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Unformatted text preview: CHEM 356, Lecture 11, Fall 2009 1 The Harmonic Oscillator. Conceptually : the 1D harmonic oscillator is no more diﬃcult to deal with than is the particle in a 1D box. Mathematically : the mathematics of the 1D harmonic oscillator is somewhat more complicated to handle. Physically : the 1D harmonic oscillator model is very important, as it serves as the simplest prototype model for molecular vibrational analysis, and for vibrational motion in simple solids (as, for example, we have seen when discussing the behaviour of the heat capacity of monatomic solids). The model is one of a particle of mass moving in one dimension, subject to the potential energy ( ) = 1 2 2 . The physical importance of this model problem can be realized if we recall from earlier lectures that the potential energy is a scalar function whose derivative (or gradient, in 2D and 3D cases) gives the force on the particle: i.e., = − ? ? = − . ( F = −∇ ) We shall follow the usual 4–step approach. i) hamiltonian: ℋ = − ℏ 2 2 ? 2 ? 2 + 1 2 2 ; For large values of ∣ ∣ , we see that ( ) → ∞ , and thus ( ) must be such that ( ) → 0 as → ∞ . This gives us two ‘terminal’ boundary conditions, namely, lim →±∞ ( ) = 0 . CHEM 356, Lecture 11, Fall 2009 2 ii) Write down the time–independent Schr¨ odinger equation: − ℏ 2 2 ? 2 ? 2 + ( ) = , which we shall rewrite in the form ?...
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- Fall '09
- 1D harmonic oscillator, Hermite DE