Quantum Physics and Applications: Quantum Mechanics and Photonics

Comprehensive practice flashcards covering Module 1 (Quantum Mechanics) and Module 4 (Photonics) for First Year B.E. (CSE stream) at Bangalore Institute of Technology, including wave-particle duality, uncertainty principle, Schrödinger equation, potential wells, tunneling, lasers, optical fibers, modulators, and interferometry.

When and by whom was quantum mechanics developed to explain atomic and subatomic phenomena?

Quantum mechanics was developed in 1925 by Erwin Schrödinger and others to provide a fundamentally new approach for understanding and solving problems involving atoms, molecules, and matter in the solid state.

What is the de Broglie hypothesis of matter waves?

Proposed by Louis de Broglie in 1924, it states that since nature loves symmetry and radiation exhibits dual nature (behaving as waves in interference/diffraction and particles in photoelectric/Compton effects), material particles such as electrons, protons, and neutrons also exhibit wave nature under suitable circumstances.

What is the formula for the de Broglie wavelength of a material particle?

$\lambda = \frac{h}{P} = \frac{h}{mv}$, where $h$ is Planck's constant, $P$ is momentum, $m$ is mass, and $v$ is velocity.

What are the alternative expressions for de Broglie wavelength in terms of kinetic energy, accelerating potential, and temperature?

In terms of kinetic energy $E$: $\lambda = \frac{h}{\sqrt{2mE}}$ In terms of accelerating potential $V$: $\lambda = \frac{h}{\sqrt{2mqV}}$ In terms of temperature $T$: $\lambda = \frac{h}{\sqrt{3mkT}}$

What is the relationship between phase velocity, group velocity, and the speed of light for a matter wave?

$v_{phase}v_{group} = c^2$, or $v_{phase} = \frac{c^2}{v_{group}}$, where $v_{group} = v_{particle}$. Because $v_{phase} > c$, it has no direct physical significance.

What is a wave packet and what does its envelope represent?

A wave packet is a wave group formed by the superposition of two or more waves having slightly different wavelengths. The envelope of the wave packet indicates the region where the particle is likely to be found.

State Heisenberg's uncertainty principle for position-momentum, energy-time, and angular momentum-angular displacement.

The principle states that the product of inherent uncertainties in simultaneous measurement satisfies:

1. Position and momentum: $\Delta x \Delta P \ge \frac{h}{4\pi}$
2. Energy and time: $\Delta E \Delta t \ge \frac{h}{4\pi}$
3. Angular momentum and angular displacement: $\Delta L \Delta \theta \ge \frac{h}{4\pi}$

How does Heisenberg's uncertainty principle prove the non-existence of electrons inside the nucleus?

For a nucleus of diameter $\approx 10^{-14}\,\text{m}$, taking $\Delta x \le 10^{-14}\,\text{m}$ yields an uncertainty in momentum $\Delta P \ge 0.5 \times 10^{-20}\,\text{Ns}$. Using $E = \frac{P^2}{2m}$, the required kinetic energy is $E \ge 85\,\text{MeV}$. However, emitted beta particles possess energies of only $3\text{ to }4\,\text{MeV}$, proving electrons cannot exist inside the nucleus.

What is a wave function ($\psi$) and what is its complex notation for a wave travelling in the positive x-direction?

A wave function $\psi$ is a variable quantity that mathematically describes the wave characteristics of a particle and contains all information about a quantum system. In complex notation, it is given by $\psi = A e^{i(kx - \omega t)}$.

What are the momentum operator and the Hamiltonian operator in quantum mechanics?

The momentum operator is $\hat{P} = \frac{h}{2\pi i}\frac{\partial}{\partial x}$, and the total energy or Hamiltonian operator is $\hat{H} = -\frac{h^2}{8\pi^2 m}\frac{\partial^2}{\partial x^2} + U$.

Define an eigenfunction and an eigenvalue.

An eigenfunction is a wave function that, when operated on by an operator, returns the original wave function multiplied by a constant. The characteristic real constant obtained is called the eigenvalue.

What is the physical significance and mathematical formula for the expectation value of position for a single particle?

The expectation value represents the average result of a measurement after performing a large number of identical experiments. For a normalized wave function, it is given by $\langle x \rangle = \int_{-\infty}^{\infty} x |\psi|^2\,dx$.

