Physics 1st Year (Unit 3)


Syllabus (Theory)

Introduction to Quantum

Wave function


Download PPT Notes PDF of Introduction to Quantum (Unit 3)


Lecture 3.1 Introduction to Quantum

Quantum mechanics is a physical science dealing with the behavior of matter and energy on the scale of atoms and subatomic particles/waves.

It also forms the basis for the contemporary understanding of how very large objects such as stars and galaxies, and cosmological events such as the Big Bang, can be analyzed and explained.

Quantum mechanics is the foundation of several related disciplines including nanotechnology, condensed matter physics, quantum chemistry, structural biology, particle physics, and electronics.

The term “quantum mechanics” was first coined by Max Born in 1924.

The acceptance by the general physics community of quantum mechanics is due to its accurate prediction of the physical behaviour of systems, including systems where Newtonian mechanics fails.

Even general relativity is limited — in ways quantum mechanics is not — for describing systems at the atomic scale or smaller, at very low or very high energies, or at the lowest temperatures.

Through a century of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical.

The foundations of quantum mechanics date from the early 1800s, but the real beginnings of QM date from the work of Max Planck in 1900.

Albert Einstein and Niels Bohr soon made important contributions to what is now called the “old quantum theory.”

However, it was not until 1924 that a more complete picture emerged with Louis de Broglie’s matter-wave hypothesis and the true importance of quantum mechanics became clear.

Some of the most prominent scientists to subsequently contribute in the mid-1920s to what is now called the “new quantum mechanics” or “new physics” were Max Born, Paul Dirac, Werner Heisenberg, Wolfgang Pauli, and Erwin Schrödinger.

Later, the field was further expanded with work by Julian Schwinger, Sin-Itiro Tomonaga and Richard Feynman for the development of Quantum Electrodynamics in 1947 and by Murray Gell-Mann in particular for the development of Quantum Chromodynamics.

The interference that produces colored bands on bubbles cannot be explained by a model that depicts light as a particle.

It can be explained by a model that depicts it as a wave.

The drawing shows sine waves that resemble waves on the surface of water being reflected from two surfaces of a film of varying width, but that depiction of the wave nature of light is only a crude analogy.

Early researchers differed in their explanations of the fundamental nature of what we now call electromagnetic radiation.

Some maintained that light and other frequencies of electromagnetic radiation are composed of particles, while others asserted that electromagnetic radiation is a wave phenomenon.

In classical physics these ideas are mutually contradictory.

Ever since the early days of QM scientists have acknowledged that neither idea by itself can explain electromagnetic radiation.

Despite the success of quantum mechanics, it does have some controversial elements.

For example, the behaviour of microscopic objects described in quantum mechanics is very different from our everyday experience, which may provoke some degree of incredulity.

Most of classical physics is now recognized to be composed of special cases of quantum physics theory and/or relativity theory.

Dirac brought relativity theory to bear on quantum physics so that it could properly deal with events that occur at a substantial fraction of the speed of light.

Classical physics, however, also deals with mass attraction (gravity), and no one has yet been able to bring gravity into a unified theory with the relativized quantum theory.

Lecture 3.5 Wave function

Wave function

·         The quantity that characterizes the de–Broglie wave or matter wave is called the wave function

·         It is usually denoted as Ψ = (x, y, z, t). 

·         This gives complete information about the state of a physical system at a particular time. 

·         It is also called the state function and represents the probability amplitude. If ‘Ψ’ is large, the probability of finding the particle is also large and if ‘Ψ ‘ is small then the probability of finding the particle is small. 

·         The wave function gives the likelihood of finding the particle at a given instant and at a given position inside the wave packet.

·         |ψ(x)|2 determines the probability (density) that an object in the state ψ(x) will be found at position x.

Physical significance of wave function

·         In any physical wave if ‘A’ is the amplitude of the wave, then the energy density i.e., energy per unit volume is equal to ‘A2

·         Similar interpretation can be made in case of mater wave also. In matter wave, if ‘Ψ ‘is the wave function of matter waves at any point in space, then the particle density at that point may be taken as proportional to ‘Ψ2’.

·         Thus, Ψ2 is a measure of particle density. According to Max Born Ψ*Ψ = Ψ2 gives the probability of finding the particle in the state ‘Ψ’. i.e., ‘Ψ2’ is a measure of probability density.

·         The probability of finding the particle in a volume dv (= dx dy dz) is given by dv or dx dy dz.

·         Since the particle has to be present somewhere, total probability of finding the particle somewhere is unity i.e., particle is certainly to be found somewhere in space. i.e.,

∫ ∫ ∫ +∞ −∞ +∞ −∞ +∞ −∞ Ψ 2dx dy dz = 1.

Such condition is called Normalization condition. A wave function which satisfies this condition is known as normalized wave function.

·         Two wave functions that are perpendicular to each other and must satisfy the following equation:


These wave functions are known as orthogonal wave function.

Properties of wave functions:

 There are certain properties that an acceptable wave function ‘Ψ‘ must satisfy:

·         In order to avoid infinite probabilities, Ψ must be finite for all values of x, y, z. 

·         In order to avoid multiple values of the probability, Ψ must be single valued. i.e., for each set of x, y and z, Ψ must have a unique value. 

·         For finite potentials, Ψ and x, y, z, ∂Ψ/∂x, ∂Ψ/∂y, ∂Ψ/∂z must be continuous in all regions. 

·         In order to normalize the wave function, Ψ must approach to zero as ‘x’ approaches to ± infinity.


·         The quantity that characterizes the de–Broglie wave or matter wave is called the wave function

·         It is usually denoted as Ψ = (x, y, z, t). 

