Question

A question on Solid State Basics, please give an answer in the form of text, not only formulas.

What is meant by the 'normal mode' of vibrations? How is the leap to quantum mechanics from classical mechanics made with this concept?

Answer 1

Basically, Normal modes arises when we impose a periodic potential to the system of the particles as in case of phonons. Where the all particle oscillates with same frequency and have same phase but their amplitude of oscillation varies and these waves are standing waves.

Now the differene between Classical and Quantum Mechanics in very basic way is that, As you had solved and find the energy of harmonic oscillator classically and quantum mechanically as well.

Note that the Energy distribution in Classical mechanics is Continous while in Quantum mechanics Energy get Quantised. So why there is quantization in quantum mechanics not in classical?, the straighforward answer is that classical measurement scale of energy is very large compare to quantum.

Another example is that let us assume that we have a Finite potential well having potential 'V'. First of we try to analyse using classical mechanics, lets say there is a particle in the Finite potential well having energy 'E' such that E V. Now classically I ask that, is there any possibility so that the particle can cross the well?, the answer is no unless we apply any external force to the particle.

But Quantum mechanically we find that the particle have some probability( ~ exponentially decaying solution ) means that the particle can cross the potential even when Energy E < V.

There are several example given in many Quantum mechanics.