⚛️Intro to Quantum Mechanics I Unit 3 – Quantum Mechanics: Mathematical Foundations

Key Concepts and Terminology

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic scales
• Wave-particle duality refers to the concept that particles can exhibit wave-like properties and vice versa
• Quantum states represent the possible configurations or conditions of a quantum system
• Observables are physical quantities that can be measured in a quantum system (position, momentum, energy)
• Operators are mathematical tools used to represent observables and manipulate wave functions
• The Schrödinger equation is the fundamental equation of quantum mechanics that describes the time evolution of a quantum system
• Probability amplitudes are complex numbers that represent the likelihood of a quantum system being in a particular state
• Measurement in quantum mechanics causes the wave function to collapse into a definite state

Mathematical Prerequisites

• Complex numbers are essential in quantum mechanics as they are used to represent wave functions and probability amplitudes
  • Complex numbers consist of a real part and an imaginary part ($a + bi$, where $i = \sqrt{-1}$)
  • Operations on complex numbers include addition, subtraction, multiplication, and division
• Linear algebra is crucial for understanding the mathematical formalism of quantum mechanics
  • Vectors represent quantum states and wave functions
  • Matrices represent operators and transformations
  • Eigenvalues and eigenvectors are used to describe the possible outcomes of measurements and the corresponding states
• Differential equations, particularly partial differential equations (PDEs), are used to formulate and solve the Schrödinger equation
• Probability theory is necessary to interpret the results of quantum measurements and calculate expectation values
• Fourier analysis is used to study the relationship between position and momentum representations of wave functions

Quantum States and Wave Functions

• A quantum state is a complete description of a quantum system and is represented by a wave function $\Psi(x, t)$
• Wave functions are complex-valued functions that contain all the information about a quantum system
  • The modulus squared of a wave function, $|\Psi(x, t)|^2$, represents the probability density of finding the particle at position $x$ at time $t$
  • Wave functions must be normalized, meaning that the integral of $|\Psi(x, t)|^2$ over all space equals 1
• The superposition principle states that a quantum system can exist in multiple states simultaneously until a measurement is made
  • A superposition is a linear combination of different quantum states
  • Quantum interference occurs when multiple wave functions interact, leading to constructive or destructive interference patterns
• The uncertainty principle, formulated by Heisenberg, states that certain pairs of physical properties (position and momentum) cannot be simultaneously known with arbitrary precision
  • Mathematically, $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\Delta x$ is the uncertainty in position, $\Delta p$ is the uncertainty in momentum, and $\hbar$ is the reduced Planck's constant

Operators and Observables

• Operators are mathematical tools used to represent physical quantities (observables) and manipulate wave functions
  • Operators act on wave functions to extract information or transform them into new wave functions
  • Common operators include the position operator $\hat{x}$, momentum operator $\hat{p} = -i\hbar \frac{\partial}{\partial x}$, and energy operator (Hamiltonian) $\hat{H}$
• Observables are physical quantities that can be measured in a quantum system
  • Observables are represented by Hermitian (self-adjoint) operators, which ensure real eigenvalues corresponding to the possible measurement outcomes
  • The eigenvalues of an observable represent the possible values that can be obtained upon measurement
  • The eigenfunctions (eigenstates) of an observable represent the wave functions that correspond to specific eigenvalues
• The commutator of two operators, $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$, determines whether the observables represented by the operators can be simultaneously measured with arbitrary precision
  • If the commutator is zero, the observables are said to commute and can be simultaneously measured
  • If the commutator is non-zero, the observables are said to be incompatible and cannot be simultaneously measured with arbitrary precision (uncertainty principle)

Schrödinger Equation

• The Schrödinger equation is the fundamental equation of quantum mechanics that describes the time evolution of a quantum system
  • The time-dependent Schrödinger equation is given by $i\hbar \frac{\partial \Psi(x, t)}{\partial t} = \hat{H} \Psi(x, t)$, where $\hat{H}$ is the Hamiltonian operator
  • The time-independent Schrödinger equation is given by $\hat{H} \Psi(x) = E \Psi(x)$, where $E$ is the energy eigenvalue
• The Hamiltonian operator $\hat{H}$ represents the total energy of the system and is the sum of the kinetic and potential energy operators
  • In one dimension, $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)$, where $m$ is the mass of the particle and $V(x)$ is the potential energy
• Solving the Schrödinger equation involves finding the wave functions $\Psi(x, t)$ that satisfy the equation for a given potential energy $V(x)$
  • The solutions to the time-independent Schrödinger equation are the energy eigenfunctions (stationary states) and their corresponding energy eigenvalues
  • The general solution to the time-dependent Schrödinger equation is a linear combination of the stationary states, with time-dependent coefficients determined by the initial conditions
• The Schrödinger equation can be solved analytically for simple potentials (infinite square well, harmonic oscillator) but requires numerical methods for more complex systems