·         |ψ(x)| 2 determines the probability (density) that an object in the state ψ(x) will be found at position x.

Practice questions

1.      Enlist the properties of acceptable wave function.

2.      What you understand by normalised and orthogonal wave function.



It was observed that the wave function of a particle of fixed energy E could most naturally be written as a linear combination of wave functions of the form

Ψ(x,t) = Aei(kx−ωt)

representing a wave travelling in the positive x direction, and a corresponding wave travelling in the opposite direction, so giving rise to a standing wave.

for a free particle of momentum p = ℏk and energy E = ℏω. With this in mind, we can then note that 

2Ψ /∂x2 = −k2Ψ

which can be written, using E = p2/2m = ℏ2k2/2m

-ℏ2 /2m ∂2Ψ /∂x2 = p2/ 2m Ψ………. (1)

Similarly, ∂Ψ/ ∂t = −iωEΨ

which can be written, using E = ℏω

i/ω ∂Ψ/ ∂t = = EΨ………..(2)

We now generalize this to the situation in which there is both a kinetic energy and a potential energy present, then

E = p2/2m + V (x)

so that EΨ = p2/ 2m Ψ + V (x)Ψ

where Ψ is now the wave function of a particle moving in the presence of a potential V (x). But if we assume that the results Eq. (6.3) and Eq. (6.5) still apply in this case then we have

− ℏ2/ 2m ∂2ψ/ ∂x2 + V (x)Ψ = iℏ ∂ψ/ ∂t


1.      Wave function can be written as Ψ(x,t) = Aei(kx−ωt)

2.      Time dependent wave equation is 

2/ 2m ∂2ψ/ ∂x2 + V (x)Ψ = iℏ ∂ψ/ ∂t

Practice questions

1.      Give energy and momentum operator.

2.      Derive time independent energy operator.



A very important special case of the Schrodinger equation is the situation when the potential energy term does not depend on time. In fact, this particular case will cover most of the problems

As the name implies, this is the situation when the potential depends only on position.

If ᴪ(x,t) = ᴪ(x), then the Schrodinger equation becomes 

Ψ(x,t) = Aei(kx) e −iωt ………. (1)

ᴪo = Aei(kx)

Ψ(x,t) = ᴪo e −iωt ………….(2)

Using time dependent equation

− ℏ2/ 2m ∂2ψ/ ∂x2 + V (x)Ψ = iℏ ∂ψ/ ∂t…….(3)

Double Differentiate 2 with respect to x, which results into

2ψ/ ∂x2 = ∂2ψo/ ∂x2 e −iωt ………(4)

Differentiate 1 with respect to time t

∂ψ/ ∂t = -iωᴪo e −iωt …………..(5)

Put 4 and 5 equations in equation 3 

− ℏ2/ 2m ∂2ψo/ ∂x2 e −iωt + V (x) ᴪo e −iωt = iℏ -iωᴪo e −iωt

Extracting common terms i.e. e −iωt

− ℏ2/ 2m ∂2ψo/ ∂x2 + V (x) ᴪo = ℏω ᴪo

 This is one-dimensional time independent Schrodinger wave equation. 


Time independent Schrodinger equation is given as 

− ℏ2/ 2m ∂2ψo/ ∂x2 + V (x) ᴪo = ℏω ᴪo

Practice questions

1.      Give energy and momentum operator.

2.      Derive time independent Schrodinger equation.

Lecture 3.9 Practice problems



·         Find the wavelength of a proton that is moving at 1.00% of the speed of light.

Solution. Formula used- ⅄ = h/mv 

·         h = 6.62607015 × 10−34 joule second

·         m = 1.6726219 × 10-27 kg 

·         v = 1/100 * 3 ˟ 10 8 m/s

·         Wavelength = 1.32 ˟ 10 -13  m


·         What is the velocity of a 0.400-kg billiard ball if its wavelength is 7.50 fm?

Solution ⅄ = h/mv 

·         m = 0.4 kg     

·          λ = 0.75 ˟ 10 -15 m 

Velocity = …..m/s


·         An electron of mass 9.11 × 10−31 kg moves at nearly the speed of light. Using a velocity of 3.00 × 108 m/s, calculate the wavelength of the electron.

SOLUTION. mass (m) = 9.11 × 10−31 kg

·         Planck’s constant (h) = 6.6262 10−34 × J · s

·         velocity (v) = 3.00 × 108 m/s

·         =h/mv=6.626×10−34J⋅s/(9.11×10−31 kg)×(3.00×108 m/s)=2.42×10−12 m


·         The momentum of photon A is pA and the momentum of photon B is pB. If pB = 1/3 pA, which of the following of the de Broglie wavelength is correct?


We can use the formula: λ = h/p.

In this case, we know that p = 7.6 × 10-35 kg∙m/s.

Thus, λ = (6.63 × 10-34) / (7.6 × 10-35) = 8.72 eV


·         If the position of an electron in an atom is measured to an accuracy of 0.0100 nm, what is the electron’s uncertainty in velocity? (b) If the electron has this velocity, what is its kinetic energy in eV?


Using the equals sign in the uncertainty principle to express the minimum uncertainty, we have


Solving for ΔpΔp and substituting known values gives

Δp=Δp=h/4πΔx==6.63×10−34J⋅s/4π(1.00×10−11m) =5.28×10−24kg⋅m/s



Solving for Δv and substituting the mass of an electron gives



Although large, this velocity is not highly relativistic, and so the electron’s kinetic energy is

KEe= 1/2mv2=1/2(9.11×10−31kg)(5.79×106m/s)=95.5eV


·         An atom in an excited state temporarily stores energy. If the lifetime of this excited state is measured to be 1.0×10−10s1.0×10−10s, what is the minimum uncertainty in the energy of the state in eV?