What is the Born interpretation of the wave function?

Proposed by Max Born in 1926, it states that while $\psi$ has no direct physical meaning, the square of its absolute value, $|\psi|^2 = \psi^*\psi$, represents the probability density (probability of finding the particle per unit volume).

What are the four essential properties of a well-behaved wave function (eigenfunction)?

1. $\psi$ must be single-valued everywhere.
2. $\psi$ must be finite everywhere.
3. $\psi$ and its first derivatives ($\frac{\partial\psi}{\partial x}$, $\frac{\partial\psi}{\partial t}$) must be continuous everywhere.
4. For bound states, $\psi$ (or $\psi^*\psi$) must vanish at infinity.

State the 1D and 3D time-independent Schrödinger wave equations.

In 1D: $\frac{d^2\psi}{dx^2} + \frac{8\pi^2 m}{h^2}(E - V)\psi = 0$ In 3D: $\nabla^2\psi + \frac{8\pi^2 m}{h^2}(E - V)\psi = 0$, where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$.

What are the energy eigenvalues and normalized wave functions for a particle in a 1D infinite potential well of width $a$?

Energy eigenvalues: $E_n = \frac{n^2 h^2}{8ma^2}$, where $n = 1, 2, 3, \dots$ Normalized wave functions: $\psi_n = \sqrt{\frac{2}{a}}\sin\left(\frac{n\pi x}{a}\right)$.

What is the zero-point energy of a particle in a 1D infinite potential well?

The lowest allowed energy (ground state energy, $n=1$) is $E_{zero-point} = \frac{h^2}{8ma^2}$.

Where is a particle most likely to be found in the ground state ($n=1$) and first excited state ($n=2$) of a 1D infinite potential well?

For $n=1$: The probability density is maximum at the center, $x = \frac{a}{2}$. For $n=2$: The probability density is zero at $x = 0$, $x = \frac{a}{2}$, and $x = a$, and reaches maximum values at $x = \frac{a}{4}$ and $x = \frac{3a}{4}$.

What is the total energy expression for a particle in a 3D cubical box of side $a$?

$E_n = \frac{h^2}{8ma^2}(n_x^2 + n_y^2 + n_z^2)$, where $n_x, n_y, n_z$ are positive integers ($1, 2, 3, \dots$).

What is degeneracy in quantum mechanics, and how does it apply to the 3D potential well?

Degeneracy occurs when different combinations of quantum numbers yield the same energy eigenvalue but distinct wave functions. The ground state $(1,1,1)$ is non-degenerate ($E = \frac{3h^2}{8ma^2}$), while the first excited state ($E = \frac{6h^2}{8ma^2}$) is 3-fold degenerate due to combinations $(1,1,2)$, $(1,2,1)$, and $(2,1,1)$.

How does the energy spectrum of a particle in a finite potential well differ from that in an infinite potential well?

Because the wave functions penetrate exponentially into the classically forbidden barrier regions ($x < 0$ and $x > a$), the effective wavelength is longer. Consequently, the momentum and energy levels ($E_n$) are lower for each $n$ than in an infinite well of the same width.

What is quantum tunneling, and what are some physical phenomena that rely on it?

Quantum tunneling is the phenomenon where a particle has a finite probability of penetrating and emerging through a potential energy barrier even when its energy $E$ is less than the barrier height $V$. Examples include alpha particle decay in radioactive nuclei (where an alpha particle strikes the barrier $\approx 10^{38}$ times before escaping), tunnel diodes, scanning tunneling microscopes (STM), and nuclear fusion in the sun.

What does the acronym LASER stand for, and who constructed the first laser?

LASER stands for Light Amplification by Stimulated Emission of Radiation. The first laser was built by Theodore H. Maiman in 1960.

What are the five primary characteristics of a laser beam?

1. Directionality: Extremely low divergence.
2. Monochromaticity: Photons share the same frequency.
3. Coherence: Emitted light waves are locked in phase.
4. Intensity: High power and intensity generated by continuous reflections.
5. Focusability: Can be brought to a sharp focus using a lens.

Define induced absorption and state its rate equation.

