Quantum Mechanics

It  is a key branch of physics that explains how matter and energy behave at the atomic and subatomic levels. Unlike classical mechanics, it introduces ideas like wave-particle duality, quantization, and the uncertainty principle. Developed in the early 20th century to explain things that classical physics couldn't, such as blackbody radiation, the photoelectric effect, and atomic spectra, quantum mechanics describes particles through wavefunctions, which are governed by the Schrödinger equation. This theory forms the foundation of modern physics and plays a crucial role in technologies ranging from electronics to quantum computing.

1.0Basics of Quantum Mechanics

• Quantum mechanics is the part of physics that focuses on the behavior of tiny particles like atoms, electrons, and photons. It uncovers a strange, counterintuitive world where particles act like waves, events happen based on probability, and the act of observing something can actually change it. Far from being just a theoretical idea, quantum mechanics is the backbone of many modern technologies, such as semiconductors, lasers, and even quantum computers.

2.0Wave-Particle Duality

• Light behaves like a wave in diffraction experiments, yet it knocks electrons off metals in the photoelectric effect — as if it were a particle. Similarly, electrons show interference patterns when passed through a double-slit experiment.This duality was encapsulated by de Broglie's hypothesis:

$\lambda =\frac{h}{p}$

$\lambda \to Wavelength,h\to Planc{k}^{\prime }sConstant,p\to Momentum$

3.0The Uncertainty Principle

• According to This principle we cannot precisely know both the position (x) and momentum (p)of a particle simultaneously. It’s a fundamental limit, not just an observational error.

$\Delta x\cdot \Delta p\ge \frac{h}{2}$

$\Delta x=\text{Uncertainty in position},\Delta p=\text{uncertainty in momentum}$

$\hslash =\text{reduced Planck’s constant or Dirac Constant}=\frac{h}{2\pi }$

4.0Superposition and Entanglement

• A quantum particle can persist in a superposition of multiple states. When measured, it "collapses" into one. For example, a qubit (quantum bit) can be in state:

$|\psi ⟩=\alpha |0⟩+\beta |1⟩$

• Entanglement occurs when particles become correlated in such a way that the state of one instantaneously affects the other, no matter the distance.

5.0The Schrödinger Equation

• The Schrödinger equation revolutionized physics by introducing a probabilistic model for particles and predicting phenomena that classical physics could not, such as quantum tunneling, energy quantization, and wave-particle duality.
• This is the foundational equation of non-relativistic quantum mechanics:

$i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$

• In 1-d ,for a particle of mass m in a potential Vx:

$i\hslash \frac{\partial \psi (x,t)}{\partial t}=\left(-\frac{{\hslash }^2}{2m}\frac{{\partial }^2}{\partial x^2}+V(x)\right)\psi (x,t)$

6.0Operators and Observables

• An operator is a rule or function that acts on a wavefunction ψ to give another function. Observables correspond to Hermitian operators, ensuring real-valued measurement outcomes.

Common Operators

7.0The Wavefunction and Probability

Wavefunction

• The wavefunction $\psi (x,t)$ does not represent a physical object itself but rather the probability amplitude of a particle’s position, momentum, or other properties. It’s a solution to the Schrödinger equation, and its square modulus gives the probability density.

${\left|\psi (x,t)\right|}^2=\text{Probability density at position }x\text{ and time }t$

Probability Interpretation

• The Born Rule tells us that ${\left|\psi (x,t)\right|}^{2}dx$ gives the probability of locating the particle between x and x+dx.
• The total probability of locating the particle anywhere must equal 1.