Solving the uncertainty principle for ΔEΔE and substituting known values gives


Now converting to eV yields



·         A proton is confined in an infinite square well of width 10 fm. (The nuclear potential that binds protons and neutrons in the nucleus of an atom is often approximated by an infinite square well potential.

·         Calculate the energy and wavelength of the photon emitted when the proton undergoes a transition from the first excited state (n = 2) to the ground state (n = 1). 

·         •In what region of the electromagnetic spectrum does this wavelength belong?


E= n 22 /8mL2

The energy E and wavelength λ of a photon emitted as the particle makes a transition from the n = 2 state to the n = 1 state are 

E = E2 − E1 = 3h 2 /8mL2 

λ = hc /E

For a proton (m=938 MeV/c 2 ), E = 6.15MeV and λ = 202 fm. The wavelength is in the gamma ray region of the spectrum.


1.       What is the probability of locating a particle of mass m between x=L/4 and x=L/2 in a 1-D box of length LL? Assume the particle is in the n=1 energy state.

2.       Calculate the electronic transition energy of acetylaldehyde (the stuff that gives you a hangover) using the particle in a box model. Assume that aspirin is a box of length 300pm that contains 4 electrons.

3.       Determine the minimum uncertainties in the positions of the following objects if their speeds are known with a precision of 1.0×10−3m/s1.0×10−3m/s: (a) an electron and (b) a bowling ball of mass 6.0 kg.

4.       A sodium atom makes a transition from the first excited state to the ground state, emitting a 589.0-nm photon with energy 2.105 eV. If the lifetime of this excited state is 1.6×10−8s1.6×10−8s, what is the uncertainty in energy of this excited state? What is the width of the corresponding spectral line. 

5.       How Long Are de Broglie Matter Waves? Calculate the de Broglie wavelength of: (a) a 0.65-kg basketball thrown at a speed of 10 m/s, (b) a nonrelativistic electron with a kinetic energy of 1.0 eV

6.       If an electron and a proton are traveling at the same speed, which one has the shorter de Broglie wavelength?

7.       If a particle is accelerating, how does this affect its de Broglie wavelength?

8.       Why is the wave-like nature of matter not observed every day for macroscopic objects?

9.       What is the wavelength of a neutron at rest? Explain.

10.   What is the energy of an electron whose de Broglie wavelength is that of a photon of yellow light with wavelength 590 nm? (b) What is the de Broglie wavelength of an electron whose energy is that of the photon of yellow light?

11.   The de Broglie wavelength of a neutron is 0.01 nm. What is the speed and energy of this neutron?

12.   What is the wavelength of an electron that is moving at a 3% of the speed of light?


Lecture 3.2.1 Nanotechnology


Imagine a world where microscopic medical implants patrol our arteries, diagnosing ailments and fighting disease; where military battle-suits deflect explosions; where computer chips are no bigger than specks of dust; and where clouds of miniature space probes transmit data from the atmospheres of Mars or Titan.

Nanomaterials are typically between 0.1 and 100 nanometres (nm) in size – with 1 nm being equivalent to one billionth of a metre (10-9 m).

This is the scale at which the basic functions of the biological world operate – and materials of this size display unusual physical and chemical properties. These profoundly different properties are due to an increase in surface area compared to volume as particles get smaller – and also the grip of weird quantum effects at the atomic-scale.

Nanotechnology is a field of research and innovation concerned with building ‘things’ – generally, materials and devices – on the scale of atoms and molecules. A nanometre is one-billionth of a metre: ten times the diameter of a hydrogen atom. The diameter of a human hair is, on average, 80,000 nanometres. At such scales, the ordinary rules of physics and chemistry no longer apply. For instance, materials’ characteristics, such as their colour, strength, conductivity and reactivity, can differ substantially between the nanoscale and the macro.

History of Nanotechnology

·         The idea of nanotechnology was born in 1959 when physicist Richard Feynman gave a lecture exploring the idea of building things at the atomic and molecular-scale.

·         However, experimental nanotechnology did not come into its own until 1981, when IBM scientists in Zurich, Switzerland, built the first scanning tunnelling microscope (STM). This allows us to see single atoms by scanning a tiny probe over the surface of a silicon crystal.

·         In 1990, IBM scientists discovered how to use an STM to move single xenon atoms around on a nickel surface – in an iconic experiment, with an inspired eye for marketing, they moved 35 atoms to spell out “IBM”.

·         Other significant advances were made in 1985, when chemists discovered how to create a soccer-ball-shaped molecule of 60 carbon atoms, which they called buckminsterfullerene (also known as C60 or Buckyball’s). 

·         In 1991, tiny, super-strong rolls of carbon atoms known as carbon nanotubes were created. These are six times lighter, yet 100 times stronger than steel.

Potential of nanotechnology


·         Nanoscale additives to or surface treatments of fabrics can provide lightweight ballistic energy deflection in personal body Armor, or can help them resist wrinkling, staining, and bacterial growth.

·         Clear nanoscale films on eyeglasses, computer and camera displays, windows, and other surfaces can make them water- and residue-repellent, antireflective, self-cleaning, resistant to ultraviolet or infrared light, antifog, antimicrobial, scratch-resistant, or electrically conductive.

·         Nanoscale materials are beginning to enable washable, durable “smart fabrics” equipped with flexible nanoscale sensors and electronics with capabilities for health monitoring, solar energy capture, and energy harvesting through movement.

·         Nano-engineered materials in automotive products include high-power rechargeable battery systems; thermoelectric materials for temperature control; tires with lower rolling resistance; high-efficiency/low-cost sensors and electronics; thin-film smart solar panels; and fuel additives for cleaner exhaust and extended range.

