Quantum Mechanics

Quantum Mechanics By: HarleenKaur Lecturer in Physics P.G.G.C.G. Sector-11, Chandigarh.

Physics Paper C BSc II CHAPTER 1 ORIGIN OF QUANTUM MECHANICS

Need for Development of Quantum Mechanics Origin of Quantum mechanics had its roots at the failure of Newton’s laws of motion and Special theory of relativity to explain the behavior of microscopic world of atom. Some key phenomena which are responsible for development of quantum mechanics are:- Black body radiation, Specific heat of solids, Photoelectric effect, Compton effect, Hydrogen atom spectra etc.

Photoelectric Effect - The phenomenon of ejection of electrons from the surface of the metals when light of certain threshold frequency strikes it.

The Experiment: By varying the voltage on a negatively charged grid between the ejecting surface and the collector plate, Lenard was able to: • Determine that the particles had a negative charge. • Determine the kinetic energy of the ejected particles. He found that by turning up the voltage he reached a point where the current in his circuit ceased to flow because no more electrons were making it to the other side of the gap. This was noted and dubbed the stopping voltage.

Larger frequency, means smaller wavelength, and larger Energy=h.

Lenard’s Findings: • Thus he theorized that stopping voltage must be equal to the maximum kinetic energy of the ejected particles, or: KEmax = eVstopping Perplexing Observations: • The intensity of light had no effect on energy • There was a threshold frequency for ejection Classical physics failed to explain this, Lenard won the Nobel Prize in Physics in 1905.

Larger light intensity means larger number of photons at a given frequency (Energy)

Einstein Interpretation A new theory of light: • Electromagnetic waves carry discrete energy packets • The energy per packet depends on wavelength, explaining Lenard’s threshold frequency. • More intense light corresponds to more photons, not higher energy photons.

Einstein’s Relations: Einstein predicted that a graph of the maximum kinetic energy versus frequency would be a straight line, given by the linear relation: KE = h - Φ where Φ is known as work function and it is characteristics of given metal. …Therefore light energy comes in multiples of h.

Graph of KEmax vs. frequency

Photoelectric Effect - • The photoelectric effect provides evidence for the particle nature of light. • It also provides evidence for quantization. • If light shines on the surface of a metal, there is a point at which electrons are ejected from the metal. • The electrons will only be ejected once the threshold frequency is reached . • Below the threshold frequency, no electrons are ejected. • Above the threshold frequency, the number of electrons ejected depend on the intensity of the light.

de-Broglie Hypothesis or Wave-Particle Duality Wave-Particle Duality set the stage for the 20th century Quantum Mechanics. • Just as light possesed both particle and wave properties then matter particles must also possess wave properties. • The wave associated with matter particles are known as Matter waves or de-Broglie waves. • De-Broglie wavelength is given by: where h = Planck’s constant p=mv is the momentum of the particle.

de-Broglie wavelength associated with an electron • de-Broglie wavelength for an electron accelerated through potential difference V is given by relation • As value of Planck’s constant is very small (h=6.62x10-34 Js), the order of the wavelength associated with ordinary objects is negligible . Thus, Wave-Particle duality is not appreciable for objects that we interact in daily life(macroscopic objects). A0

Wave Packet • Wave packet is formed by the superposition of waves having slightly different frequency and amplitude. They interfere constructively in such a way that amplitude of resultant wave motion is very large in small region and negligibly small in rest of space. • A wave packet is localized – a good representation of particle!

Phase velocity and Group velocity • Phase velocity – Velocity of an individual wave comprising the wave packet. It describes the rate at which the phase of the wave propagates in space. It is always greater than velocity of light and can’t be accepted. • Group velocity – Velocity of a wave packet. It is equal to the velocity of particle. It describes the rate at which the envelope of the wave packet propagates. where =2 is the angular velocity. k=2/ is the propagation vector.

Heisenberg Uncertainty Principle It is the direct consequence of Wave-Particle duality. It states that: It is impossible to measure both position and momentum of a particle simultaneously with accuracy. The product of the uncertainties is approximately of the order of h/2. x.px  h/2 where x is the uncertainty in measurement of position and pxis the uncertainty in measurement of momentum.

HEISENBERG-BOHR thought experiment: • It shows that a measurement itself introduces the uncertainty. • When we “look” at an object we see it through the photons that are detected by the microscope. • These are the photons that are scattered in the angle < 2θ • Momentum of electron is changed

HEISENBERG-BOHR thought experiment: Trying to determine the electron position, we introduce the uncertainty of the momentum

HEISENBERG-BOHR thought experiment: Image is not the point, but the diffraction pattern The uncertainty of the position is approximately the width of the central maximum

Time - Energy Uncertainty relation • Heisenberg’s Uncertainty principle is applicable to all “conjugate variables”. • The Time-Energy Uncertainty principle states: Et  h/2 where E is the uncertainty in measurement of energy and t is the uncertainty in measurement of time. • It explains the natural broadening of spectral lines. Since every excited state has a finite life span, t will be finite so that no spectral line is infinitely sharp (E0). • Energy conservation law can be violated but for the short time interval! • As value of Planck’s constant is very small, Uncertainty principle is relevant only for microscopic objects.

Applications of Uncertainty Principle • Non-existence of electrons in the nucleus. • Existence of Protons, neutrons and alpha particles in the nucleus. • Zero point energy- The minimum energy of the system at 0K. It gives finite value of zero point energy. • Binding energy of an electron in atom.

Problems • An electron and a proton have the same de-Broglie wavelength. Prove that the energy of the electron is greater. • Calculate de-Broglie wavelength of a proton moving with 1/10th velocity of light. • The electron in hydrogen atom may be thought of as confined to a radius of 5x10-11m.Calculate the minimum uncertainty in the momentum of the electron. Also calculate the minimum kinetic energy of the electron. • A radar pulse lasts for 0.25s.What is the order of uncertainty in the energy of photons?

Quantum Mechanics