2.2 Schrödinger Equation and Wave Functions

Quantum mechanics gets weird when we zoom in super close. The Schrödinger equation is like the rulebook for this tiny world, telling us how particles behave as waves. It's the foundation for understanding atoms, molecules, and all sorts of quantum phenomena.

Wave functions are the mathematical tools we use to describe these quantum particles. They're not something we can directly observe, but they give us probabilities for where particles might be and what properties they might have. It's a whole new way of thinking about reality at the smallest scales.

Time-Dependent and Time-Independent Schrödinger Equations

Derivation and Formulation

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• The time-dependent Schrödinger equation is a partial differential equation that describes the evolution of a quantum system over time, given by $iℏ\frac{∂Ψ(r,t)}{∂t} = ĤΨ(r,t)$, where $Ĥ$ is the Hamiltonian operator, $Ψ(r,t)$ is the wave function, $ℏ$ is the reduced Planck's constant, and $i$ is the imaginary unit
• The time-independent Schrödinger equation is an eigenvalue equation that describes the stationary states of a quantum system, given by $ĤΨ(r) = EΨ(r)$, where $E$ is the energy eigenvalue corresponding to the eigenstate $Ψ(r)$
• The Hamiltonian operator $Ĥ$ consists of the kinetic energy operator $T̂ = -\frac{ℏ^2}{2m}∇^2$ and the potential energy operator $V̂(r)$, such that $Ĥ = T̂ + V̂(r)$

Relation and Separation of Variables

• The time-independent Schrödinger equation can be derived from the time-dependent equation by assuming that the wave function is separable into spatial and temporal components, $Ψ(r,t) = ψ(r)φ(t)$, and applying the separation of variables technique
  • This assumption allows the time-dependent equation to be split into two separate equations, one for the spatial component and another for the temporal component
  • The spatial equation is the time-independent Schrödinger equation, while the temporal equation describes the phase evolution of the wave function over time

Wave Functions and Their Properties

Physical Interpretation and Probability

• The wave function $Ψ(r,t)$ is a complex-valued function that contains all the information about a quantum system and its evolution over time
• The probability density of finding a particle at a given position $r$ and time $t$ is given by the square modulus of the wave function, $|Ψ(r,t)|^2$
  • This interpretation, known as the Born rule, connects the abstract wave function to observable quantities in quantum mechanics
  • The probability of finding the particle in a small volume element $dV$ around position $r$ at time $t$ is given by $|Ψ(r,t)|^2 dV$

Mathematical Requirements and Normalization

• The wave function must be continuous, single-valued, and square-integrable to ensure that the probability density is well-defined and normalizable
  • Continuity ensures that the wave function does not have any abrupt jumps or discontinuities
  • Single-valuedness means that the wave function assigns a unique value to each point in space and time
  • Square-integrability guarantees that the integral of $|Ψ(r,t)|^2$ over all space is finite
• The normalization condition requires that the integral of the probability density over all space equals 1, ensuring that the total probability of finding the particle somewhere in space is 100%
  • Mathematically, this is expressed as $∫|Ψ(r,t)|^2 dr = 1$
  • Normalization allows the wave function to be interpreted as a probability amplitude, with its square modulus representing the probability density

Expectation Values and Operators

• The expectation value of an observable $A$ is given by $⟨A⟩ = ∫Ψ^*(r,t)ÂΨ(r,t)dr$, where $Â$ is the corresponding operator and $Ψ^*(r,t)$ is the complex conjugate of the wave function
  • The expectation value represents the average value of the observable over many measurements on identically prepared systems
  • Examples of observables include position ($\hat{x}$), momentum ($\hat{p} = -iℏ\frac{∂}{∂x}$), and energy ($\hat{H}$)
• Operators in quantum mechanics are mathematical objects that correspond to physical observables and act on wave functions to extract information about the system
  • Hermitian operators have real eigenvalues and orthogonal eigenfunctions, ensuring that the observables they represent have real-valued measurable quantities
  • The commutator of two operators, $[Â, B̂] = ÂB̂ - B̂Â$, determines whether the observables they represent can be simultaneously measured with arbitrary precision (commuting operators) or are subject to the Heisenberg uncertainty principle (non-commuting operators)

