Harmonic oscillator potential curve

FIGURE 8.2 The simple harmonic oscillator potential curve with energy levels and wavefunctions 17 (y) shown to v = 5. 

If an H 2 molecule had a pure harmonic oscillator potential curve with force constant k = 570 N m and equilibrium bond length R of 0.74 A, estimate the probability of finding the ground state molecule with a bond length of R = 2.00 0.01 A. 

Figure 6.4 Potential energy curve and energy levels for a diatomic molecule behaving as an anharmonic oscillator compared with those for a harmonic oscillator (dashed curve)...

Fig. 2.1. Approximate potentials for the nuclear shell model. The solid curve represents the 3-dimensional harmonic oscillator potential, the dashed curve the infinite square well and the dot-dashed curve a more nearly realistic Woods-Saxon potential, V(r) = — V0/[l + exp (r — R)/a ] (Woods Saxon 1954). Adapted from Cowley (1995).

Figure 9.7 Vibrational energy levels determined from solution of the one-dimensional Schrodinger equation for some arbitrary variable 6 (some higher levels not shown). In addition to the energy levels (horizontal lines across the potential curve), the vibrational wave functions are shown for levels 0 and 3. Conventionally, the wave functions are plotted in units of (probability) with the same abscissa as the potential curve and an individual ordinate having its zero at the same height as the location of the vibrational level on the energy ordinate - those coordinate systems are explicitly represented here. Note that the absorption frequency typically measured by infrared spectroscopy is associated with the 0 —> 1 transition, as indicated on the plot. For the harmonic oscillator potential, all energy levels are separated by the same amount, but this is not necessarily the case for a more general potential...

The BQ1 term alone, with B positive, would give a potential resembling the harmonic oscillator potential in Figure 6.4 (dashed curve) but with steeper sides. The inclusion of the AQ2 term, with A negative, adds an upside-down parabola at Q = 0 and the result is a W-shaped potential. The barrier height b is given by... 

Calculation of Morse and Harmonic Oscillator Potential Energy Curves... 

We postulate the potential energy curve of a dissociating harmonic oscillator reactant as that shown in Fig. 1. It is a truncated harmonic oscillator potential with a finite number of equally spaced energy levels such that the level N is the last... 

Fig. 3a. Time of flight spectrum for 700 nK atoms. Fig 3b. Time evolution of the temperature of Cs atoms during adiabatic expansion (points). The solid curve is the evolution expected if the atoms were expanding in a weakening harmonic oscillator potential.

Figure Bl.2.3. Comparison of the harmonic oscillator potential energy curve and energy levels (dashed lines) with those for an anharmonic oscillator. The harmonic oscillator is a fair representation of the true potential energy curve at the bottom of the well. Note that the energy levels become closer together with increasing vibrational energy for the anharmonic oscillator. The anharmonicity has been greatly exaggerated.

Figure 25.2 Potential energy curve for a diatomic molecule (full line) compared to that for the harmonic oscillator (dashed curves). Note differences in vibrational levels.

FIGURE 4.5 Potential energy for vibration of a diatomic molecule (solid curve) and for a harmonic oscillator (dashed curve). Also shown are the bound-state vibrational energy levels for the diatomic molecule. In contrast to the harmonic-oscillator, a diatomic molecule has only a finite number of bound vibrational levels... 

A comparison of the Morse potential (blue) and the harmonic oscillator potential (green), showing the effects of anharmonicity of the potential energy curve, where is the depth of the well. [ Mark M Sa moza/CCC-BY-SA 3.0/G FDL /Wikimedia Commons reproduced from http //en.wikipedia.org/wiki /Morse potential (accessed December 27, 2013).]... 

FIGURE 16-3 Potential-energy diagrams, (a) harmonic oscillator, (b) Curve 1, harmonic oscillator curve 2. anharmonic motion. 

Figurell.8 The parabola shows the harmonic oscillator potential. The horizontal lines show the energy levels, which are equally spaced for the quantum harmonic oscillator. The lowest level has energy liv/2. This is called the zero-point energy. The curve on each horizontal line shows tp x), the particle distribution probability for each energy level.

Our goal with this derivation is to find a simple and reasonably accurate solution to the energies and wavefunctions of atoms vibrating in a chemical bond. We will first show that the potential energy curve of a chemical bond can be crudely approximated by the harmonic oscillator potential energy jkx . Then we will turn once more to our friends at the 19th-century French Academy for the solution to the differential equation. The final step is relating the mathematical solution to the physical parameters of the molecule. A DERIVATION SUMMARY appears after Eq. 8.24. 

There is a fair match between the harmonic oscillator potential and the real molecular potential energy curve we saw in Chapter 5 at low potential energy (this is our requirement that R should be near R in the Taylor series). The match becomes worse as we go up in energy and the potential becomes anharmonic. At low vibrational energy, the harmonic oscillator is a good approximation to most molecular vibrations. 

FIGURE 14.29 The Morse potential is a better fit to the potential energy curve of a real molecule than is the harmonic oscillator potential energy surface, superimposed... 

The above results are based upon a harmonic oscillator potential energy curve, which of course does not give a particularly good representation of the potential energy curves for actual molecules. We could include additional terms of 14-18 in the expression for the potential energy a more satisfactory procedure is to assume some appropriate... 

Let us consider again the potential diagram shown in Figure 3.2. For a harmonic oscillator, the curves are parabolas described by Equation (3.3). Both in the initial and the final state, the dipoles of the solvent fluctuate. Hence within the framework of our model which does not take into account the dielectric saturation, it is natural to assume the same frequency o) both in the initial and in the final states. Thus, the curves Uj and U are actually the same, but with different coordinates of the minima (U and qQ). The point of intersection of the potential curves is found from the condition Ui(q ) = Uf(q ) ... 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

Owing to the effects of mechanical anharmonicity - to which we shall refer in future simply as anharmonicity since we encounter electrical anharmonicity much less frequently -the vibrational wave functions are also modified compared wifh fhose of a harmonic oscillator. Figure 6.6 shows some wave functions and probabilify densify functions (iA A ) for an anharmonic oscillator. The asymmefry in and (iA A ) 5 compared wifh fhe harmonic oscillator wave functions in Figure f.i3, increases fheir magnitude on the shallow side of the potential curve compared with the steep side. 

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