sets of functions

  • the set of eigenfunctions for a particle in a box has a property called completeness

physical example of completeness

  • consider loudspeaker cone producing sound
    • the cone diaphragm movement is completely described by the observed position in time
    • equivalently it is also described by the set of the various frequencies that drive it
    • this is together called completeness

formally

  • the position of the diaphragm as a function of time can be replaced by
    • a set of different frequency components with different amplitudes
      • the phases of the different components has to be known
      • i.e. where the sine waves start
    • this replacement is done using fourier analysis
    • the method of representing the motion in terms of different frequency components
      • is called fourier series
  • fourier analysis is just a specific case of a more general mathematical concept

completeness of eigenfunctions

  • consider some random function of a particle in a box that is formally zero at the walls
  • this function can be written as a sum of
    • weighted sum of EFs (eigenfunctions)
  • EFs are a mathematical set of functions that can be used
    • to completely describe any function that sits between the walls
    • in terms of the heights of the function at each different horizontal position

fourier series over time

  • suppose we are interested in the behavior of some function
    • such a loudspeaker cone displacement, which generates sound
    • from time \(0\) to time \(t_0\)
      • presuming it starts and ends at \( 0 \) displacement
  • the corresponding fourier series would be \[ f(t) = \sum_{n=1}^{\infty} a_n \left( \frac{n\pi t}{t_0} \right) \]
  • i.e. any function \(f(t)\) can be written as a sum of sine waves of different frequencies
    • the frequencies differ by the factor \(n\)
    • \(a_n\) are the amplitudes of corresponding component sine waves

fourier series over space

  • consider some function \(f(z)\) over the distance \(L_z\)
    • from \(z = 0 \) to \( z = L_z \)
    • and that started and ended at 0 height
    • i.e. amplitude is \(0\) at the walls
  • the fourier series is given by \[ f(z) = \sum_{n=1}^{\infty} a_n \sin{ \left( \frac{n \pi z }{L_z}\right)} \]
    • the frequencies differ by the factor \(n\)
    • \(a_n\) are the amplitudes of corresponding component sine waves

normalized eigenfuntions

  • consider the set of normalized eigenfunctions derived for a particle in a box \[ \psi_n(z) = \sqrt{ \frac{2}{L_z}} \sin{\left( \frac{n\pi z}{L_z} \right) } \]
  • with a substitution, the normalized waves can be used as the building blocks of the fourier series \[ f(z) = \sum_{n=1}^{\infty} a_n \sin{\left( \frac{n\pi z}{L_z} \right)} = \sum_{n=1}^{\infty} b_n \psi_n(z) \] \[ \text{ where } b_n = a_n \sqrt{ \frac{L_z}{2}} \]
  • this \( b_n\) based function is a mathematical statement that is true for any function between \(0\) and \(L_z\)
    • whose amplitude is \(0\) at the walls at \(0\) and \(L_z\)
    • is not necessarily a solution to the particle in a box problem

expansion in eigenfunctions

  • restating the mathematics of normalized wave functions \[ f(z) = \sum_{n=1}^{\infty} b_n \psi_n(z) \]
    • is the expansion of \( f(z) \) in the complete set of normalized eigenfunctions \( \psi_n(z) \)
  • fourier series is connected to the normalized set of any eigenfunctions
    • by the means of completeness property
  • this is a purely mathematical statement
    • can be applied to any other set of eigenfunctions
  • a set of functions such as the \( \psi_n(z) \) can be used to represent a function \(f(z)\)
    • \( \psi_n(z) \) acts as ‘basis set of functions’ or simply ‘basis’ of its space
    • keeping in mind \(f(z)\) is some function
      • in the space of the set of functions \( \psi_n(z) \)
      • and whose amplitude is \(0\) at the walls
  • the set of expansion coefficients \(b_n\) (amplitudes) in \(f(z)\)
    • is the representation of \(f(z)\)
    • in the basis \(\psi_n(z)\)
    • this works because of the completeness of the set of basis functions

orthogonality of functions

  • in addition to being a complete set, the eigenfunctions are also “orthogonal” to one another
    • orthogonality in this sense is least like the other

orthogonal functions

  • two non-zero functions \( g(z) \) and \( h(z) \) defined from \(0\) to \(L_z\) if \[ \int_{0}^{L_z} g^{*}(z) h(z) dz = 0 \]

  • all functions in a set of eigenfunctions such as \( \psi_n(z) \) are orthonormal to each other

orthogonality and parity

  • functions with opposite parity are always orthogonal
    • even parity functions are orthogonal to odd parity functions
    • such as the first two particle-in-a-box eigenfunctions
  • with sine functions like in the particle-in-a-box eigenfunctions
    • even functions with same parity are orthogonal

orthonormality - kronecker delta

  • kronecker delta
    • \( \delta_{nm} = 0 \text{ for } n \neq m \)
    • \( \text{ else } \delta_{nn} = 1 \)
  • the orthogonality and the normalization as one condition \[ \int_{0}^{L_z} \psi_n^{*}(z) \psi_m(z)dz = \delta_{nm} \]
    • the orthonormality conditions
  • a set of functions that is both
    • normalized and
    • mutually orthogonal
    • is orthonormal
  • orthonormal sets allow one particularly useful operation
    • finding expansion coefficients

expansion coefficients

  • to write the function \( f(x) \) in terms of a complete set of orthonormal functions \( \psi_n(x) \) \[ f(x) = \sum_{n} c_n \psi_n(x) \]
    • where \( c_n \): expansion coefficients
  • to find \( c_n \), pre-multiply by \( \psi_m^{*}(x) \) and integrate \[ \begin{align} \int \psi_m^{*}(x) f(x) dx & = \int \psi_m^{*}(x) \left[ \sum_n c_n \psi_n(x) \right] dx \\
    & = \sum_{n} c_n \int \psi_m^{*} (x) \psi_{n}(x) dx \\
    & = \sum_{n} c_n \delta_{mn} \\
    & = c_m \end{align} \]
  • any function in a orthonormal complete set is expandable
    • \( \int\psi_{m}^{*}(x)f(x)dx \)
    • also called the ‘overlap integral’