·         Nanostructured ceramic coatings exhibit much greater toughness than conventional wear-resistant coatings for machine parts. Nanotechnology-enabled lubricants and engine oils also significantly reduce wear and tear, which can significantly extend the lifetimes of moving parts in everything from power tools to industrial machinery.

·         Nanoparticles are used increasingly in catalysis to boost chemical reactions. This reduces the quantity of catalytic materials necessary to produce desired results, saving money and reducing pollutants. Two big applications are in petroleum refining and in automotive catalytic converters.

Electronics and IT Applications

·         Transistors, the basic switches that enable all modern computing, have gotten smaller and smaller through nanotechnology. At the turn of the century, a typical transistor was 130 to 250 nanometres in size. In 2014, Intel created a 14-nanometre transistor, then IBM created the first seven nanometre transistor in 2015, and then Lawrence Berkeley National Lab demonstrated a one nanometre transistor in 2016 Smaller, faster, and better transistors may mean that soon your computer’s entire memory may be stored on a single tiny chip.

·         Using magnetic random-access memory (MRAM), computers will be able to “boot” almost instantly. MRAM is enabled by nanometre‐scale magnetic tunnel junctions and can quickly and effectively save data during a system shutdown or enable resume‐play features.

·         Ultra-high definition displays and televisions are now being sold that use quantum dots to produce more vibrant colours while being more energy efficient.

Medical and Healthcare Applications 

·         Nanotechnology is being studied for both the diagnosis and treatment of atherosclerosis, or the build-up of plaque in arteries. In one technique, researchers created a nanoparticle that mimics the body’s “good” cholesterol, known as HDL (high-density lipoprotein), which helps to shrink plaque. 

·         Nanotechnology researchers are working on a number of different therapeutics where a nanoparticle can encapsulate or otherwise help to deliver medication directly to cancer cells and minimize the risk of damage to healthy tissue. This has the potential to change the way doctors treat cancer and dramatically reduce the toxic effects of chemotherapy.

·         Research in the use of nanotechnology for regenerative medicine spans several application areas, including bone and neural tissue engineering. For instance, novel materials can be engineered to mimic the crystal mineral structure of human bone or used as a restorative resin for dental applications. Researchers are looking for ways to grow complex tissues with the goal of one day growing human organs for transplant. Researchers are also studying ways to use graphene nanoribbons to help repair spinal cord injuries; preliminary research shows that neurons grow well on the conductive graphene surface. 

·         Nanomedicine researchers are looking at ways that nanotechnology can improve vaccines, including vaccine delivery without the use of needles. Researchers also are working to create a universal vaccine scaffold for the annual flu vaccine that would cover more strains and require fewer resources to develop each year.

Energy Applications

·         Nanotechnology is improving the efficiency of fuel production from raw petroleum materials through better catalysis. It is also enabling reduced fuel consumption in vehicles and power plants through higher-efficiency combustion and decreased friction.

·         Nanotechnology is also being applied to oil and gas extraction through, for example, the use of nanotechnology-enabled gas lift valves in offshore operations or the use of nanoparticles to detect microscopic down-well oil pipeline fractures. 

·         Researchers are investigating carbon nanotube “scrubbers” and membranes to separate carbon dioxide from power plant exhaust.

·         Nanotechnology is already being used to develop many new kinds of batteries that are quicker-charging, more efficient, lighter weight, have a higher power density, and hold electrical charge longer. 

·         An epoxy containing carbon nanotubes is being used to make windmill blades that are longer, stronger, and lighter-weight than other blades to increase the amount of electricity that windmills can generate.

Environmental Remediation

·         Nanotechnology could help meet the need for affordable, clean drinking water through rapid, low-cost detection and treatment of impurities in water. 

·         Engineers have developed a thin film membrane with nanopores for energy-efficient desalination. This molybdenum disulphide (MoS2) membrane filtered two to five times more water than current conventional filters.

·         Nanoparticles are being developed to clean industrial water pollutants in ground water through chemical reactions that render the pollutants harmless. This process would cost less than methods that require pumping the water out of the ground for treatment.

Future Transportation Benefits

·         Nanoscale sensors and devices may provide cost-effective continuous monitoring of the structural integrity and performance of bridges, tunnels, rails, parking structures, and pavements over time. Nanoscale sensors, communications devices, and other innovations enabled by nanoelectronics can also support an enhanced transportation infrastructure that can communicate with vehicle-based systems to help drivers maintain lane position, avoid collisions, adjust travel routes to avoid congestion, and improve drivers’ interfaces to onboard electronics. 

·         “Game changing” benefits from the use of nanotechnology-enabled lightweight, high-strength materials would apply to almost any transportation vehicle. For example, it has been estimated that reducing the weight of a commercial jet aircraft by 20 percent could reduce its fuel consumption by as much as 15 percent. A preliminary analysis performed for NASA has indicated that the development and use of advanced nanomaterials with twice the strength of conventional composites would reduce the gross weight of a launch vehicle by as much as 63 percent. Not only could this save a significant amount of energy needed to launch spacecraft into orbit, but it would also enable the development of single stage to orbit launch vehicles, further reducing launch costs, increasing mission reliability, and opening the door to alternative propulsion concepts.


·         Nanomaterials are typically between 0.1 and 100 nanometres (nm) in size.

·         The idea of nanotechnology was born in 1959 when physicist Richard Feynman gave a lecture exploring the idea of building things at the atomic and molecular-scale.

·         Nanoscale additives to or surface treatments of fabrics can provide lightweight ballistic energy deflection.

