Correct Answer: (B)
To find the wavefunction of a particle in a one-dimensional infinite potential well (also known as a quantum box), we solve the time-independent Schrödinger equation:
\[ - \frac{\hbar^2}{2m} \frac{d^2 \psi(x)}{dx^2} = E \psi(x) \]For an infinite square well defined from \( x = 0 \) to \( x = L \), the potential is:
\[ V(x) = \begin{cases} 0 & \text{for } 0 < x < L \\ \infty & \text{otherwise} \end{cases} \]The boundary conditions require:
\[ \psi(0) = 0, \quad \psi(L) = 0 \]Solving the differential equation with these boundary conditions gives normalized eigenfunctions:
\[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{n \pi x}{L} \right), \quad n = 1, 2, 3, \dots \]So, for the ground state \( n = 1 \), the wavefunction is:
\[ \psi_1(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{\pi x}{L} \right) \]This satisfies:
Hence, option (B) is correct.
Correct Answer: (B)
Direct derivation from Schrödinger equation gives:
\[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \]Correct Answer: (C)
The standing wave inside the box has wavelength:
\[ \lambda_n = \frac{2L}{n} \]Correct Answer: (C)
Outside the finite well, where \( E < V_0 \), the wavefunction decays exponentially:
\[ \psi(x) \sim e^{-\kappa x}, \quad \kappa = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}} \]Correct Answer: (B)
Quantum tunneling allows particles to penetrate classically forbidden regions due to wavefunction non-zero amplitude beyond the barrier.
Correct Answer: (C)
Transmission probability for \( E < V_0 \):
\[ T \approx e^{-2ka}, \quad k = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}} \]Correct Answer: (C)
\( k \) is the decay constant in the evanescent wave solution of Schrödinger equation under the barrier:
\[ k = \sqrt{\frac{2m(V_0 - E)}{\hbar^2}} \]Correct Answer: (C)
Bound states in finite wells satisfy \( E < V_0 \), with exponentially decaying tails outside the well.
Correct Answer: (B)
First excited state wavefunction:
\[ \psi_2(x) = \sqrt{\frac{2}{L}} \sin\left( \frac{2\pi x}{L} \right) \]At \( x = L/2 \): \( \psi_2(L/2) = 0 \), so the probability is zero.
Nil
A quantum dot is a semiconductor nanostructure where electrons are confined in all three spatial dimensions (x, y, z), making it a 0-dimensional (0D) system. This confinement leads to discrete, atom-like energy levels. Unlike quantum wells (2D) and nanowires (1D), quantum dots exhibit complete spatial confinement.
In quantum dots, electrons are spatially confined in all directions. This spatial confinement leads to the quantization of energy levels, similar to particles in a box. The boundary conditions restrict the allowed wavelengths of the electron wavefunctions, and hence the energy levels become discrete.
A nanowire confines charge carriers in two directions (e.g., x and y), allowing free motion only along one axis (z-direction). This makes it a 1D system. In contrast, quantum wells confine in one direction (2D system) and quantum dots confine in all three directions (0D system).
In a particle-in-a-box model (ideal quantum confinement), the energy levels are given by \[ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} \] Therefore, the energy spacing between adjacent levels increases as \( \frac{1}{L^2} \), where \( L \) is the confinement length. Smaller the quantum dot, larger the energy spacing.
A nanowire allows free motion of electrons only along its long axis (say z-direction), while confinement in x and y directions creates quantization in those directions. This results in a 1D quantum system. It's not the atomic length but the dimensional restriction that defines the 1D nature.
A superlattice is a periodic structure consisting of alternating layers of two or more different semiconductor materials (like GaAs and AlAs), typically a few nanometers thick. The periodic potential modulation leads to new electronic band structures and mini-bands due to quantum mechanical tunneling.
In heterostructures, different materials have different band gaps. The alignment of conduction and valence bands at the interface determines how electrons and holes are confined and how they move, affecting device behavior such as tunneling, recombination, or carrier transport.
Quantum confinement arises when the size of a nanostructure becomes comparable to or smaller than the de Broglie wavelength of electrons. This restriction leads to discrete energy states. Neither surface tension nor thermal motion are responsible for this quantization.
