[QMSE] W03 - Completeness and Orthogonality

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

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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 $$f\left(t\right)=\sum _{n=1}^{\infty }{a}_n\left(\frac{n\pi t}{t_0}\right)$$
• i.e. any function $f\left(t\right)$ 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 $f\left(z\right)$ 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 $$f\left(z\right)=\sum _{n=1}^{\infty }{a}_n\sin \left(\frac{n\pi z}{L_z}\right)$$
  • the frequencies differ by the factor $n$
  • $a_n$ are the amplitudes of corresponding component sine waves

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normalized eigenfuntions

• consider the set of normalized eigenfunctions derived for a particle in a box $${\psi }_n\left(z\right)=\sqrt{\frac{2}{L_z}}\sin \left(\frac{n\pi z}{L_z}\right)$$
• with a substitution, the normalized waves can be used as the building blocks of the fourier series $$f\left(z\right)=\sum _{n=1}^{\infty }{a}_n\sin \left(\frac{n\pi z}{L_z}\right)=\sum _{n=1}^{\infty }{b}_n{\psi }_n\left(z\right)$$ $$\text{where}{b}_n=a_n\sqrt{\frac{L_z}{2}}$$
• 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 $$f\left(z\right)=\sum _{n=1}^{\infty }{b}_n{\psi }_n\left(z\right)$$
  • is the expansion of $f\left(z\right)$ in the complete set of normalized eigenfunctions ${\psi }_n\left(z\right)$
• 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 ${\psi }_n\left(z\right)$ can be used to represent a function $f\left(z\right)$
  • ${\psi }_n\left(z\right)$ acts as ‘basis set of functions’ or simply ‘basis’ of its space
  • keeping in mind $f\left(z\right)$ is some function
    • in the space of the set of functions ${\psi }_n\left(z\right)$
    • and whose amplitude is $0$ at the walls
• the set of expansion coefficients $b_n$ (amplitudes) in $f\left(z\right)$
  • is the representation of $f\left(z\right)$
  • in the basis ${\psi }_n\left(z\right)$
  • this works because of the completeness of the set of basis functions

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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 $g\left(z\right)$ and $h\left(z\right)$ defined from $0$ to $L_z$ if $${\int }_0^{L_z}{g}^{\ast }\left(z\right)h\left(z\right)dz=0$$

• all functions in a set of eigenfunctions such as ${\psi }_n\left(z\right)$ 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
  • ${\delta }_{nm}=0\text{for}n\ne m$
  • $\text{else}{\delta }_{nn}=1$
• the orthogonality and the normalization as one condition $${\int }_0^{L_z}{\psi }_n^{\ast }\left(z\right){\psi }_m\left(z\right)dz={\delta }_{nm}$$
  • 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 $f\left(x\right)$ in terms of a complete set of orthonormal functions ${\psi }_n\left(x\right)$ $$f\left(x\right)=\sum _n{c}_n{\psi }_n\left(x\right)$$
  • where $c_n$: expansion coefficients
• to find $c_n$, pre-multiply by ${\psi }_m^{\ast }\left(x\right)$ and integrate $$\begin{array}{cc}\int {\psi }_m^{\ast }\left(x\right)f\left(x\right)dx& =\int {\psi }_m^{\ast }\left(x\right)\left[\sum _n{c}_n{\psi }_n\right(x\left)\right]dx\\ & =\sum _n{c}_n\int {\psi }_m^{\ast }\left(x\right){\psi }_n\left(x\right)dx\\ & =\sum _n{c}_n{\delta }_{mn}\\ & =c_m\end{array}$$
• any function in a orthonormal complete set is expandable
  • $\int {\psi }_m^{\ast }\left(x\right)f\left(x\right)dx$
  • also called the ‘overlap integral’