes should be included. They can be clipped from the adobe pdf file in the same folder. The restoring force F on the oscillating species is proportional to the magnitude of its displacement from equilibrium where the force is zero; the directions of the force and the displacement are opposite to each other. Put the diagram of the equilibrium position, the displacement and the restoring force be indicated. There are two pictures below the animations in the pdf file that can be put in here. The color background needs to be edited to blue as was done in earlier lectures. The restoring force on the oscillator undergoing harmonic motion is proportional to the displacement of the oscillator from its equilibrium position. For a diatomic molecule, the equilibrium distance is not zero but the bond distance at zero Kelvin in the limit of no vibration. The restoring force is therefore proportional to the difference between the instantaneous value for the bond distance (x) and the equilibrium bond distance x e. The proportionality constant k is known as the spring constant or the force constant. Formally

3 F x or F x( xx e ) F kx or F kx The restoring force of the harmonic oscillator for any given displacement is also equal to the negative derivative of the potential energy of the oscillator for that displacement, Hence it can be calculated immediately as F k x or F k x V x ; V(x) 1 kx constant or 1 kx constant The constant is the value of the potential energy in the absence of any displacement and in the case of molecule it is the potential energy of the molecule with no vibration. Clearly, these may not be zero, but for the study of harmonic oscillators in what follows, we ignore them as they contribute constant values to all our calculation. They will be set to zero. We shall use the following expression for the potential energy with the understanding that x stands for xalso in the case of molecular motion. 1 V( x) kx We can plot the potential energy as a function of the displacement and it is an even function and is a parabola centred at x 0A larger value of k means that the restoring force is high even for small displacement. It means, for a diatomic molecule, a stronger bond. A smaller value of k means for example, a weaker bond for an oscillating diatomic molecule. Illustrate the different values of k and the shape of the parabola with a figure. Please ask me. I have not included this in this directory. The kinetic energy of the oscillator is T 1 p mx m where x is the velocity of the oscillator with mass m and momentum p. For a diatomic molecule with atoms of mass m 1 and m, the mass of the oscillator is the reduced mass of the molecule,

4 1 1 1 mm ; 1 m m m m m m 1 1 The Hamiltonian (in classical mechanics, the total energy) for the harmonic oscillator therefore, is p 1 H T V kx m This is a classical mechanical expression. Graphs are given below for different total energies of the oscillator. From the graphs, it is possible to obtain precise values for both the kinetic energy and the potential energy of the harmonic oscillator at any point during its motion and for any value of the total energy of oscillator. It is also important to note that the oscillator can have any energy depending upon the amplitude of vibration and it can be measured. All these possibilities are shown to be false in the realm of quantum mechanics.. The Schrödinger equation for a harmonic oscillator To describe the harmonic oscillator using quantum mechanics, use the operator form for p; this has been done earlier in the box problem; the quantum mechanical Hamiltonian (which is an operator) is, therefore, Hˆ d 1 kx mdx Solving the harmonics oscillator problem in quantum mechanics means that we obtain eigenvalues and eigenvectors for the oscillator by solving the Schrödinger equation d 1 mdx Hˆ kx E. We don't solve the equation here. We discuss only the solutions. A supplement describes the method of solving the equation. 3. Solutions of the Schrödinger equation for a harmonic oscillator: The solutions are constructed using Hermite polynomials and an exponential (Gaussian) function (In the process of solving the differential equation above, one gets the Hermite s

5 differential equation whose solutions are known as Hermite polynomials.) The polynomials are infinite in number and are classified as polynomial of first order, second order, third order etc. as given below: Denote Hermite polynomials (functions of a dimensionless coordinate x) by Hv ( x), v 1,, 3,... The functional forms for the polynomials are given below for the first few of them. H ( x) 1 0 H ( x) x 1 ( ) 4 3 3( ) ( ) ( ) ( ) ( ) H x x H x x x H x x x H x x x x H x x x x H x x x x x H ( x) 56x 3594x 13, 440x 13, 440x 1680 There is a recursion relation between these polynomials which can be used to generate the above ones as well as higher ones if necessary. The relation is given here, H ( x) xh ( x) vh ( x) v1 v v1 The harmonic oscillator energies (eigenvalues) and wave functions are given as E v k 1 v m 1 1 k h v ; ; v 0, 1,, m x / ( x) N H x e, v v v km and N v 1 v v! 1/4

6 Important: The harmonic oscillator energies are quantized and are equidistant. This statement means that the difference between any two adjacent energy levels is the same for all energy levels. (The energy levels are equally spaced). E 1 E 0 h E E 1 E 3 E E 4 E 3 E v1 E v h Also, the lowest value of quantum number allowed is 0, unlike the particle in a box. Plots of eigenvalues, eigenfunctions and probabilities Label all the plots with axis and quantum numbers.