As established in the particle-in-a-box model, the energy spacing \( \Delta E \propto \frac{1}{L^2} \). Therefore, if the quantum dot's size is halved (\( L \rightarrow \frac{L}{2} \)), the energy spacing increases by a factor of: \[ \left(\frac{1}{\frac{L}{2}}\right)^2 = \frac{1}{\frac{L^2}{4}} = \frac{4}{L^2} \] Hence, the energy level spacing quadruples.
GaAs/AlGaAs is a widely used material system in the fabrication of quantum wells and heterostructures. These materials have similar lattice constants, minimizing strain, and their different band gaps create potential wells for carrier confinement, enabling devices like lasers, HEMTs, and photodetectors.
Explanation: Electron beam lithography (EBL) is a top-down technique used to fabricate nanoscale structures. It works by focusing an electron beam on a resist-coated substrate and writing patterns with nanoscale precision. This is in contrast to bottom-up approaches like sol-gel or self-assembly. EBL enables precise control over feature dimensions, which is essential in creating quantum nanostructures.
Explanation: When the size of a nanostructure becomes comparable to the de Broglie wavelength of electrons, quantum confinement occurs. As a result, electrons cannot occupy arbitrary energy levels but instead are restricted to discrete levels. This quantization is a direct consequence of the Schrödinger equation under confinement, where allowed energy levels are solutions to boundary conditions.
Explanation: Single-electron tunneling occurs when electrons tunnel one by one through a nanoscale junction, typically in a structure called a single-electron transistor. This phenomenon is observed when the Coulomb charging energy \( E_C = \frac{e^2}{2C} \) exceeds thermal energy \( kT \). It results in discrete charge transfer and current quantization.
Explanation: For single-electron tunneling to be observed, the charging energy \( E_C = \frac{e^2}{2C} \) must exceed the thermal energy \( kT \). Hence, a small capacitance is required, which increases \( E_C \). This makes the addition of a single electron energetically significant, preventing continuous electron flow and enabling Coulomb blockade.
Explanation: Metal nanoclusters have sizes comparable to the Fermi wavelength. At this scale, electrons are confined, resulting in discrete energy levels rather than continuous bands. This leads to size-dependent electronic, optical, and chemical properties, unlike bulk metals which have delocalized conduction bands.
Explanation: In semiconducting nanoparticles, as particle size decreases, quantum confinement increases, raising the energy of the lowest unoccupied molecular orbital (LUMO) and lowering the highest occupied molecular orbital (HOMO). This increases the effective band gap \( E_g \), leading to a blue shift in absorption and emission spectra, which causes visible color changes.
Explanation: Rare gas atoms (e.g., Ar, Ne) are chemically inert and do not form covalent or ionic bonds. The weak attractive force between atoms is van der Waals interaction. These forces arise due to temporary dipoles induced in the electron clouds of adjacent atoms.
Explanation: The Langmuir-Blodgett technique involves creating a monolayer of molecules on a liquid surface and then transferring this monolayer onto a solid substrate. This technique enables self-assembly of nanostructures with high uniformity and control over layering and orientation.
Explanation: Nanoparticles have a very high surface area to volume ratio, which increases the number of active sites for catalytic reactions. This makes them highly effective catalysts, often requiring smaller amounts of material compared to bulk catalysts while achieving superior activity.
Explanation: Molecular clusters consist of a specific number of atoms or molecules bonded together. Their properties, such as ionization potential, electronic transitions, and optical absorption, are highly dependent on the cluster size. This leads to tunable functionalities, making them ideal for sensors, catalysis, and quantum devices.
A quantum dot confines electrons in all three spatial dimensions. According to quantum mechanics, this spatial confinement results in the quantization of energy levels. The Schrödinger equation solved for a 0D potential well yields discrete eigenvalues, i.e., the electrons can only occupy certain quantized energy states.
A nanowire has quantum confinement in two spatial directions, allowing electron motion only along its length. This makes it a quasi-one-dimensional system, where transport is effectively limited to a single axis due to size constraints in the transverse directions.
Bottom-up approaches like Chemical Vapor Deposition (CVD) build nanostructures by assembling atoms or molecules. In CVD, gaseous precursors react or decompose on a substrate to form the desired material layer-by-layer, allowing precise control over composition and thickness.
Top-down fabrication refers to techniques that begin with bulk material and pattern or etch it to create nanoscale features. Electron Beam Lithography (EBL) is a prime example: it uses focused electron beams to write patterns on a resist, enabling nanoscale resolution.
