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One problem with this classical formulation is that it is not general. We cannot use it, for example, to describe vibrations of diatomic molecules, where quantum effects are important. A first step toward a quantum formulation is to use the classical expression $k=m\text{\hspace{0.05em}}{\omega}^{\text{\hspace{0.05em}}2}$ to limit mention of a “spring” constant between the atoms. In this way the potential energy function can be written in a more general form,
Combining this expression with the time-independent Schrӧdinger equation gives
To solve [link] —that is, to find the allowed energies E and their corresponding wave functions $\psi (x)$ —we require the wave functions to be symmetric about $x=0$ (the bottom of the potential well) and to be normalizable. These conditions ensure that the probability density $|\psi (x){|}^{\text{\hspace{0.05em}}2}$ must be finite when integrated over the entire range of x from $\text{\u2212}\infty $ to $+\infty $ . How to solve [link] is the subject of a more advanced course in quantum mechanics; here, we simply cite the results. The allowed energies are
The wave functions that correspond to these energies (the stationary states or states of definite energy) are
where $\beta =\sqrt{m\text{\hspace{0.05em}}\text{\hspace{0.05em}}\omega \text{/}\hslash}$ , ${N}_{n}$ is the normalization constant, and ${H}_{n}(y)$ is a polynomial of degree n called a Hermite polynomial . The first four Hermite polynomials are
A few sample wave functions are given in [link] . As the value of the principal number increases, the solutions alternate between even functions and odd functions about $x=0$ .
Several interesting features appear in this solution. Unlike a classical oscillator, the measured energies of a quantum oscillator can have only energy values given by [link] . Moreover, unlike the case for a quantum particle in a box, the allowable energy levels are evenly spaced,
When a particle bound to such a system makes a transition from a higher-energy state to a lower-energy state, the smallest-energy quantum carried by the emitted photon is necessarily hf . Similarly, when the particle makes a transition from a lower-energy state to a higher-energy state, the smallest-energy quantum that can be absorbed by the particle is hf . A quantum oscillator can absorb or emit energy only in multiples of this smallest-energy quantum. This is consistent with Planck’s hypothesis for the energy exchanges between radiation and the cavity walls in the blackbody radiation problem.
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