·         Transistors, the basic switches that enable all modern computing, have gotten smaller and smaller through nanotechnology.

·         Nanotechnology is being studied for both the diagnosis and treatment of atherosclerosis, or the build-up of plaque in arteries.

·         Researchers are investigating carbon nanotube “scrubbers” and membranes to separate carbon dioxide from power plant exhaust.

Practice questions 

1.      Define nanomaterials and compare this size with other materials.

2.      Enlist some applications of nanotechnology in transportations.

3.      How nanotechnology is being used in using energy efficiently.

Lecture 3.2.2 Properties of nanomaterials

Properties of nanomaterials

Inverse Hall-Petch effect

 It is well established that decreasing the grain size results in an increased hardness and strength as grain boundaries pose an impediment to dislocation motion (dislocation ‘pileup mechanism- the usually accepted mechanism!). But when grain size reduces to tens of nanometres (< 15 nm) the grain is not able to support a dislocation pile-up. Hence the trend of increasing hardness/strength with a decrease in grain size is broken in the nanocrystalline materials. They are even reporting in literature of decreasing strength with decreasing grain size at very small grain sizes (< 5 nm). 

Giant Magnetoresistance

Giant magnetoresistance is the dramatic decrease in the electrical resistance on the application of magnetic field, in an otherwise antiferromagnetic hybrid. Giant magnetoresistance is seen in a hybrid consisting of a non-magnetic- nano-film placed between ferromagnetic layers. In the absence of external magnetic fields the magnetic layers are anti-ferromagnetically coupled. The resistance of this state is very high. On application of a magnetic field, the spin vectors in the ferromagnetic layer tends to align in parallel leading to a drastic reduction in resistivity. 


 Ferromagnetic materials consist of magnetic domains within which the spins are parallel. When the particle size is reduced to very small sizes (typically less than 20 nm) the entire particle becomes a single domain. On further reduction in size (about less than 5 nm) the spins get thermally disordered in the absence of magnetic fields. When an external magnetic field is applied the spins are able to align in the direction of applied magnetic field, making them behave as super paramagnets (i.e., in the absence of external field the particle is paramagnetic and in the presence of a field all the spins are aligned in parallel, leading to a large increase in magnetization). This is an interesting example in which a ferromagnetic material in bulk behaves like a paramagnet when particle size is made very small. 


As the surface roughness is increased from micron-scale to nano-scale, the actual contact area of the surface decreases (assuming the apparent contact to be constant). The tip of each asperity supports part of the water droplet. This shifts hydrophobicity to higher level of super hydrophobicity; wherein, contact angles of greater than 165° can be obtained. The normal (maximum) contact angle obtained in the case of hydrophobicity is about 120° (with the best available substrate of least surface energy). The phenomenon of super hydrophobicity can lead to the development of non-wetting clothes, self-cleaning windows, non bio-fouling surfaces etc. 

Super surface activity 

With decreasing dimension of the particles, the number of surface atoms increases drastically (calculation shown later in an example). This leads to a significant energy contribution to the system from the unsatisfied bonds of the surface atoms. Hence, the surface becomes extremely ‘active’ due to the high available surface energy. This effect finds applications in: adsorption of toxic gases, catalysis, etc. 

Targeted drug delivery 

Nanoparticles can be loaded with specific sensor and drug molecule(s). The drug can be transported to the required site through blood stream. On detection of the affected tissue/cells/area by the functionalized surface group the drug is released locally on desired target (targeted drug delivery).

Achieving superhydrophobic/antibiofouling surfaces 

Nanoscale surface roughness enhances material properties like hydrophilicity or hydrophobicity, which is otherwise unattainable (by exploring different materials). The phenomenon of super hydrophobicity relies on achieving Cassie Baxter state (allowing support of water droplet on nano-roughness, without actually wetting the entire surface). Correspondingly, these water-repelling surfaces do not allow fungal/algae growth on their surfaces by rejecting water deposition on their surface. 

Rapid catalysis 

Surface activity of nanoparticles enhances multifold (few orders of magnitude) because of two reasons: (i) the surface area available to react increases as we go down in size, and (ii) the unsatisfied bonds lead to instability of nanoparticle itself. Hence the synergistic combination of enhanced surface with high energetic associated with nanoparticles enhances their catalytic activity dramatically.


Functionalization is the addition of one (or more) functional groups on the surface of a material (or particles). Usually this surface modification is achieved by chemical synthesis methods to impart certain properties to the surface (e.g enhance affinity of surface for a particular species or make the surface water repellent). It is easier to functionalize nanoparticles as they possess higher surface activity. Functionalized nanoparticles find applications in rapid catalysis, targeted drug delivery, sensors etc. 

Nano porous membrane filters 

Membrane filters sieve out harmful bacteria and are permeable only to the molecules which can pass through the nano-porous membrane. These are utilized in the filtering of water to get bacteria-free water. 

Non-wetting clothing 

Non-wetting clothes have been developed by coating nanoparticles on the fabric. If you spill coffee on your trousers made of such a material, it will just flow away without leaving a stain. This layer of nanoparticles is transparent and is invisible to eye! 

Scratch resistant lenses 

Nanometre sized alumina particles can be used as a feedstock for thermal spraying or spark plasma sintering to form fully dense pellets of transparent ceramic. These optical components being made of a ceramic provides for superior strength, hardness and thermal shock resistance as compared to their glass/plastic counterparts. Wear resistant optical components can be utilized in aerospace applications.

Spin Valves 

Spin valves essentially utilizes the giant magnetoresistance (GMR) effect. An applied magnetic field can be used to switch the material showing GMR effect from a high resistance state to a low resistance state. This makes these devices behave as ‘spin valves’ or micro-switches. Spin valves have been used in the fabrication of spin valve transistors (using silicon emitter and collector), which can be used in the detection of magnetic fields. 

