5.6: The Harmonic-Oscillator Wavefunctions involve Hermite Polynomials
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For a diatomic molecule, there is only one vibrational mode, so there will be only a single set of vibrational wavefunctions with associated energies for this system. For polytatomic molecules, there will be a set of wavefunctions with associated energy associated with each vibrational mode.
The Hamiltonian operator, the general quantum mechanical operator for energy, includes both a kinetic energy term, \(\hat {T}\), and a potential energy term, \(\hat {V}\).
\[ \hat {H} = \hat {T} + \hat {V} \label {5.6.2}\]
For the free particle and the particle in a box, the potential energy term used in the Hamiltonian was zero. As shown in Equation \(\ref{5.6.2}\), the classical expression for the energy of a harmonic oscillator includes both a kinetic energy term and the harmonic potential energy term. Transforming this equation into the corresponding Hamiltonian operator gives,
\[\hat {H} (q) = \dfrac {1}{2 \mu} \hat {P}^2_q + \dfrac {k}{2} \hat {q}^2 \label {5.6.3}\]
where \(\hat {q}\) is the operator for the length of the normal coordinate, and \(\hat {P}_q\) is the momentum operator associated with the normal coordinate. \(\mu\) is an effective (reduced) mass, and \(k\) is an effective force constant, and these quantities will be different for each of the normal modes (vibrations).
Substituting the definitions for the operators yields
\[\hat {H} (q) = -\dfrac {\hbar ^2}{2\mu} \dfrac {d^2}{dq^2} + \dfrac {k}{2} q^2 \label {5.6.4}\]
since the operator for position or displacement is just the position or displacement. The time-independent Schrödinger Equation then becomes
\[- \dfrac {\hbar ^2}{2\mu} \dfrac {d^2 \psi _v (q)}{dq^2} + \dfrac {k}{2} q^2 \psi _v (q) = E_v \psi _v (q) \label {15.6.5}\]
or upon rearranging
\[ \dfrac {d^2 \psi _v (q)}{dq^2} + \dfrac {2 \mu}{\hbar ^2} \left ( E_v - \dfrac {k}{2} q^2 \right ) \psi _v (q) = 0 \label {15.6.6}\]
This differential equation is not straightforward to solve. Rather than fully develop the details of the solution, we will outline the method used because it represents a common strategy for solving differential equations. The steps taken to solve Equation \(\ref{15.6.6}\) are to simplify the equation by collecting constants in the parameter \(\beta\)
\[ \beta ^2 = \dfrac {\hbar}{\sqrt { \mu k}} \label {15.6.7} \]
and then changing the variable from \(q\) to \(x\) where
\[x = \dfrac {q}{\beta} \label{scale}\]
so that
\[ \dfrac {d^2}{dq^2} = \dfrac {1}{\beta^2} \dfrac {d^2}{dx^2} \label {15.6.8}\]
After substituting Equations \(\ref{15.6.7}\) and \(\ref{15.6.8}\) into Equation \(\ref{15.6.6}\) the differential equation for the harmonic oscillator becomes
\[ \dfrac {d^2 \psi _v (x)}{dx^2} + \left ( \dfrac {2 \mu \beta ^2 E_v}{\hbar ^2} - x^2 \right ) \psi _v (x) = 0 \label {15.6.9}\]
A common strategy for solving differential equations, which is employed here, is to find a solution that is valid for large values of the variable and then develop the complete solution as a product of this asymptotic solution and a power series. Since the potential energy approaches infinity as \(x\) and the coordinate \(q\) approach infinity, the wavefunction must approach zero. The function that has this property and satisfies the differential equation for large values of \(x\) is the exponential function
\[ \lim_{x \rightarrow \infty} \psi(x) \exp \left ( \dfrac {-x^2}{2} \right ) \label {15.6.10}\]
The general expression for a power series is
\[ \sum _{n=0}^\infty c_n x^n \label {15.6.11}\]
which can be truncated after the first term, after the second term, after the third term, etc. to produce a set of polynomials. There is one polynomial for each value of \(v\) where \(v\) can be equal to any integer value including zero.