Solving the Schrödinger Equation

Simple Quantum Systems

• The infinite square well potential is a simple quantum system with $V(x) = 0$ for $0 < x < L$ and $V(x) = ∞$ elsewhere, leading to energy eigenvalues $E_n = \frac{n^2h^2}{8mL^2}$ and eigenfunctions $ψ_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{nπx}{L})$, where $n$ is a positive integer
  • This system represents a particle confined to a one-dimensional box with impenetrable walls
  • The energy levels are quantized, and the eigenfunctions exhibit nodes and antinodes within the well
• The harmonic oscillator potential is another simple quantum system with $V(x) = \frac{1}{2}kx^2$, leading to energy eigenvalues $E_n = (n + \frac{1}{2})ℏω$ and eigenfunctions expressed in terms of Hermite polynomials, where $ω = \sqrt{\frac{k}{m}}$ is the angular frequency and $n$ is a non-negative integer
  • This system represents a particle subject to a restoring force proportional to its displacement from equilibrium, such as a mass attached to a spring
  • The energy levels are evenly spaced, and the eigenfunctions exhibit increasing numbers of nodes with increasing energy

Boundary Conditions and Matching

• The particle in a box with finite potential walls can be solved by applying appropriate boundary conditions and matching the wave functions and their derivatives at the potential boundaries
  • The wave function must be continuous and differentiable at the boundaries, ensuring a smooth transition between the regions
  • The energy eigenvalues are determined by transcendental equations that arise from the matching conditions, and the eigenfunctions are piecewise-defined functions that decay exponentially outside the well
• The hydrogen atom can be solved in spherical coordinates by separating the Schrödinger equation into radial and angular parts, leading to energy eigenvalues $E_n = -\frac{13.6 eV}{n^2}$ and eigenfunctions expressed in terms of spherical harmonics and associated Laguerre polynomials
  • The boundary conditions require the wave function to be finite at the origin and approach zero at infinity, ensuring a bound state solution
  • The separation of variables leads to quantum numbers ($n$, $l$, $m_l$) that characterize the energy, angular momentum, and its projection on a chosen axis

Energy Eigenstates from Solutions

Stationary States and Eigenfunctions

• The energy eigenstates are the stationary states of a quantum system, characterized by their energy eigenvalues and corresponding eigenfunctions
  • In a stationary state, the probability density does not change over time, although the wave function may acquire a phase factor
  • The eigenfunctions are the solutions to the time-independent Schrödinger equation for a given potential energy function
• The eigenfunctions form a complete orthonormal set, meaning that any arbitrary state of the system can be expressed as a linear combination of the eigenfunctions
  • Completeness ensures that the eigenfunctions span the entire Hilbert space of possible states
  • Orthonormality means that the eigenfunctions are mutually orthogonal ($∫ψ_n^*(x)ψ_m(x)dx = δ_{nm}$) and normalized ($∫|ψ_n(x)|^2dx = 1$)

Probability Amplitudes and Measurements

• The coefficients in the linear combination represent the probability amplitudes of measuring the system in each eigenstate, and their squared moduli give the probabilities of finding the system in those states
  • The probability amplitudes are complex numbers that encapsulate the relative phases and magnitudes of the eigenstates in the superposition
  • The act of measurement collapses the wave function onto one of the eigenstates, with the probability given by the squared modulus of the corresponding probability amplitude
• The ground state is the lowest energy eigenstate, while excited states correspond to higher energy eigenvalues
  • The ground state is the most stable configuration of the system, and it is the state that the system naturally tends to occupy in the absence of external perturbations
  • Excited states can be accessed by absorbing energy from the environment or through controlled excitations, such as electromagnetic radiation or particle collisions

Energy Spectra and Degeneracy

• The energy spectrum can be discrete (as in bound systems like the infinite square well or the hydrogen atom) or continuous (as in unbound systems like free particles or scattering states)
  • Discrete spectra arise when the system is confined by a potential energy function that allows only certain energy values, leading to quantized energy levels
  • Continuous spectra occur when the system is not confined, and the energy can take on any value within a certain range
• The degeneracy of an energy level refers to the number of distinct eigenstates sharing the same energy eigenvalue, which can occur due to symmetries in the system
  • Degenerate states have different eigenfunctions but the same energy, and they can be linearly combined to form alternative basis sets
  • The presence of degeneracy can lead to interesting physical phenomena, such as the Zeeman effect (splitting of energy levels in a magnetic field) or the fine structure of atomic spectra (due to relativistic corrections and spin-orbit coupling)