Induced absorption occurs when an atom in lower energy level $E_1$ absorbs an incident photon of energy $h\nu = E_2 - E_1$ and transitions to excited level $E_2$ ($A + h\nu \rightarrow A^*$). The rate is $R_{abs} = B_{12} U_\nu N_1$, where $B_{12}$ is the Einstein coefficient for induced absorption.

Define spontaneous emission and state its rate equation.

Spontaneous emission occurs when an excited atom in $E_2$ transitions to lower level $E_1$ on its own without external influence, releasing a photon ($A^* \rightarrow A + h\nu$). The rate is $R_{sp} = A_{21} N_2$, where $A_{21}$ is the Einstein coefficient for spontaneous emission.

Define stimulated emission and state its rate equation.

Stimulated emission occurs when an incident photon of energy $h\nu = E_2 - E_1$ triggers an excited atom to transition to $E_1$, emitting a second identical photon in phase, frequency, and direction ($A^* + h\nu \rightarrow A + 2h\nu$). The rate is $R_{st} = B_{21} U_\nu N_2$, where $B_{21}$ is the Einstein coefficient for stimulated emission.

What are the Einstein relations between A and B coefficients, and what is their significance?

The relations are:

1. $B_{12} = B_{21}$: Probability of upward transition equals downward transition, meaning optical amplification cannot be achieved in a two-level system without a third metastable level.
2. $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$: The ratio of spontaneous to stimulated emission is proportional to $\nu^3$, making laser action difficult at high frequencies such as X-rays.

What are the main differences between spontaneous emission and stimulated emission?

Spontaneous emission occurs without incident radiation, produces incoherent photons with random phase, random direction, and a range of frequencies, and does not cause amplification. Stimulated emission occurs only under incident radiation, produces coherent photons with identical phase, direction, and frequency, and leads to optical amplification.

What is population inversion and how is it achieved?

Population inversion is a non-equilibrium state in which the population of the upper energy level ($N_2$) exceeds that of the lower energy level ($N_1$). It is achieved by pumping atoms from lower to higher levels using optical, electrical, or chemical energy.

What is a metastable state and what is its typical lifetime compared to an ordinary excited state?

A metastable state is a long-lived upper energy level where atoms accumulate to establish population inversion. Atoms remain in a metastable state for about $10^{-6}\text{ to }10^{-3}\,\text{s}$, whereas ordinary excited states have lifetimes of only about $10^{-9}\,\text{s}$.

What are the three essential requisites of a laser system?

1. Excitation (Pumping) Source: Supplies energy (optical, electrical, or chemical) to achieve population inversion.
2. Active Medium: Supports population inversion and lasing action.
3. Laser Cavity: A pair of mirrors (one fully reflecting, one partially reflecting) bounding the active medium to reflect photons along the cavity axis and amplify light.

Describe the construction and working characteristics of a GaAs semiconductor laser diode.

It consists of a single GaAs crystal with an n-section doped with tellurium and a p-section doped with zinc, having dimensions $\approx 1\,\text{mm}$ and junction width $1\,\mu\text{m}\text{ to }100\,\mu\text{m}$. Forward biasing injects high concentrations of electrons and holes that recombine to emit photons. Because the band gap of GaAs is $1.4\,\text{eV}$, it emits radiation at a wavelength of $8400\,\text{\AA}$.

What is a single photon source and what are its main quantum applications?

A single photon source emits exactly one photon at a time on demand or with high probability. It is essential for Quantum Key Distribution (QKD) in quantum communications, photonic qubits in optical quantum computing, and sensitivity enhancement in quantum sensing.

Why are optical attenuators used in single photon sources for Quantum Key Distribution (QKD)?

Attenuators attenuate laser pulses to reduce the mean photon number per pulse to $0.1$, preventing eavesdroppers from splitting off a photon to steal transmitted cryptographic key information.

What is the Pockels effect and what materials exhibit it?

The Pockels effect is a linear electro-optic phenomenon where the refractive index and birefringence of a crystal change linearly with an applied electric field ($\Delta n \propto E$). It occurs in crystals such as monopotassium phosphate ($\text{KH}_2\text{PO}_4$), potassium dideuterium phosphate ($\text{KD}_2\text{PO}_4$), lithium niobate ($\text{LiNbO}_3$), beta-barium borate (BBO), and barium titanate ($\text{BaTiO}_3$).

What is the Kerr effect and how does a Kerr cell function?