${\int }_{-\infty }^{+\infty }{\left|\psi (x,t)\right|}^{2}dx=1$

This is called Normalization

8.0Derivation of Schrödinger’s Equation

• Start with the classical energy of a particle:

$E=\frac{p^2}{2m}+V(x)$

Using de Broglie’s relation $P=-i\hslash \frac{d}{dx}$ and energy operator $E=-i\hslash \frac{d}{dt}$

$\hat{H}\psi =E\psi \Rightarrow \left(\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)\right)\psi (x)=E\psi (x)$

This is time independent Schrödinger’s Equation

9.0Particle in a 1D Infinite Potential Well

Potential

Vx=0    0<x<l or { ∞ elsewhere }

Within this well:

$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}=E\psi \Rightarrow \frac{d^2\psi }{d{x}^2}+k^2\psi =0\text{where}k=\frac{\sqrt{2mE}}{\hslash }$

General Solution

$\psi (x)=A\sin (kx)+B\cos (kx)$

Boundary Conditions:

$\psi (0)=0\Rightarrow B=0\text{and}\psi (L)=0\Rightarrow kL=n\pi \Rightarrow k=\frac{n\pi }{L}$

Quantized Energy Levels:

$E_n=\frac{n^2{\pi }^2{\hslash }^2}{2m{L}^2}\text{for}n=1,2,3,\dots$

10.0Quantum Harmonic Oscillator

• The quantum harmonic oscillator models a particle subject to a restoring force proportional to its displacement—just like a mass on a spring. It's one of the most important solvable problems in quantum mechanics.

$V(x)=\frac{1}{2}m{\omega }^2{x}^2$

Using the Schrödinger’s Equation

$-\frac{{\hslash }^2}{2m}\frac{d^2\psi }{d{x}^2}+\frac{1}{2}m{\omega }^2{x}^2\psi =E\psi$

The solution involves Hermite polynomials Hnx,and the energy levels are:

$E_n=\left(n+\frac{1}{2}\right)\hslash \omega$

Note-The zero point energy at n=0, which reflects the uncertainty principle.

11.0Scattering Theory in Quantum Mechanics

• Scattering theory studies how quantum particles (like electrons or neutrons) interact with targets or potentials. It helps understand particle collisions, atomic structures, and nuclear reactions.

Wavefunction

• After interaction, the total wavefunction is a sum of an incident wave and a scattered wave.
• Asymptotic Form

$\psi (\vec{r})\approx e^{ikz}+f(\theta )\frac{e^{ikr}}{r}$

$e^{ikz}:incidentplanewave$

f:scattering amplitude

$\frac{e^{ikr}}{r}:sphericalscatteredwave$

• Differential Cross Section:

$\frac{d\sigma }{d\Omega }={\left|f(\theta )\right|}^2$

Measures the probability of scattering into a specific angle.

• Born Approximation:
  Used for weak potentials to simplify calculations.
• Partial Wave Analysis:
  Decomposes the wavefunction into angular momentum components (important for spherical potentials).

12.0Special Relativity and Quantum Mechanics

• Special relativity and quantum mechanics are both essential for understanding the behavior of objects at high velocities and on the quantum scale. Special relativity deals with objects moving near the speed of light, while quantum mechanics explains the behavior of subatomic particles.

Special Relativity:

Postulates

1. The laws of physics hold true in all inertial reference frames.
2. The speed of light is constant for all observers in a vacuum.

Relativity Formula:

$E=m{c}^2$

$E\to Energyofaparticle,m\to Restmass,c\to Speedoflightinvacuum$

Lorentz Factor (Time Dilation and Length Contraction):

$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(v\to \text{Relative velocity of the object,c }\to \text{speed of light})$

Schrödinger Equation (Non-relativistic):

$i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$

$\left(\hslash \to \text{Reduced Planck Constant},\psi \to \text{wavefunction},\hat{H}:\text{Hamiltonian operator}\right)$

Dirac Equation (for relativistic particles):

$\left(i{\gamma }^{\mu }{\partial }_{\mu }-m\right)\psi =0$

${\gamma }^{\mu }\to \text{Dirac matrices},\psi \to \text{Dirac spinor},m\to \text{Rest mass of the particle}$

Energy-Momentum Relation (Relativistic Energy):

$E^2=p^2{c}^2+m^2{c}^4$

Energy of a Quantum Field:

$E=\sqrt{p^2{c}^2+m^2{c}^4}$

Quantum field theory unites quantum mechanics and special relativity, treating particles as quantized field excitations.

13.0Applications of Quantum Mechanics

1.Particle physics (e.g., Higgs boson)

2.High-energy physics experiments (e.g., in particle accelerators)

3.Relativistic corrections in atomic physics