3.3 Boundary conditions and normalization

The Schrödinger equation is the backbone of quantum mechanics, but it needs proper constraints to yield meaningful results. Boundary conditions and normalization are key to making the equation work in real-world scenarios.

These concepts ensure that quantum mechanical solutions make physical sense. They help us find allowed energy levels, calculate probabilities, and understand how particles behave in different potential environments. Mastering these ideas is crucial for tackling more complex quantum systems.

Boundary conditions for Schrödinger equation

Types and applications of boundary conditions

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• Constraints imposed on solutions to the Schrödinger equation at specific points or regions in space (typically at edges of potential well or at infinity)
• Bound states require wavefunction to approach zero as position approaches infinity (ψ(x) → 0 as x → ±∞)
• Unbound states must be finite and well-behaved at infinity (represented by plane waves)
• Choice of boundary conditions depends on specific potential and physical situation (infinite potential barriers, finite potential wells, periodic potentials)
• Applying boundary conditions often leads to quantization of energy levels in bound systems (fundamental feature of quantum mechanics)
• Examples of boundary conditions:
  • Infinite potential barrier: wavefunction must be zero at the barrier
  • Periodic potential: wavefunction must satisfy Bloch's theorem

Continuity requirements

• Wavefunction must be continuous everywhere
• First derivative of wavefunction must be continuous everywhere (except where potential is infinite)
• Matching conditions at potential discontinuities require wavefunction and its derivative to be continuous across boundary (ensures smooth transitions between regions)
• Examples of continuity requirements:
  • Step potential: wavefunction is continuous, but first derivative may have a discontinuity
  • Finite potential well: wavefunction and its derivative must match at well boundaries

Mathematical formulation

• Boundary conditions expressed mathematically as equations involving wavefunction and its derivatives
• For bound states: $\lim_{x \to \pm\infty} \psi(x) = 0$
• For periodic potentials: $\psi(x + a) = e^{ika}\psi(x)$ (where a is the period and k is the wave vector)
• Matching conditions at potential discontinuities: $\psi_1(x_0) = \psi_2(x_0)$ $\frac{d\psi_1}{dx}(x_0) = \frac{d\psi_2}{dx}(x_0)$ (where x_0 is the point of discontinuity)

Normalization of wavefunctions

Concept and importance

• Process of scaling wavefunction so total probability of finding particle anywhere in space equals one
• Essential for calculating expectation values and probabilities in quantum mechanics
• Ensures proper probability interpretation of wavefunction
• Normalized wavefunctions maintain normalization over time (due to unitary nature of time evolution in quantum mechanics)
• Examples of normalized wavefunctions:
  • Harmonic oscillator ground state: $\psi_0(x) = (\alpha/\pi)^{1/4} e^{-\alpha x^2/2}$ (where α is a constant related to oscillator frequency)
  • Hydrogen atom 1s orbital: $\psi_{1s}(r) = \frac{1}{\sqrt{\pi a_0^3}} e^{-r/a_0}$ (where a0 is the Bohr radius)

Mathematical formulation

• Normalization condition expressed as integral of absolute square of wavefunction over all space equaling one: $\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$
• For discrete systems, normalization condition becomes a sum: $\sum_n |\psi_n|^2 = 1$
• Normalization constant typically determined by solving normalization integral (often denoted as N or A in wavefunction expressions)
• Example calculation: For a particle in an infinite potential well of width L, unnormalized wavefunction: $\psi_n(x) = \sin(\frac{n\pi x}{L})$ Normalization constant N found by solving: $\int_0^L N^2 \sin^2(\frac{n\pi x}{L}) dx = 1$ Resulting in normalized wavefunction: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$

Special cases and alternatives

• Scattering states and other non-normalizable wavefunctions require alternative normalization schemes
• Box normalization confines system to finite volume, then takes limit as volume approaches infinity
• Delta-function normalization used for continuous energy spectra (scattering states)
• Examples of alternative normalization:
  • Free particle wavefunction: $\psi_k(x) = \frac{1}{\sqrt{2\pi}} e^{ikx}$ (normalized to delta function)
  • Plane wave in a box of length L: $\psi_k(x) = \frac{1}{\sqrt{L}} e^{ikx}$ (box normalization)