As the size of a semiconducting nanoparticle decreases, the quantum confinement becomes stronger. The band gap \( E_g \) increases approximately as \( \Delta E \propto \frac{1}{L^2} \), where \( L \) is the characteristic size. This leads to a blue shift in the absorption and emission spectra.
Pulsed Laser Deposition (PLD) is a synthesis technique in which a high-power laser pulse is focused on a target material. The energy from the pulse ablates the target, forming a plasma plume that deposits thin films on the substrate.
Superlattices are formed by alternating layers of different semiconductors, typically GaAs and AlGaAs. The periodic structure causes a modification of the electronic band structure, enabling control over carrier mobility and optical properties.
In quantum dots, the band gap depends on the size of the dot. Smaller dots have larger energy level spacings due to stronger confinement, which results in emission of photons of higher energy (shorter wavelength). This is why the color changes with size.
Molecular clusters consist of a small number of atoms or molecules, typically bonded by weak interactions. Their properties such as ionization potential, optical absorption, and reactivity vary significantly with cluster size due to quantum size effects.
Physical Vapor Deposition (PVD) is a method where atoms from a solid or liquid source are vaporized and then condensed onto a substrate. This vapor phase condensation process enables the growth of thin films and nanostructures under vacuum conditions.
Superparamagnetic nanoparticles exhibit magnetic behavior only in the presence of an external magnetic field. Due to their small size (usually < 10 nm), thermal energy is enough to randomize the magnetic moment, resulting in zero remanent magnetization once the field is removed. This contrasts with ferromagnetic particles which retain magnetization.
Superparamagnetic behavior arises when the particle size is so small that the entire nanoparticle behaves like a single magnetic domain. Typically, this occurs below a critical size (~10 nm), where thermal fluctuations cause the magnetic moment to randomly flip direction.
In GMR (Giant Magnetoresistance), electrons experience different scattering depending on their spin orientation. When adjacent ferromagnetic layers are aligned antiparallel, spin-dependent scattering is high, increasing resistance. When aligned parallel, scattering is minimized and resistance drops, producing a measurable change.
GMR structures consist of alternating ferromagnetic and nonmagnetic (spacer) layers. The relative alignment of magnetizations across the ferromagnetic layers determines the spin scattering and thus the resistance.
Ferrofluids are colloidal suspensions of magnetic nanoparticles. They are widely used in engineering applications such as damping and sealing of rotating shafts in hard drives and MRI scanners, where they respond to magnetic fields but remain liquid.
Ferrofluids remain stable under magnetic fields because they are stabilized using surfactants to prevent aggregation. The nanoparticles align with magnetic fields but return to random orientation when the field is removed.
Colossal magnetoresistance is observed in perovskite manganites such as La₁₋ₓCaₓMnO₃. These materials exhibit dramatic reductions in electrical resistance in the presence of a magnetic field, often due to magnetic phase transitions.
CMR materials transition from an insulating to a metallic state under magnetic fields. This is due to the alignment of magnetic spins that reduces spin-disorder scattering, enabling conduction.
Nanostructured thermal devices use features like nanowires or nanotubes to enhance heat transfer. These increase the surface-to-volume ratio and introduce phonon scattering that alters thermal conductivity, useful in thermoelectric and microcooling applications.
Superhydrophobic surfaces repel water due to high contact angles (>150°) and low hysteresis. They do not allow water to adhere, thanks to nanostructured roughness and often mimic natural surfaces like lotus leaves. Strong adhesion contradicts superhydrophobic behavior.
The "lotus effect" refers to the self-cleaning property of lotus leaves due to their micro- and nanostructured surface. Water droplets roll off, carrying dirt with them. This is a prime example of superhydrophobicity in biomimetics.
Biomimetics involves drawing inspiration from biological systems to design advanced materials or technologies. Examples include velcro (inspired by plant burrs), gecko-inspired adhesives, and artificial skins.
Gecko feet contain millions of microscopic setae that branch into nanoscale spatulae, allowing adhesion through van der Waals forces. This has inspired synthetic dry adhesives with similar properties using nanostructures.
Moth eyes have nanostructured anti-reflective surfaces that reduce light scattering, inspiring coatings for solar panels and lenses. This is a major example of biomimetic engineering applied to optics and energy.