Probability and Measurement

• In quantum mechanics, the outcome of a measurement is probabilistic and is determined by the wave function of the system
• The probability of measuring a particular value of an observable is given by the Born rule
  • For a discrete observable, the probability of measuring the eigenvalue $\lambda_i$ is given by $P(\lambda_i) = |\langle \psi_i | \Psi \rangle|^2$, where $|\psi_i\rangle$ is the eigenstate corresponding to $\lambda_i$ and $|\Psi\rangle$ is the state of the system
  • For a continuous observable, the probability density of measuring a value between $x$ and $x + dx$ is given by $P(x)dx = |\Psi(x)|^2 dx$
• The expectation value (average value) of an observable $\hat{A}$ in a state $|\Psi\rangle$ is given by $\langle \hat{A} \rangle = \langle \Psi | \hat{A} | \Psi \rangle$
  • For a discrete observable, $\langle \hat{A} \rangle = \sum_i \lambda_i P(\lambda_i)$, where $\lambda_i$ are the eigenvalues and $P(\lambda_i)$ are the probabilities
  • For a continuous observable, $\langle \hat{A} \rangle = \int_{-\infty}^{\infty} \Psi^*(x) \hat{A} \Psi(x) dx$
• The measurement process in quantum mechanics is described by the collapse of the wave function
  • Upon measurement, the wave function of the system instantaneously collapses into one of the eigenstates of the observable being measured
  • The probability of collapsing into a particular eigenstate is given by the Born rule
  • After the measurement, the system is described by the eigenstate corresponding to the measured eigenvalue

Applications and Examples

• The infinite square well is a simple quantum system that demonstrates the quantization of energy levels
  • The potential energy is zero inside the well and infinite outside, confining the particle to the well
  • The energy eigenvalues are given by $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$, where $n$ is a positive integer and $L$ is the width of the well
  • The corresponding eigenfunctions (stationary states) are given by $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$
• The quantum harmonic oscillator is a model for vibrational motion in molecules and the behavior of photons in a light field
  • The potential energy is given by $V(x) = \frac{1}{2}m\omega^2 x^2$, where $m$ is the mass of the particle and $\omega$ is the angular frequency
  • The energy eigenvalues are given by $E_n = (n + \frac{1}{2})\hbar\omega$, where $n$ is a non-negative integer
  • The corresponding eigenfunctions involve Hermite polynomials and Gaussian functions
• The hydrogen atom is a fundamental quantum system that demonstrates the quantization of angular momentum and the probabilistic nature of electron orbitals
  • The Schrödinger equation for the hydrogen atom is solved in spherical coordinates, leading to the concept of atomic orbitals (s, p, d, f)
  • The energy levels are given by $E_n = -\frac{13.6 eV}{n^2}$, where $n$ is the principal quantum number
  • The electron probability density is given by the modulus squared of the wave function, $|\Psi(r, \theta, \phi)|^2$, which determines the shape of the orbitals

Common Pitfalls and Tips

• Remember that wave functions are complex-valued and that the physically meaningful quantity is the modulus squared, not the wave function itself
• Pay attention to the dimensions and units of physical quantities, especially when setting up and solving the Schrödinger equation
• Be cautious when interpreting the results of measurements in quantum mechanics, as the act of measurement itself affects the system
• Keep in mind that the uncertainty principle is a fundamental limitation on the precision of simultaneous measurements of certain pairs of observables
• When solving the Schrödinger equation, make sure to impose appropriate boundary conditions and normalization constraints on the wave functions
• Practice visualizing and interpreting the shapes of wave functions and probability densities for different quantum systems
• Remember that quantum mechanics is a probabilistic theory and that the outcomes of individual measurements are inherently random
• When dealing with operators, be aware of their commutation relations and the implications for simultaneous measurability of observables