2.1 Basic principles of quantum mechanics

Quantum mechanics is the foundation of modern physics, explaining the behavior of matter and energy at the atomic scale. It introduces mind-bending concepts like wave-particle duality, uncertainty, and probability, challenging our classical intuitions about the nature of reality.

This section dives into the basic principles of quantum mechanics, including wave-particle duality, the uncertainty principle, and quantum states. We'll explore how these ideas shape our understanding of atoms, electrons, and the quantum world around us.

Wave-Particle Duality and Uncertainty

Dual Nature of Matter and Light

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• Wave-particle duality describes the fundamental property of matter and light exhibiting both wave-like and particle-like characteristics
• Light demonstrates wave behavior through phenomena such as interference and diffraction
• Light also exhibits particle-like properties through the photoelectric effect, where it interacts with matter as discrete packets of energy called photons
• Matter, including electrons and atoms, can display wave-like properties observed in electron diffraction experiments
• Louis de Broglie proposed that all matter has an associated wavelength given by the formula $λ = h/p$, where h is Planck's constant and p is momentum

Uncertainty and Probability in Quantum Mechanics

• Heisenberg uncertainty principle states the fundamental limit to the precision with which certain pairs of physical properties can be determined simultaneously
• Mathematically expressed as $Δx · Δp ≥ h/4π$, where Δx is the uncertainty in position and Δp is the uncertainty in momentum
• Applies to other complementary variables such as energy and time, with the relation $ΔE · Δt ≥ h/4π$
• Probability density represents the likelihood of finding a particle at a specific location in space
• Calculated as the square of the wavefunction's absolute value: $P(x) = |Ψ(x)|^2$
• Wavefunctions describe the quantum state of a system and contain all possible information about a particle's properties
• Solutions to the Schrödinger equation yield wavefunctions for specific quantum systems (hydrogen atom, particle in a box)

Quantum States and Numbers

Quantum Numbers and Energy Levels

• Quantum numbers describe the allowed energy states of electrons in atoms
• Principal quantum number (n) determines the main energy level and overall size of the electron orbital
• Angular momentum quantum number (l) defines the shape of the orbital (s, p, d, f)
• Magnetic quantum number (ml) specifies the orientation of the orbital in space
• Spin quantum number (ms) represents the intrinsic angular momentum of an electron (±1/2)
• Quantization of energy refers to the discrete energy levels allowed for electrons in atoms
• Bohr model demonstrates quantized energy levels in the hydrogen atom with the formula $E_n = -R_H(1/n^2)$, where RH is the Rydberg constant

Superposition and Quantum States

• Superposition principle states that a quantum system can exist in multiple states simultaneously
• Mathematically expressed as a linear combination of basis states: $|Ψ⟩ = c_1|ψ_1⟩ + c_2|ψ_2⟩ + ... + c_n|ψ_n⟩$
• Measurement of a superposition state causes the wavefunction to collapse into one of the basis states
• Quantum entanglement occurs when particles interact in ways that their quantum states cannot be described independently
• Schrödinger's cat thought experiment illustrates the concept of superposition in macroscopic systems

Quantum Mechanics Fundamentals

Schrödinger Equation and Wavefunctions

• Schrödinger equation serves as the fundamental equation of quantum mechanics
• Time-independent Schrödinger equation: $-ℏ^2/2m ∇^2Ψ + VΨ = EΨ$
• Time-dependent Schrödinger equation: $iℏ ∂Ψ/∂t = ĤΨ$
• Wavefunctions (Ψ) represent the quantum state of a system and contain all possible information about a particle's properties
• Must be continuous, single-valued, and square-integrable
• Normalization condition ensures the total probability of finding a particle somewhere in space is 1: $∫|Ψ|^2 dτ = 1$

Probability and Energy in Quantum Systems

• Probability density calculated as $P(x) = |Ψ(x)|^2$ gives the likelihood of finding a particle at a specific location
• Born interpretation states that the wavefunction's square modulus represents the probability density
• Expectation values provide the average measurement of an observable over many identical experiments
• Calculated using the formula $⟨A⟩ = ∫Ψ*ÂΨ dτ$, where  is the operator corresponding to the observable A
• Quantization of energy manifests in discrete energy levels for bound systems (atoms, molecules)
• Energy eigenvalues obtained by solving the time-independent Schrödinger equation
• Particle in a box model demonstrates quantized energy levels with $E_n = n^2h^2/8mL^2$, where L is the box length and n is the quantum number