\[ \sum _{n=0}^v c_n x^n \label {15.6.12}\]
Each of the truncations of the power series in Equation \(\ref{15.6.12}\) can be multiplied by the exponential function in Equation \(\ref{15.6.10}\) to create a family of valid solutions to the differential equation.
\[\psi _v (x) = \sum _{n=0}^v c_n x^n exp \left ( \dfrac {-x^2}{2} \right ) \label {15.6.13}\]
While polynomials in general approach \(∞\) (or \(-∞\)) as \(x\) approaches \(∞\), the decreasing exponential term overpowers the polynomial term so that the overall wavefunction exhibits the desired approach to zero at large values of \(x\) or \(-x\). The exact forms of polynomials that solve Equation \(\ref{15.6.9}\) are the Hermite polynomials, which are standard mathematical functions known from the work of Charles Hermite. The first eight Hermite polynomials, \(H_v(x)\), are given below.
- \(H_0 = 1\)
- \(H_1 = 2x\)
- \(H_2 = -2 + 4x^2\)
- \(H_3 = -12x + 8x^3\)
- \(H_4 = 12 - 48x^2 +16x^4\)
- \(H_5 = 120x - 160x^3 + 32x^5\)
- \(H_6 = -120 + 720x^2 - 480 x^4 + 64x^6\)
- \(H_7 = -1680x + 3360 x^3 - 1344 x^5 + 128 x^7\)
The Hermite polynomials like those in Table \(\PageIndex{1}\) can be produced by using the following generating function
\[ H_v (x) = (-1)^v \exp (x^2) \dfrac {d^v}{dx^v} \exp (-x^2) \label {5.6.14}\]
Generating functions provide a more economical way to obtain sets of functions compared to purchasing books of tables, and they are often more convenient to use in mathematical derivations.
Because of the association of the wavefunction with a probability density, it is necessary for the wavefunction to include a normalization constant, \(N_v\).
\[N_v = \dfrac {1}{(2^v v! \sqrt {\pi} )^{1/2}} \label {5.6.15}\]
The final form of the harmonic oscillator wavefunctions is thus
\[ \psi _v (x) = N_v H_v (x) e^{-x^2/2} \label {5.6.16}\]
The energy eigenvalues for a quantum mechanical oscillator also are obtained by solving the Schrödinger equation. The energies are restricted to discrete values
\[E_v = \left ( v + \dfrac {1}{2} \right ) \hbar \omega \label {5.6.17}\]
with \(v = 0, 1, 2, 3, \cdots \).
The energies depend both on the quantum number, \(v\), and the oscillator frequency
\[ \omega = \sqrt {\dfrac {k}{\mu}}\]
which in turn depends on the spring constant \(k\) and the reduced mass of the vibration \(\mu\).
The normalized wavefunctions for the first four states of the harmonic oscillator are shown in Figure \(\PageIndex{1}\), and the corresponding probability densities are shown in Figure \(\PageIndex{2}\). You should remember the mathematical and graphical forms of the first few harmonic oscillator wavefunctions, and the correlation of \(v\) with \(E_v\). The number of nodes in the wavefunction will help you to remember these characteristics. Also note that the functions fall off exponentially and that the symmetry alternates. For \(v\) equal to an even number, \(Ψ_v\) is gerade; for v equal to an odd number, \(Ψ_v\) is ungerade.
In completing Exercise \(\PageIndex{9}\), you should have noticed that as the quantum number increases and becomes very large, the probability distribution approaches that of a classical oscillator. This observation is very general. It was first noticed by Bohr, and is called the Bohr Correspondence Principle. This principle states that classical behavior is approached in the limit of large values for a quantum number. A classical oscillator is most likely to be found in the region of space where its velocity is the smallest. This situation is similar to walking through one room and running through another. In which room do you spend more time? Where is it more likely that you will be found?
David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")