The Kerr effect is a quadratic electro-optic phenomenon where the refractive index varies with the square of the applied electric field ($\Delta n \propto E^2$), commonly observed in polar liquids like nitrobenzene. A Kerr cell places this liquid between capacitor plates; applying voltage rotates the polarization of transmitted light, allowing amplitude modulation via an output polarizer for high-speed optical switching.

What is the operating principle of an optical fiber, and what attenuation limit did Charles K. Kao determine?

Optical fibers operate on total internal reflection (TIR), requiring light to travel from a denser medium (core, $n_1$) to a rarer medium (cladding, $n_2$) at an incident angle greater than the critical angle ($i > \theta_c$). Nobel laureate Charles K. Kao demonstrated that signal attenuation must not exceed $20\,\text{dB/km}$ for practical telecommunications.

Define Acceptance Angle and Numerical Aperture (NA) of an optical fiber.

Acceptance angle ($\theta_0$) is the maximum angle of incidence at the fiber face below which light undergoes total internal reflection in the core. Numerical aperture (NA) measures light-gathering ability and is defined as $\text{NA} = \sin(\theta_0) = \sqrt{n_1^2 - n_2^2}$ (for fiber in air, $n_0 = 1$).

What is the mathematical condition for light ray propagation inside an optical fiber?

For an angle of incidence $\theta_i$ at the fiber entrance, propagation occurs if $\theta_i < \theta_0$, which equates to $\sin(\theta_i) < \text{NA}$ or $\sin(\theta_i) < \sqrt{n_1^2 - n_2^2}$.

What is the V-number formula and how is it related to the number of propagating modes ($N$)?

The V-number is defined as $V = \frac{\pi d}{\lambda}\sqrt{n_1^2 - n_2^2} = \frac{\pi d}{\lambda}(\text{NA})$, where $d$ is core diameter and $\lambda$ is operating wavelength. For $V \gg 1$, the number of supported modes is approximately $N \approx \frac{V^2}{2}$.

Compare Step-Index Single Mode, Step-Index Multimode, and Graded-Index Multimode fibers.

1. Step-Index Single Mode: Core diameter $8\text{--}10\,\mu\text{m}$, cladding $60\text{--}70\,\mu\text{m}$, uniform core index, laser source, used in submarine cables.
2. Step-Index Multimode: Core diameter $50\text{--}200\,\mu\text{m}$, cladding $100\text{--}250\,\mu\text{m}$, uniform core index, laser/LED source, used in data links.
3. Graded-Index Multimode: Core diameter $50\text{--}200\,\mu\text{m}$, cladding $100\text{--}250\,\mu\text{m}$, parabolic radially decreasing core index (sinusoidal ray paths), used in medium-distance telecom.

What is the formula for the attenuation coefficient ($\alpha$) in optical fibers?

$\alpha = -\frac{10}{L}\log\left(\frac{P_{out}}{P_{in}}\right)\,\text{dB/km}$, where $L$ is fiber length in kilometers, $P_{in}$ is launched input power, and $P_{out}$ is output power.

What are the three main attenuation mechanisms in optical fibers?

1. Absorption loss: Intrinsic material absorption and impurity absorption (e.g., iron, chromium, cobalt, copper).
2. Scattering loss: Rayleigh scattering caused by structural inhomogeneities smaller than the wavelength.
3. Radiation loss: Macroscopic bending (wrapping on spools/corners) and microscopic bending (lateral cabling pressure).

How does a Mach-Zehnder interferometer measure relative phase shift?

An input beam is split by beam splitter $BS_1$ into reflected ($R$) and transmitted ($T$) paths steered by mirrors $M_1$ and $M_2$, then recombined at $BS_2$. Reflection introduces a $180^\circ$ phase shift, causing destructive interference at detector $D_2$ and constructive interference at $D_1$. Introducing a test sample in path $R$ shifts the relative phase, causing light to be detected at $D_2$.

What are four prominent applications of the Mach-Zehnder interferometer?

1. Electro-optic modulators.
2. Quantum computing, quantum cryptography, and quantum entanglement.
3. Precision metrology (measuring tiny length variations, vibrations, or refractive index changes).
4. Aerodynamics, plasma physics, and heat transfer (measuring gas pressure, density, and temperature).