Failures of Classical Mechanics -- The Birth of Quantum Mechanics 

Objectives: 

To understand a number of well-known experiments in which the outcome cannot be explained by classical theory.

1. Blackbody Radiation

Classical theory predicts that a blackbody emits infinite amount of energy at any temperature above 0 K.   In classical theory, radiation energy is proportional to the square of the amplitude and independent of the frequency.  Furthermore, energy is a continuous quantity.  However, Plank showed that if radiation energy is proportional to the frequency and can only take discrete values, excellent agreement with experimental observations can be obtained.  MORE... 

2. The Photoelectric effects

Classical theory predicts that increasing the intensity of the incident light beam will increase the number and the kinetic energy of the ejected electrons from the exposed metal surface.   Experimental observations show that the kinetic energy of the ejected electrons does not depend on the intensity of the light beam but on its frequency.  Using the same assumption as Plank used for the blackbody radiation problem Einstein was able to obtain agreement between theoretical prediction and experimental observations.   MORE....

3. The atomic spectra

In classical mechanics, energy is a continuous quantity.  However, atomic specra show discrete lines as functions of the frequency.  This can only be understood if atoms are only allowed to access discrete energy values and thus they absorbed and emitted radiation at discrete frequencies.  This is the most direct evidence of energy quantization.

4. Electron diffraction pattern

A characteristic property of waves is that they can interfere with one another constructively or destructively to yield one with larger or smaller amplitudes, respectively.  This is the principle behind the X-ray difraction pattern.   Davisson and Germer showed that scattering of an electron beam from a crystal surface yields a diffraction pattern similar to that of X-ray difraction.  This shows that electrons can behave like waves. 

5. Wave-Particle Duality 

The above experiments show that light normally behaves as waves but can behave as particle under certain instances while electrons normally behave as particles but also show wave-like characteristics.   This is known as the wave-particle duality of nature and can be quantitatively represented by the de Broglie's relation:

Since the Plank's constant h is very small, namely 6.626 x   J.s, only very small particles such as electrons can show noticeable dual behaviors.

 6. The Uncertainty Principle 

The wave-particle duality of nature has led to another fundamental principle known as the Uncertainly Principle developed by Heisenberg.  The Uncertainty principle states that it is not possible to determine the location and momenten of a particle simultaneously with unlimited precision.  More specifically,

This contradicts with classical mechanics where the location and momentum of a particle can be determined exactly.  However, since the Plank's constant is small such uncertainty is completely inconsequential for macroscopic objects we encounter in our daily life.

7. The Birth of Quantum Mechanics

Evidence clearly shows that in the regime of small objects classical theory, where energy can vary continuously and where waves and particles have distinct characteristics, is no longer valid.  A new fundamental framework must be developed to take its place -- that is Quantum Mechanics.    But is there a sharp boundary between Classical Mechanics and Quantum Mechanics?  Is it correct to use the mass of the object to predict if it will have quantum mechanical behaviors?

When does one need quantum mechanics?  

Atomic spectrum of He atom shows discrete energy levels and thus Helium atoms have quantum mechanical behaviors.  However, classical collision theory can accurately describe the origin of pressure arising from a container filled with Helium gas.  Thus, the mass of the object is not a clear criterion to decide when quantum mechanics is needed.  

A distinctive feature of quantum mechanics is the quantization of energy, i.e. the system has discrete energy levels.  As the energy spacing between these levels, , decreases to zero, the allowed energy levels form a continuous spectrum and thus classical mechanics becomes valid.  In fact, it can be shown from the Boltzmann distribution function that when then quantum mechanics is required.

-- ThanhTruong - 12 Jul 2007