7 The levels are equidistant as you will find by substituting 0,1,,..., in the formula for the eigenvalues given above. The harmonic oscillator model is quite important in molecular spectroscopy /infrared spectroscopy to be studied later. Important: The space available for the harmonic oscillator is all possible values of x, x to Therefore, if we want the wave functions to contain probability information in the same way we did with the particle in a box or the particle in a ring, we must ensure that the

8 wave function vanishes at the boundaries, namely at x and at x. The wave function will have to be finite in the entire region in between, and must have a unique value everywhere. Also the squares of the wave function must be such that when we add (x) dx for all values of x, i.e., we do the sum (x) dx, we must get the result of unity, since the probability of finding the oscillator anywhere in all its allowed region must be unity. The solutions for the eigenfunctions given above satisfy all of these conditions, namely, x / ( x) N H x e, where N v v v v 1 v v! 1/4 km and x x dx*( x) ( x) x ( Nv) dxexp( x ) Hv( x) x 1 Problem: Verify this by substituting the first few Hermite polynomials in the above equation. Significance of the solutions. (Why is harmonic oscillator important?) 1. The quantum mechanical harmonic oscillator has some energy even when the quantum number is zero. (i.e., it cannot have zero energy of vibration). Classically an oscillator having zero energy does not oscillate!! A state of absolute rest is possible, however, only in classical systems. Zero point energy (referring to the above) is a purely quantum mechanical concept and is fundamentally important for molecules and solids.

9 . The boundary for an oscillator is endless. What does the potential look like? It is a parabola and the classical oscillator can ONLY oscillate between the two ends of the parabola. The total energy of the oscillator is all potential energy at the turning points and all kinetic energy midway between the two points. However, in quantum mechanics the wave function outside of this classical parabola can be nonzero for all values of x, though very small. Thus, the square of the wave function which represents probability density is nonzero everywhere, except for a finite number of points. It goes to zero as x tends towards infinity. This is often stated as that quantum mechanical systems can be found in regions where they are strictly forbidden if classical mechanics were applicable. This is also known as quantum mechanical tunneling. Though tunneling can often be explained by solution of the Schrödinger equation for particles in one dimensional finite potential barrier models in physics, chemistry students are introduced to tunneling from harmonic oscillator model. It is a finite potential barrier model anyway. Experimentally tunneling phenomena have been amply demonstrated. The invention of scanning tunneling electron microscope which was credited with a Nobel Prize in physics amply demonstrates the idea. which are forbidden by classical mechanics.. Squares of Harmonic oscillator wave functions. The plots of squares of wave functions above are worth analyzing. The fraction of the area under the square of the wave function in the middle region of the potential curve decreases with increasing total energy, so the harmonic oscillator behaves more and more classically with increasing energy of the oscillator. Also the square of the wavefunction tends towards a maximum at the turning points for larger and larger values of total energy meaning that the likelihood of finding the oscillator near the turning point region increases with energy. Thus the oscillator is more likely to be found where its motion is arrested as would one expect classically. For small values of energy you notice that the oscillator is more likely to be found in the middle of the potential well. (The wave functions associated with lower energies have peaks in the centre of the parabolic curve. This is against classical intuition as it is in these regions that the oscillator has the maximum kinetic energy, classically speaking, and is least likely to be spotted!!

10 3. Neither the kinetic energy alone nor the potential energy alone can be determined exactly for the harmonic oscillator. Only average values can be obtained through many measurements for which a theoretical limiting value can be provided. Mathematically this can be stated as follows: the kinetic energy operator does not commute with the potential energy operator and therefore eigenvalues for both cannot be obtained simultaneously. 4. The diatomic molecular vibrational energy is quantized and the simplest model above explains the basic features of the vibrational spectra of most stable molecules. Replace the mass of the oscillator by the reduced mass of the diatomic molecule and the connection between the two systems is established immediately.

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