Analytical Method for Solving the Simple Harmonic Oscillator

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian ${\mathcal {H}}$, it describes how a state $|\Psi \rangle$ evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

In contrast to the elegant method described above to solve the harmonic oscillator, there is another "brute force" method to find out the eigenvalues and eigenfunctions. This method uses exapansion of the wavefunction in a power series.

Let us start with the Schrödinger equation:

$-{\frac {\hbar ^{2}}{2m}}{\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} x^{2}}}+{\frac {1}{2}}m\omega ^{2}x^{2}\psi =E\psi$

or, ${\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} x^{2}}}=(\xi ^{2}x^{2}-k)\psi$ where $k={\frac {2mE}{\hbar ^{2}}}$, $\xi ^{2}={\frac {m^{2}\omega ^{2}}{\hbar ^{2}}}$

We shall look at the asymptotic behavior

At large x, ${\frac {\mathrm {d} ^{2}\psi }{\mathrm {d} x^{2}}}\approx \xi ^{2}x^{2}\psi$

To find its solution, let us make the following ansatz:

$\psi (x)=e^{kx^{2}}$

Substituting this in the asymptotic equation, we get

$4k^{2}x^{2}+2K=\xi ^{2}x^{2}\!$

or in the large x limit,

$k=\pm \xi /2$

with this value of k,

$\psi (x)=Ae^{-{\xi x^{2}/2}}+Be^{\xi x^{2}/2}$

For $\psi (x)$ to remain finite at the origin, $B=0\!$.

So $\psi (x)=Ae^{-{\xi x^{2}/2}}$ for large $x$

Now that we have separated out the asymptotic behavior, we shall postulate that the complete solution, valid everywhere, can be written as:

$\psi (x)=h(x)e^{-{\xi x^{2}/2}}$ where $h(x)$ is some polynomial. It is clear that $h(x)$ must diverge slower than the rate at which $e^{-{\xi x^{2}/2}}$ converges for large $x$.

$\psi '=[h'(x)-\xi xh(x)]e^{-{\xi x^{2}/2}}$

$\psi ''=[h''(x)-2\xi xh'(x)+(\xi ^{2}x^{2}-\xi )h(x)]e^{-{\xi x^{2}/2}}$

Putting this back in the differential equation, we get

$h''(x)-2\xi xh'(x)+(k-\xi )h(x)=0\!$

let us try a series solution for $h(x)$

$h(x)=\sum _{j=0}^{\infty }a_{j}x^{j}$ $h'(x)=\sum _{j=0}^{\infty }ja_{j}x^{j-1}$ $h''(x)=\sum _{j=0}^{\infty }j(j-1)a_{j}x^{j-2}$

Substituting and equating coefficient of each power on both sides, we get the recursion relation

$\sum _{j=0}^{\infty }[(j+1)(j+2)a_{j+2}-(2j\xi +\xi -k)]=0$

$a_{j+2}={\frac {2j\xi +\xi -k}{(j+1)(j+2)}}a_{j}$x

unless this terminates after a finite number of terms, the whole solution will blow up at $x=\pm \infty .$ So

$2n\xi +\xi -k=0\!$

where

$n$ is a non-negative integer.

$k=\xi (2n+1)\!$

Since $k$ depends on the energy, we get

$E=\hbar \omega \left(n+{\frac {1}{2}}\right)$ $n=0,1,2,3,...$.

Once $k$ is constrained as above, we have

$a_{j+2}={\frac {-2\xi (n-j)}{(j+1)(j+2)}}a_{j}$.

Hence the series starts with either $a_{0}$ or $a_{1}$, and will be even or odd, respectively. These are called Hermite polynomials. The properly normalized eigenfunctions are

$\psi _{n}(x)=\left({\frac {m\omega }{\pi \hbar }}\right)^{1/4}{\frac {1}{\sqrt {2^{n}n!}}}H_{n}(x)e^{-{\xi x^{2}/2}}$

Example

Reference

Introduction to Quantum Mechanics, 2nd ed. , by D. J. Griffiths