The solutions to the Schrödinger equation for the quantum harmonic oscillator are a set of wavefunctions labeled by the vibrational quantum number \(v = 0,1,2,\ldots\). Each wavefunction has a characteristic mathematical form that reflects the physical constraints of the problem.
Every harmonic oscillator wavefunction can be written as the product of three distinct parts:
1. Normalization constant
The first factor is a normalization constant, which ensures that the total probability of finding the particle somewhere along the coordinate \(x\) is equal to one:
\[ \int_{-\infty}^{\infty} |\psi_v(x)|^2\,dx = 1. \]
This constant depends on the quantum number \(v\) as well as on the physical parameters of the oscillator (the reduced mass and force constant).
2. Hermite polynomial
The second factor is a Hermite polynomial, \(H_v(x)\). This polynomial determines the number of nodes and the overall shape of the wavefunction.
The degree of the Hermite polynomial is equal to the vibrational quantum number \(v\). As \(v\) increases, the wavefunction acquires more nodes and becomes more oscillatory.
Hermite polynomials
The functions \(H_v(x)\) that appear in the harmonic oscillator wavefunctions are called Hermite polynomials. They are a specific set of orthogonal polynomials that arise naturally when solving the Schrödinger equation for a quadratic potential.
Orthogonal polynomials will have several important properties and relationships. These include
- A generation function
- An orthogonality relationship
- One or more recursion formula(e)
The generation function for the Hermite Polynoials is
\[H_v(y) = (-1)^v e^{y^2} \frac{d^v}{dy^v} e^{-y^2}\]
The Hermite polynomials are orthogonal on the interval \((-\infty,\infty)\) by including the weighting function \(e^{-y^2}\):
\[ \int_{-\infty}^{\infty} H_v(y)\,H_{v'}(y)\,e^{-y^2}\,dx = \delta_{vv'}\,2^v\,v!\,\sqrt{\pi}. \]
And finally, the Hermite polynolials obey the recursion formula
\[ H_{v+1}(y) = 2y H_v(y) - 2v H_{v-1}(y)\]
Another important relationship (which is especially useful for deriving selection rules) is
\[ \frac{d}{dy} H_v(y) = 2v H_{v-1}(y) \]
Each Hermite polynomial is labeled by a non-negative integer \(v\), which corresponds directly to the vibrational quantum number. The degree of the polynomial is equal to \(v\).
First few Hermite polynomials
| \(v\) | \(H_v(y)\) | Parity |
|---|---|---|
| \(0\) | \(H_0(y)=1\) | Even |
| \(1\) | \(H_1(y)=2y\) | Odd |
| \(2\) | \(H_2(y)=4y^2-2\) | Even |
| \(3\) | \(H_3(y)=8y^3-12y\) | Odd |
| \(4\) | \(H_4(y)=16y^4-48y^2+12\) | Even |
These polynomials determine the number of nodes in the wavefunction. The wavefunction for quantum number \(v\) has exactly \(v\) nodes (zeros), not counting the points at infinity. You will also notice that the functions are even or odd depending on whether \(v\) is even or odd!
Physical role in the wavefunctions
On their own, Hermite polynomials grow rapidly with \(x\). When multiplied by the exponential decay factor \(e^{-\alpha x^2}\), however, the resulting product remains finite and normalizable.
The Hermite polynomials therefore control the oscillatory structure of the wavefunction, while the exponential term controls its overall spatial extent.
Big idea: Hermite polynomials encode the quantized oscillatory behavior of the harmonic oscillator, with each higher polynomial corresponding to a higher-energy vibrational state.
3. Exponential decay factor
The final factor is an exponential function of the form \(e^{-\alpha x^2}\), where \( \alpha \) is a positive constant.
This exponential factor ensures that the wavefunction vanishes as \(x \rightarrow \pm\infty\). Without this term, the wavefunction would diverge and could not be normalized.
Physically, this reflects the fact that the probability of finding the particle very far from the equilibrium position is extremely small.
Putting these pieces together, the harmonic oscillator wavefunctions have the general form
\[ \psi_v(x) = N_v \times H_v(\alpha^\frac{1}{2} x) \times e^{-\frac{\alpha x^2}{2}}. \]
where \( \alpha = \sqrt{k \times \mu} \) and \( N_v = \sqrt{\frac{\sqrt{\frac{\alpha}{\pi}}}{2^v \times v!}} \)
Normalization constant from Hermite orthogonality
The normalized harmonic oscillator wavefunctions are most cleanly written in terms of the dimensionless coordinate \(y\), where the wavefunction has the form
\[ \psi_v(y)=N_v\,H_v(y)\,e^{-y^2/2}. \]
The normalization condition is
\[ \int_{-\infty}^{\infty} |\psi_v(y)|^2\,dy = 1. \]
Substituting \(\psi_v(y)\) gives
\[ \int_{-\infty}^{\infty} \left(N_v\,H_v(y)\,e^{-y^2/2}\right) \left(N_v\,H_v(y)\,e^{-y^2/2}\right)\,dy = N_v^2\int_{-\infty}^{\infty} H_v^2(y)\,e^{-y^2}\,dy = 1. \]
Now apply the Hermite orthogonality relation (setting \(v'=v\), so \(\delta_{vv}=1\)):
\[ \int_{-\infty}^{\infty} H_v(y)\,H_{v'}(y)\,e^{-y^2}\,dy = \delta_{vv'}\,2^v\,v!\,\sqrt{\pi} \quad \Rightarrow \quad \int_{-\infty}^{\infty} H_v^2(y)\,e^{-y^2}\,dy = 2^v\,v!\,\sqrt{\pi}. \]
Substitute this into the normalization condition:
\[ N_v^2\left(2^v\,v!\,\sqrt{\pi}\right)=1 \quad\Rightarrow\quad N_v^2=\frac{1}{2^v\,v!\,\sqrt{\pi}}. \]
Taking the positive square root (so the overall sign is conventional) gives the normalization constant:
\[ N_v=\frac{1}{\sqrt{2^v\,v!\,\sqrt{\pi}}}. \]
Big idea: the exponential factor \(e^{-y^2/2}\) produces the weight \(e^{-y^2}\) in \(|\psi_v|^2\), allowing the Hermite orthogonality relation to determine the normalization constant directly.
Interactive: visualize harmonic oscillator vibrational wavefunctions
Select the vibrational quantum number \(v\) and choose whether to plot \(\psi_v(x)\) and/or \(|\psi_v(x)|^2\). The horizontal axis uses a dimensionless coordinate \(y = \sqrt{\alpha}\,x\), so the curves are comparable across different molecules.
Tip: The wavefunction alternates between even and odd parity as \(v\) increases, and the number of nodes in \(\psi_v\) is exactly \(v\).
Big idea: the structure of the harmonic oscillator wavefunctions reflects both mathematical requirements (normalizability) and physical behavior (increasing complexity with increasing vibrational energy).