• the WM harmonic oscillator
    • an exactly solvable problem
    • problem slightly more complicated than “particle-in-infinitely-deep-box”
  • many kinds of quantum mechanical systems are explained based on this model
    • a good first approximation for many physical systems
  • example one
    • tiny mass on spring to model atom vibration in a molecule or crystalline solids
    • these atom vibrations maybe described in QM as phonon modes
  • example two
    • the idea of photons finds basis in the idea of harmonic oscillator

quantum mechanical harmonic oscillator

  • the classical simple harmonic motion is explored
    • with potential and potential differences instead of forces
    • this can be done within CM using hamiltonian mechanics
    • in QM working directly with forces is avoided
      • SE is also setup that way from the beginning

quantum mechanical solution

  • we have a mass \(m\) on a spring
    • \( m \) is small
  • we are going to use \(z\) for our coordinate here
    • simple-harmonic-oscillator
  • the potential from the restoring force \(F\) is \[ \begin{align} F & = -kx \rightarrow \text{ spring restoring force } \\
    V(z) & = \int_{0}^{z} -F dz_0 \\
    & = \int_{0}^{z} -K z_0 dz_0 \\
    & = \frac{1}{2} K z^2 \\
    & = \frac{1}{2} m \omega^2 z^2 \\
    \end{align} \]
    • parabola

harmonic oscillator schrodinger equation

\[ \begin{align} V(z) = \frac{1}{2} m \omega^2 z^2 \\\ \end{align} \]

  • schrodinger equation (time-independent) is \[ \begin{align} -\frac{\hbar^2}{2m} \frac{d^2 \psi}{dz^2} + \frac{1}{2} m\omega^2 z^2 \psi = E \psi \end{align} \]
    • for convenience, we define a dimensionless distance unit \[ \begin{align} \xi = \sqrt{ \frac{m\omega}{h} } z \end{align} \]
  • so the schrodinger’s equation becomes \[ \begin{align} \frac{1}{2} \frac{d^2 \psi }{d \xi^2} - \frac{\xi^2}{2} \psi = -\frac{E}{h\omega}\psi \end{align} \]
    • one specific solution to this equation \[ \begin{align} \frac{1}{2} \frac{d^2 \psi}{d \xi^2} - \frac{\xi^2}{2} \psi & = -\frac{E}{ \hbar \omega} \psi \\
      \psi & \propto \exp{\left( \frac{-\xi^2}{2} \right) } \\
      \end{align} \]
    • with a corresponding energy \( E = \frac{\hbar \omega}{2} \)
  • this suggests we look for solutions of the form \[ \psi_{n}(\xi) = A_n \exp{ \left( \frac{-\xi^2}{2} \right) } H_n(\xi) \]
    • where: \( H_n(\xi) \): is a set of functions to be determined

\[ \begin{align} \frac{d^2 H_n(\xi)}{d\xi^2} - 2 \xi \frac{dH_n(\xi)}{d\xi} + \left( \frac{2E}{h\omega} - 1 \right) H_n (\xi) & = 0 \\
\end{align} \]

  • this is the defining differential equation for the hermite polynomials
    • solutions for hermite polynomials exist only if \[ \frac{2E}{\hbar\omega} - 1 = 2n \qquad \text{ n = 0,1,2,… } \]
    • in terms of energy, solutions only exist for energy levels \[ E_n = \left( n + \frac{1}{2}\right) \hbar \omega \qquad \text{ n = 0,1,2,… } \]
  • so, the allowed energy levels are equally spaced separated by an amount \( \hbar\omega \)
    • \(\omega\): classical oscillation frequency
    • like the potential well, there is a “zero-point energy”
      • \( E_0 = \frac{\hbar\omega}{2} \)

hermite polynomial

  • list of hermite polynomials
    • \( H_0 = 1 \rightarrow \) even
    • \( H_1(\xi) = 2\xi \rightarrow \) odd
    • \( H_2(\xi) = 4\xi^2 - 2 \rightarrow \) even
    • \( H_3(\xi) = 8\xi^3 - 12\xi \rightarrow \) odd
    • \( H_4(\xi) = 16\xi^4 - 48\xi^2 + 12 \rightarrow \) even
  • properties
    • they have “define parity” i.e. they are either even or odd
    • they satisfy a “recurrence relation”
      • successive hermite polynomials can be calculated
        • if the previous two are known
      • i.e. \( H_n(\xi) = 2\xi H_{n-1}(\xi) - 2(n-1)H_{n-2}(\xi) \)
  • normalizing \[ \psi_n(\xi)= A_n \exp{\left( \frac{-\xi^2}{2} \right) H_n(\xi) } \]
    • gives \[ \begin{align} A_n & = \sqrt{ \frac{1}{\sqrt{\pi} 2^n n!}} \\
      \xi & = z \sqrt{ \frac{m\omega}{\hbar} } \\
      E_n & = \left( n + \frac{1}{2}\right) \hbar \omega \qquad \text{ n = 0,1,2,… } \\
      \\
      \text{ expressing it } & \text{in terms of the original coordinates } \\
      \psi_n(z) & = \sqrt{ \frac{1}{2^n n!} \sqrt{\frac{m\omega}{\pi \hbar}}} \exp{\left( -z^2 \frac{m\omega}{2\hbar} \right)} H_n\left( z \sqrt{\frac{m\omega}{\hbar}} \right) \end{align}
      \]
  • harmonic oscillator eigensolutions are graphically seen as follows harmonic-osicalltor-eigensolutions
    • in this above plot, the inttersections of the parabola and the dashed lines
      • give the “classical turning points”
      • where a classical mass of that energy turns round and goes back downhill
    • \( \omega \) is the angular frequency of a classical oscillator
      • with the same mass and
      • the same parabolic potential energy
  • in QM eigenstates found in this particle in a potential problem
    • there is no oscillation
    • they are static solutions, albeit, probability distributions
    • if the modulus squared of any of the wave functions
  • to find and analyze the actual oscillations, schrodinger’s time equation has to be used