2.1.2 Quantum Mechanics: a short review

Table of contents - Glossary - Study Aids - ¬ ®

In this section:

Introduction
The photoelectric effect
Blackbody radiation
Radiation from a hydrogen atom
The Schödinger equation

Introduction

Quantum mechanics emerged in the beginning of the twentieth century as a new discipline because of the need to explain phenomena which could not be explained using Newtonian mechanics. These phenomena include the photoelectric effect, blackbody radiation and the rather complex radiation from an excited hydrogen gas. It is these and other experimental observations which lead to the concepts of quantization of light into photons, the particle-wave duality, the de Broglie wavelength and the fundamental equation describing quantum mechanics, namely the Schödinger equation.

The photoelectric effect

The photoelectric effect is by now the "classic" experiment which demonstrates the quantized nature of light: when applying monochromatic light to a metal in vacuum one finds that electrons are released from the metal. This experiment confirms the notion that electrons are confined to the metal, but can escape when provided sufficient energy, for instance in the form of light. However the surprising fact is that when illuminating with long wavelengths (typically > 400 nm) no electrons are emitted from the metal even if the light intensity is increased. On the other hand one easily observes electron emission at ultra-violet wavelengths for which the number of electrons emitted does vary with the light intensity. A more detailed analysis reveals that the maximum kinetic energy of the emitted electrons varies linearly with the inverse of the wavelength, for wavelengths shorter than the maximum wavelength which yields electron emission.

The experiment is illustrated with the figure below:

photoele.gif

Fig. 1.1.1

Experimental set-up to measure the photoelectric effect

Light is incident on one of two electrodes to which an external voltage is applied. The external voltage is adjusted so that the current due to the photoemitted electrons becomes zero. This voltage corresponds to the maximum kinetic energy of the electrons in units of electron volt. That voltage is measured for different wavelengths and plotted as a function of the inverse of the wavelength. The resulting plot is shown in the figure below:

photoel1.gif

Fig. 1.1.2

Maximum kinetic energy of electrons emitted from a metal by light as a function of the inverse of the wavelength of the light

Albert Einstein explained this experiment by postulating that the energy of light is quantized, namely that light consists of individual particles called photons, so that the kinetic energy of the electrons equals the energy of the photons minus the energy, qF_M, required to extract the electrons from the metal. The workfunction, F_M, quantifies the potential which the electrons have to overcome to leave the metal. The slope of the curve was measured to be 1.24 eV/micron which yielded the following relation for the photon energy, E_ph:

where h is Planck's constant, n is the frequency of the light, c is the speed of light in vacuum and l is the wavelength of the light.

While other light related phenomena such as the interference of two coherent light beams demonstrate the wave characteristics of light, it is the photoelectric effect which demonstrates the quantized particle-like behavior which lead to the particle-wave duality concept. This concept states that particles observed in an appropriate environment behave as waves, while waves can also behave as particles. This concepts applies to all waves and particles. For instance coherent electron beams also yield interference patterns similar to those of light beams.

It is the wave-like behavior of particles which led to the de Broglie wavelength: since particles have wave-like properties there is an associated wavelength which is given by:

where l is the wavelength, h is Planck's constant and p is the particle momentum. One early hint that this was the correct expression for the wavelength of a particle is the fact that the ground state energy and the bohr radius of a hydrogen atom can be calculated using classical mechanics provided one assumes the above relation between the electron momentum and wavelength.

Blackbody radiation

Another experiment which could simply not be explained without quantum mechanics is the blackbody radiation experiment: By simply heating an object to high temperatures one finds that it radiates energy in the form of infra-red, visible and ultra-violet light. The appearance is that of a red glow at temperatures around 800° C which becomes brighter at higher temperatures and eventually looks like white light. The spectrum of the radiation is continuous which led scientists to believe that classical electro-magnetic theory should apply. However all attempts to describe this phenomenon failed until Max Planck developed the blackbody radiation theory based on the assumption that the energy associated with light is quantized and the energy quantum or photon energy equals: E_ph = hn.

Radiation from an excited hydrogen gas

The spectrum of electromagnetic radiation from an excited hydrogen gas consists of a set of discreet wavelengths. While the discreet nature of the emitted wavelengths can easily be associated with energy levels, E_n described by:

one still has to explain why the possible energy values are not continuous as one would expect in any "classical" system and further more how these specific values are obtained.

Niels Bohr provided a part of the puzzle, by assuming that electrons behave within the hydrogen atom as a wave rather than a particle, so that the trajectories of electrons around the proton are limited to those with a length which equals an integer number of wavelengths. The Bohr model also assumes that the momentum of the particle is linked to its wavelength by:

where this wavelength is called the de Broglie wavelength. The model further assumes a circular trajectory and that the centrifugal force equals the electrostatic force, or:

solving for the radius of the trajectory one finds the Bohr radius:

and the corresponding energy is obtained by adding the kinetic energy and the potential energy of the particle, yielding:

While the Bohr model does provide the correct answer it leaves a lot of unanswered questions and more importantly does not provide a general method to solve any problem of this type.

The Schrödinger equation

A general procedure to solve quantum mechanical problems was proposed by Erwin Schrödinger. Starting from a classical description of the total energy, E, which equals the sum of the kinetic energy, T, and potential energy, V, or:

he converted this expression into a wave equation by defining a wavefunction, Y, and multiplied each term in the equaton with that wavefunction:

To incorporate the de Broglie wavelength of the particle we introduce an operator which provides the momentum squared when applied to a plane wave:

and without claiming that this is an actual proof we now simply replace the momentum squared, p², by this operator yielding Schrödinger's equation.

2.1.1 ¬ ® 2.1.3