Unusual behaviour of nanomaterials 

Surface to volume ratio

The surface area to volume ratio for a material or substance made of nanoparticles has a significant effect on the properties of the material. Firstly, materials made up of nanoparticles have a relative larger surface area when compared to the same volume of material made up of bigger particles. 

For example, let us consider a sphere of radius r .

The surface area of the sphere will be 4πr 

The volume of the sphere = 4/3πr 3 

Therefore the surface area to the volume ratio will be 4πr 2 /(4/3πr ) = 3/r 

It means that the surface area to volume ratio increases as the radius of the sphere decreases and vice versa. It also means that when a given volume of material is made up of smaller particles, the surface area of the material increases. Therefore, as particle size decreases, a greater proportion of the particles are found at the surface of the material. For example, a particle of size 3 nm has 50% of its particles on the surface; at 10 nm, 20% of its particles are on the surface; and at 30 nm, 5% of its particles are on the surface. Therefore, materials made of nanoparticles have a much greater surface area per unit volume ratio compared with the materials made up of bigger particles. This leads to nanoparticles being more chemically reactive. As chemical reactions occur between particles that are on the surface, a given mass of nanomaterial will be much more reactive than the same mass of material made up of large particles. This means that materials that are inert in their bulk form are reactive when produced in their nanoparticle form.

Electron/Quantum confinement

Electron confinement or quantum confinement is another process that occurs in nanoparticles. We know that when atoms are isolated, then their energy levels are discrete but when they are in packed form and in large numbers (in solid form), the energy levels split and form bands (band means group of levels). Nanoparticles represents intermediate stage.

As we have discussed the problem of particle in a box that when the dimensions of such box is of the order of the de Broglie wavelength of electrons or mean free path of the electrons, then the energy levels of the electrons change. This process is called electron confinement or quantum confinement. This process results from the electrons and holes being squeezed into a dimension that approaches a critical quantum measurement known as excit or Bohr radius. It can affect the electrical, magnetic and optical properties of nanoparticles.

Quantum well / 2 D dimensional nanomaterials

quantum well is a potential well with only discrete energy values.

The classic model used to demonstrate a quantum well is to confine particles, which were initially free to move in three dimensions, to two dimensions, by forcing them to occupy a planar region. The effects of quantum confinement take place when the quantum well thickness becomes comparable to the de Broglie wavelength of the carriers (generally electrons and holes), leading to energy levels called “energy sub bands”, i.e., the carriers can only have discrete energy values.

If one dimension is reduced to the nano-range while the other two dimensions remain large, then we obtain a structure known as quantum well. This reduction in dimension produces confinement of the electrons that also refers to the number of degrees of freedom in the electron momentum.

Lecture 3.2.3 Quantum computing

Quantum computing

Quantum computing is the use of quantum-mechanical phenomena such as superposition and entanglement to perform computation. Computers that perform quantum computations are known as quantum computers.

A quantum computer harnesses some of the almost-mystical phenomena of quantum mechanics to deliver huge leaps forward in processing power. Quantum machines promise to outstrip even the most capable of today’s—and tomorrow’s—supercomputers.

They won’t wipe out conventional computers, though. Using a classical machine will still be the easiest and most economical solution for tackling most problems. But quantum computers promise to power exciting advances in various fields, from materials science to pharmaceuticals research. Companies are already experimenting with them to develop things like lighter and more powerful batteries for electric cars, and to help create novel drugs.

Quantum computers, on the other hand, use qubits, which are typically subatomic particles such as electrons or photons. Generating and managing qubits is a scientific and engineering challenge. Some companies, such as IBM, Google, and Rigetti Computing, use superconducting circuits cooled to temperatures colder than deep space. Others, like IonQ, trap individual atoms in electromagnetic fields on a silicon chip in ultra-high-vacuum chambers. In both cases, the goal is to isolate the qubits in a controlled quantum state.

Qubits have some quirky quantum properties that mean a connected group of them can provide way more processing power than the same number of binary bits. One of those properties is known as superposition and another is called entanglement. 

All computing systems rely on a fundamental ability to store and manipulate information. Current computers manipulate individual bits, which store information as binary 0 and 1 states. Quantum computers leverage quantum mechanical phenomena to manipulate information. To do this, they rely on quantum bits, or qubits.

Here, learn about the quantum properties leveraged by qubits, how they’re used to compute, and how quantum systems scale.

There are a few different ways to create a qubit. One method uses superconductivity to create and maintain a quantum state. To work with these superconducting qubits for extended periods of time, they must be kept very cold. Any heat in the system can introduce error, which is why quantum computers operate at temperatures close to absolute zero, colder than the vacuum of space.

Quantum vocabulary and terminology

Following is a brief primer of quantum computing concepts and terms.

Qubits not bits. Quantum computers do calculations with quantum bits, or qubits, rather than the digital bits in traditional computers. Qubits allow quantum computers to consider previously unimaginable amounts of information.

Superposition. Quantum objects can be in more than one state at the same time, a situation depicted by Schrödinger’s cat, a fictional feline that is simultaneously alive and dead. For example, a qubit can represent the values 0 and 1 simultaneously, whereas classical bits can only be either a 0 or a 1.

Entanglement. When qubits are entangled, they form a connection to each other that survives no matter the distance between them. A change to one qubit will alter its entangled twin, a finding that baffled even Einstein, who called entanglement “spooky action at a distance.”

