User:RedAcer/math

$$\hbar$$$$ \omega_i$$ $$\qquad{\hbar}$$&omega; 2$$\hbar$$&omega; + 3&omega;
 * $$E_m = ({1/2}+m)\hbar\omega_i$$

$$\sum_{k=1}^N k^2$$

xy x y   $$a \qquad b$$


 * $$\frac1 {n^2} \le \frac{1}{n-1} - \frac{1}{n}, \quad n \ge 2,$$

Linear   Chain


 * n-1 n  n+1  &larr;  d  &rarr;

$$\cdots$$o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o++++++o$$\cdots$$
 * &rarr;&rarr;&rarr;&rarr;&rarr;&rarr;
 * $$u_{n-1} \qquad\quad u_n \qquad\quad u_{n+1}$$

Where 'n' labels the n'th atom, 'd' is the distance between atoms when the chain is in equilibrium and $$u_n$$ the displacement of the n'th atom from it's equilibrium position.

The harmonic oscillator eigenvalues or energy levels for the mode $$\omega_k$$ are :


 * $$E_n = ({1\over2}+n)\hbar\omega_k  \quad\quad\quad n=0,1,2,3 ......$$

If we ignore the zero-point energy then the levels are evenly spaced at :
 * $$\hbar\omega, \quad 2\hbar\omega ,\quad 3\hbar\omega \quad ......$$

So a minimum amount of energy $$\hbar\omega$$ must be supplied to the harmonic oscillator(or normal mode) to move it to the next energy level, in comparison to the photon this quantum of vibrational energy is called a phonon.

All quantum system show wave-like and particle-like properties. The particle-like properties of the phonon are best understood using the methods of second-quantisation described in another section.