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Quantum Mechanics
V H Satheeshkumar Department of Physics and Center for Advanced Research and Development Sri Bhagawan Mahaveer Jain College of Engineering Jain Global Campus, Kanakapura Road Bangalore 562 112, India. October 12, 2008

Who can use this?
The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is the second of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU for the first-semester (September 2008 - January 2009) BE students of all branches. Any student interested in exploring more about the course may visit the course homepage at For those who are looking for the economy of studying this: this chapter is worth 20 marks in the final exam! Cheers ;-)

Syllabus as prescribed by VTU
Heisenberg’s uncertainty principle and its physical significance, Application of uncertainty principle; Wave function, Properties and Physical significance of a wave function, Probability density and Normalisation of wave function; Setting up of a one dimensional time independent, Schr¨dinger wave equation, Eigen values and eigen function, o Application of Schr¨dinger wave equation : Energy eigen values for a o free particle, Energy eigen values of a particle in a potential well of infinite depth.

• Arthur Beiser, Concepts of Modern Physics, 6th Edition, Tata McGraw-Hill Publishing Company Limited, ISBN- 0-07-049553-X. ———————————— A This document is typeset in Free Software L TEX2e distributed under the terms of the GNU General Public License.





Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to provide accurate descriptions for many previously unexplained phenomena such as black body radiation, photoelectric effect and Compton effect. The term quantum mechanics was first coined by Max Born in 1924. Within the field of engineering, quantum mechanics plays an important role. The study of quantum mechanics has lead to many new inventions that include the laser, the diode, the transistor, the electron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also use quantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quantum mechanics. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to perform certain computational tasks exponentially faster than the regular computers. This chapter attempts to give you an elementary introduction to the topic.


Heisenberg’s uncertainty principle

We know from the wave-particle duality that every particle has wave-like properties. These wave properties of particles will prevent us from measuring the exact attributes of the particles. This limitation related to the measurements at microscopic level is known as the uncertainty principle. The uncertainty principle states that it is impossible to specify simultaneously the position and momentum of a particle, such as an electron, with precision. The theory further states that a more accurate determination of one quantity will result in a less precise measurement of the other, and that the product of both uncertainties is always greater than or equal to Planck’s constant divided by 4π. That is h . (1) ∆x · ∆px ≥ 4π This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called the indeterminacy principle. The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involved in the simultaneous measurements of angular displacement & angular momentum and energy & time; h , 4π h ∆t · ∆E ≥ . 4π ∆θ · ∆l ≥ Sometimes h 2π

(2) (3)

is written as . In that case...
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