Types of qubits. At the core of the quantum computer is the qubit, a quantum bit of information typically made from a particle so small that it exhibits quantum properties rather than obeying the classical laws of physics that govern our everyday lives. A number of types of qubits are in development:

·         Superconducting qubits, or transmons. Already in use in prototype computers made by Google, IBM and others, these qubits are made from superconducting electrical circuits.

·         Trapped atoms. Atoms trapped in place by lasers can behave as qubits. Trapped ions (charged atoms) can also act as qubits.

·         Silicon spin qubits. An up-and-coming technology involves trapping electrons in silicon chambers to manipulate a quantum property known as spin.

·         Topological qubits. Still quite early in development, quasi-particles called Majorana fermions, which exist in certain materials, have the potential for use as qubits. 


·         As early as 1959 the American physicist and Nobel laureate Richard Feynman noted that, as electronic components begin to reach microscopic scales, effects predicted by quantum mechanics occur—which, he suggested, might be exploited in the design of more powerful computers.

Figure 3: Caltech physicist and Nobel Laureate Richard Feynman [3]

·         During the 1980s and ’90s the theory of quantum computers advanced considerably beyond Feynman’s early speculations. In 1985 David Deutsch of the University of Oxford described the construction of quantum logic gates for a universal quantum computer, and in 1994 Peter Shor of AT&T devised an algorithm to factor numbers with a quantum computer that would require as few as six qubits (although many more qubits would be necessary for factoring large numbers in a reasonable time). When a practical quantum computer is built, it will break current encryption schemes based on multiplying two large primes; in compensation, quantum mechanical effects offer a new method of secure communication known as quantum encryption. However, actually building a useful quantum computer has proved difficult.

·         In 1998 Isaac Chuang of the Los Alamos National Laboratory, Neil Gershenfeld of the Massachusetts Institute of Technology (MIT), and Mark Kubinec of the University of California at Berkeley created the first quantum computer (2-qubit) that could be loaded with data and output a solution. Although their system was coherent for only a few nanoseconds and trivial from the perspective of solving meaningful problems, it demonstrated the principles of quantum computation. Rather than trying to isolate a few subatomic particles, they dissolved a large number of chloroform molecules (CHCL3) in water at room temperature and applied a magnetic field to orient the spins of the carbon and hydrogen nuclei in the chloroform. (Because ordinary carbon has no magnetic spin, their solution used an isotope, carbon-13.) A spin parallel to the external magnetic field could then be interpreted as a 1 and an antiparallel spin as 0, and the hydrogen nuclei and carbon-13 nuclei could be treated collectively as a 2-qubit system. In addition to the external magnetic field, radio frequency pulses were applied to cause spin states to “flip,” thereby creating superimposed parallel and antiparallel states. 

Future research 

Quantum cryptography

While the problem of noise is a serious challenge in the implementation of quantum computers, it isn’t so in quantum cryptography, where people are dealing with single qubits, for single qubits can remain isolated from the environment for significant amount of time. Using quantum cryptography, two users can exchange the very large numbers known as keys, which secure data, without anyone able to break the key exchange system. Such key exchange could help secure communications between satellites and naval ships. But the actual encryption algorithm used after the key is exchanged remains classical, and therefore the encryption is theoretically no stronger than classical methods

Lecture 3.3.4 Methods to Synthesis of Nanomaterials

Methods to Synthesis of Nanomaterials:

In general, top-down and bottom-up are the two main approaches for nanomaterials synthesis.

·         Top-down: size reduction from bulk materials.

·         Bottom-up: material synthesis from atomic level.


·         Top-down routes are included in the typical solid –state processing of the materials. 

·         This route is based with the bulk material and makes it smaller, thus breaking up larger particles by the use of physical processes like crushing, milling or grinding.

·         Usually this route is not suitable for preparing uniformly shaped materials, and it is very difficult to realize very small particles even with high energy consumption. 

·         The biggest problem with top-down approach is the imperfection of the surface structure. Such imperfection would have a significant impact on physical properties and surface chemistry of nanostructures and nanomaterials. 

·         It is well known that the conventional top-down technique can cause significant crystallographic damage to the processed patterns.


·         Bottom –up approach refers to the build-up of a material from the bottom: atom-by-atom, molecule-by-molecule or cluster-by-cluster. 

·         This route is more often used for preparing most of the nano-scale materials with the ability to generate a uniform size, shape and distribution.

·          It effectively covers chemical synthesis and precisely controlled the reaction to inhibit further particle growth.  

·         Although the bottom-up approach is nothing new, it plays an important role in the fabrication and processing of nanostructures and nanomaterials.


 Sol-Gel Method

·         The sol-gel method is a versatile process used for synthesizing various oxide materials.

·         This synthetic method generally allows control of the texture, the chemical, and the morphological properties of the solid. This method also has several advantages over other methods, such as allowing impregnation or coprecipitation, which can be used to introduce dopants. The major advantages of the sol-gel technique include molecular scale mixing, high purity of the precursors, and homogeneity of the sol- gel products with a high purity of physical, morphological, and chemical properties.

·         In a typical sol-gel process, a colloidal suspension, or a sol, is formed from the hydrolysis and polymerization reactions of the precursors, which are usually inorganic metal salts or metal organic compounds such as metal alkoxides. 

·         A general flowchart for a complete sol-gel process is shown in Figure 2.

·         Any factor that affects either or both of these reactions is likely to impact the properties of the gel. These factors, generally referred to as sol-gel parameters, includes type of precursor, type of solvent, water content, acid or base content, precursor concentration, and temperature. These parameters affect the structure of the initial gel and, in turn, the properties of the material at all subsequent processing steps.

