The time-independent Schrödinger equation is a cornerstone of quantum mechanics. It describes how particles behave in static environments, helping us understand energy levels and wavefunctions. This powerful tool lets us solve problems like particles in boxes, quantum tunneling, and atomic structures.

By separating time from space, we can focus on the spatial part of quantum systems. This approach reveals stationary states, where energy doesn't change over time. It's key for understanding atomic orbitals, molecular bonds, and many other quantum phenomena.

Time-Independent Schrödinger Equation

Derivation and Fundamentals

• General Schrödinger equation describes quantum state evolution over time and space as a partial differential equation
• Separation of variables technique splits wavefunction into spatial and temporal components
  • Spatial component: ψ(x)
  • Temporal component: e^(-iEt/ħ)
• Energy eigenvalue equation emerges from separation of variables
  • Leads to time-independent Schrödinger equation
• Time-independent Schrödinger equation functions as an eigenvalue equation
  • Hamiltonian operator acts on spatial wavefunction
  • General form: $Hψ = Eψ$
• Stationary states concept intimately related to time-independent Schrödinger equation
  • Represents quantum states with constant energy over time
• Potential energy term V(x) in time-independent Schrödinger equation impacts solutions
  • Determines shape and behavior of wavefunctions
  • Influences energy levels of the system

Mathematical Structure and Properties

• Time-independent Schrödinger equation expressed as second-order differential equation
  • One-dimensional form: $-\frac{ħ^2}{2m}\frac{d^2ψ}{dx^2} + V(x)ψ = Eψ$
• Boundary conditions crucial for determining unique solutions
  • Ensure wavefunction continuity and smoothness
  • Example: ψ(x) → 0 as x → ±∞ for bound states
• Hermitian nature of Hamiltonian operator ensures real eigenvalues (energy levels)
• Solutions form a complete set of orthogonal eigenfunctions
  • Allows expansion of arbitrary states in terms of energy eigenstates
• Discrete energy spectrum for bound states in finite potentials
  • Continuous energy spectrum for scattering states or unbound particles

Solving the Time-Independent Schrödinger Equation

Analytical Techniques

• Power series method for solving second-order differential equations
  • Expands wavefunction as infinite series: $ψ(x) = \sum_{n=0}^{\infty} a_nx^n$
  • Useful for potentials with polynomial form (harmonic oscillator)
• WKB approximation for slowly varying potentials
  • Provides approximate solutions in classical and tunneling regions
  • Wavefunction form: $ψ(x) ≈ A\exp(\pm\frac{i}{\hbar}\int\sqrt{2m(E-V(x))}dx)$
• Separation of variables in multi-dimensional problems
  • Reduces complexity by separating spatial coordinates
  • Example: Hydrogen atom (radial and angular parts)

Applications to Simple Potentials

• Infinite square well potential illustrates particle in a box
  • Energy levels: $E_n = \frac{n^2π^2ħ^2}{2mL^2}$, where n = 1, 2, 3, ...
  • Wavefunctions: $ψ_n(x) = \sqrt{\frac{2}{L}}\sin(\frac{nπx}{L})$
• Finite square well introduces concept of tunneling
  • Bound states exist below well depth
  • Scattering states occur above well depth
• Quantum harmonic oscillator models vibrating molecules and solid-state systems
  • Energy levels: $E_n = (n + \frac{1}{2})ħω$, where n = 0, 1, 2, ...
  • Wavefunctions involve Hermite polynomials
• Delta function potential demonstrates bound states in attractive potentials
  • Single bound state for attractive delta potential
  • Energy: $E = -\frac{mα^2}{2ħ^2}$, where α strength of delta potential
• Hydrogen atom solution introduces spherical harmonics and radial wavefunctions
  • Energy levels: $E_n = -\frac{13.6 eV}{n^2}$, where n = 1, 2, 3, ...
  • Wavefunctions: product of radial function and spherical harmonic

Wavefunction Interpretation

Probabilistic Nature

• Wavefunction serves as complex-valued probability amplitude
• Probability density given by squared magnitude of wavefunction: $P(x) = |ψ(x)|^2$
• Normalization ensures total probability equals 1
  • $\int_{-\infty}^{\infty} |ψ(x)|^2 dx = 1$
• Born interpretation connects wavefunction to measurement probabilities
  • Probability of finding particle in region [a,b]: $P(a≤x≤b) = \int_a^b |ψ(x)|^2 dx$

Mathematical Properties

• Wavefunction continuity ensures smooth probability distributions
• Single-valuedness requirement reflects unique physical state
• Square-integrability condition for normalizable wavefunctions
  • $\int_{-\infty}^{\infty} |ψ(x)|^2 dx < \infty$
• Expectation values calculated using wavefunction
  • For observable A: $\langle A \rangle = \int_{-\infty}^{\infty} ψ^(x)Aψ(x)dx$
• Orthogonality of eigenfunctions: $\int_{-\infty}^{\infty} ψ_m^(x)ψ_n(x)dx = δ_{mn}$
• Completeness allows expansion of arbitrary states: $f(x) = \sum_n c_nψ_n(x)$
• Uncertainty principle emerges from wavefunction properties
  • $ΔxΔp ≥ \frac{ħ}{2}$

Applications of the Time-Independent Schrödinger Equation

One-Dimensional Systems

• Effective potential reduces three-dimensional problems to one-dimensional form
  • Example: Central force problem in spherical coordinates
• Step potentials analyze particle behavior at potential discontinuities
  • Transmission coefficient: $T = \frac{4k_1k_2}{(k_1+k_2)^2}$
  • Reflection coefficient: $R = (\frac{k_1-k_2}{k_1+k_2})^2$
• Potential barriers introduce quantum tunneling phenomenon
  • Tunneling probability: $T ≈ e^{-2κL}$, where κ = √(2m(V₀-E)/ħ²)
• Bound states occur for E < V(x) asymptotically
  • Discrete energy levels in finite wells
• Scattering states arise for E > V(x) asymptotically
  • Continuous energy spectrum
• Numerical methods solve complex one-dimensional potentials
  • Finite difference method discretizes Schrödinger equation
  • Shooting method iteratively finds eigenenergies

Advanced Concepts

• Parity conservation in symmetric one-dimensional potentials
  • Even parity: ψ(-x) = ψ(x)
  • Odd parity: ψ(-x) = -ψ(x)
• Connection between energy eigenstates and stationary states
  • Time evolution: ψ(x,t) = ψ(x)e^(-iEt/ħ)
• Perturbation theory for approximately solvable systems
  • Adds small corrections to known solutions
• Variational method estimates ground state energy
  • Uses trial wavefunctions to find upper bound on energy