Continuity and differentiability of wavefunctions

Importance and physical significance

• Continuity of wavefunction ensures probability density remains well-defined (no abrupt jumps or discontinuities)
• First derivative continuity ensures finite kinetic energy (except at points of infinite potential)
• Discontinuities in wavefunction or its derivative can lead to unphysical results (infinite forces or energies)
• Second derivative of wavefunction relates to curvature (important in Schrödinger equation, connects wavefunction to potential energy)
• Requirements stem from need for wavefunction to be solution to Schrödinger equation (second-order differential equation)
• Examples of continuity importance:
  • Particle in a box: wavefunction must be continuous at boundaries, leading to quantized energy levels
  • Tunneling through potential barrier: continuity of wavefunction and its derivative determines transmission probability

Mathematical formulation

• Continuity of wavefunction: $\lim_{x \to x_0^-} \psi(x) = \lim_{x \to x_0^+} \psi(x)$
• Continuity of first derivative: $\lim_{x \to x_0^-} \frac{d\psi}{dx}(x) = \lim_{x \to x_0^+} \frac{d\psi}{dx}(x)$
• Schrödinger equation relates second derivative to potential: $-\frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} + V(x)\psi = E\psi$
• Examples of mathematical constraints:
  • Step potential: $\psi_1(x_0) = \psi_2(x_0)$ and $\frac{1}{m_1}\frac{d\psi_1}{dx}(x_0) = \frac{1}{m_2}\frac{d\psi_2}{dx}(x_0)$
  • Delta function potential: wavefunction continuous, but first derivative has a discontinuity

Special cases and exceptions

• Regions with abruptly changing potential energy (step potentials) wavefunction remains continuous, but first derivative may have discontinuity
• Infinite potential barriers allow discontinuity in first derivative of wavefunction
• These requirements play crucial role in determining allowed energy levels and matching solutions across different regions in piecewise-defined potentials
• Examples of special cases:
  • Infinite square well: wavefunction must be zero at boundaries, leading to sine wave solutions
  • Delta function potential: wavefunction continuous, but derivative has a jump discontinuity proportional to potential strength

Infinite vs finite potential wells

Infinite potential well characteristics

• Wavefunction must be zero at boundaries, leading to standing wave solutions with discrete energy levels
• Energy levels given by $E_n = \frac{n^2h^2}{8mL^2}$ (n quantum number, m particle mass, L well width)
• Wavefunctions are sine waves: $\psi_n(x) = \sqrt{\frac{2}{L}} \sin(\frac{n\pi x}{L})$
• No tunneling possible (particle completely confined within well)
• Examples of infinite well applications:
  • Particle in a box model for conjugated molecules
  • Approximation for deeply bound states in atoms or nuclei

Finite potential well properties

• Wavefunction extends into classically forbidden regions (decays exponentially outside well)
• Requires matching wavefunction and its derivative at well boundaries (leads to transcendental equations for energy levels)
• Number of bound states depends on well depth and width (at least one bound state always present)
• Energy levels lower than those of infinite well of same width (due to tunneling effect)
• Examples of finite well applications:
  • Quantum dots in semiconductor nanostructures
  • Nuclear shell model for light nuclei

Comparison and problem-solving approaches

• Infinite well simpler to solve analytically, finite well often requires numerical or graphical methods
• Finite well more realistic model for many physical systems (allows for tunneling and finite confinement)
• As well depth increases, finite well solutions approach those of infinite well
• Solving finite well problems involves:
  1. Writing wavefunctions for inside and outside well
  2. Applying continuity conditions at boundaries
  3. Solving resulting transcendental equation for energy levels
• Examples of comparative analysis:
  • Ground state energy in finite well always lower than in infinite well of same width
  • Number of nodes in wavefunction relates to quantum number in both cases, but spacing differs