·         After gelation, the wet gel can be optionally aged in its mother liquor, or in another solvent, and washed. The time between the formation of a gel and its drying, known as aging, is also an important parameter. A gel is not static during aging but can continue to undergo hydrolysis and condensation.

Figure 2: Sol-Gel and Drying Flowchart

Table 1.1 Important Parameters in the Various Steps of a Sol- Gel Process

 ·         Furthermore, syneresis, which is the expulsion of solvent due to gel shrinkage, and coarsening, which is the dissolution and reprecipitation of particles, can occur. These phenomena can affect both the chemical and structural properties of the gel after its initial formation.

·         Then it must be dried to remove the solvent. 

·         Table 3.1 showed a summary of the key steps in a sol-gel process which includes the aim of each step along with experimental parameters that can be manipulated.

Applications of sol-gel method

Applications for sol-gel process derive from the various special shapes obtained directly from the gel state (monoliths, films, fibers, and monosized powders) combined with compositional and microstructural control and low processing temperatures. Compared with other methods, such as the solid-state method, the advantages of using sol-gel process include.

·         The use of synthetic chemicals rather than minerals enables high purity materials to be synthesized.

·         It involves the use of liquid solutions as mixtures of raw materials. Since the mixing is with low viscosity liquids, homogenization can be achieved at a molecular level in a short time.

·          Since the precursors are well mixed in the solutions, they are likely to be equally well-mixed at the molecular level when the gel is formed; thus on heating the gel, chemical reaction will be easy and at a low temperature.

·         Changing physical characteristic such as pore size distribution and pore volume can be achieved.

·          Incorporating multiple components in a single step can be achieved.

·         Producing different physical forms of samples is manageable.


Nanocomputing is a term used for the representation and manipulation of data by computers smaller than a microcomputer. Current devices are already utilizing transistors with channels below 100 nanometres in length.

The current goal is to produce computers smaller than 10 nanometers. Future developments in nanocomputing will provide resolutions to the current difficulties of forming computing technology at the nanoscale. For example, current nanosized transistors have been found to produce a quantum tunnelling effect where electrons ‘tunnel’ through barriers, making them unsuitable for use as a standard switch.

The increased computing power formed by Nano computers will allow for the solution of exponentially difficult real-world problems. Nanocomputing also has the advantage of being produced to fit into any environment, including the human body, whilst being undetectable to the naked eye. The small size of devices will allow for processing power to be shared by thousands of nanocomputers. Nanocomputing in the form of DNA nanocomputers and quantum computers will require different technology than current microcomputing techniques but supply their own benefits.

DNA nanocomputing

Nanocomputing can be produced by a number of nanoscale structures including biomolecules such as DNA and proteins. As DNA functions through a coding system of four nucleobases it is suited for application in data processing. DNA nanocomputers could produce faster problem solving through the ability to explore all potential solutions simultaneously. This is in contrast to conventional computers which solve problems by exploring solution paths one at a time in a series of steps.

Solutions to difficult problems would no longer be constrained by processing time. DNA has the ability to provide this level of computing ability at the nanoscale because of the endless possible rearrangements of DNA through gene-editing technology. The large number of random genetic code combination can be used for processing solutions simultaneously, necessary for solving exponentially difficult real world problems.

Quantum computing

Quantum computing provides computational power at the nanoscale with abilities that reach beyond the limitations of conventional computers. This is because quantum computers store and manipulate data through the utilization of subatomic particles dynamics. Binary computers process single pieces of information as a binary state, either a 1 or a 0. Subatomic particles have two states, but can also exist in any superposition of states. This means they are governed by the laws of quantum mechanics rather than classical physics allowing them to compute solutions to problems with greater speed whilst requiring less space.



Future applications of Nano Computing may include:

·         The simulation of drug response that is more efficient than current medical trials. This will lead to the faster development of new drugs.

·         Greater understanding of disease development through improved computational models.

·         Improved transportation logistics across the world.

·         Improved financial modelling to avoid economic downturns.

·         The development of driverless cars with the ability to process real world driving problems faster than human drivers.

·         The rapid processing of large amounts of astronomical data for discovering new planets.

·         The production of quantum simulations for modeling the behaviour of subatomic particles without the need for creating the extreme conditions necessary for observing these particles.

·         Improved machine learning for artificial intelligence progression


Development of commercial devices for nano-computing would require solving several ‘hard’ problems. Some of these problems are discussed below. 

·         The “interconnect problem” is one of the most challenging issues. There are two aspects of the interconnect problem. First, the minimum contact resistance to make connection between nanodevices and the external world is 6 K Ohms or more, which is very high. The second aspect concerns the large number of wires that would be needed to connect such complex devices. Adequate spacing between these wires would be needed to prevent crosstalk and capacitive coupling. 

·         Current efforts are focused mostly on developing basic nano-devices that could serve as the basic building-blocks in assembling larger nano-systems. However, the nature of such integration is not yet known. Mapping of the functionality of the traditional silicon-based circuits into nano-electronics paradigm would be another challenge.

·          Developing circuit models for nanodevices that could be used for integration into CAD tools for design verification and simulation will require significant effort. 

·         It would be quite a challenge to develop design and test strategies for such dense systems.

·          Cost-effective manufacturing processes will have to be developed for mass production of nano-computers based on nano-technology. Cost-effective self-assembly of nano-devices would have to be developed.


·         Sol Gel is chemical method.

·         Sol gel comes under Bottom up approach.

·         Quantum computing provides computational power at the nanoscale with abilities that reach beyond the limitations of conventional computers.

·         Quantum computing provides improved transportation logistics across the world.

Practice questions

1.Explain Sol Gel process.

2.Define Bottom up Approach.

3.What is quantum Computing?

Introduction to Quantum

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