Schrodinger Equation

Schrodinger in 1926 wrote 4 papers in 6 weeks, each with a new solution of his equation. By year end quantum mechanics had a mathematically rigorous foundation. Two years later Dirac relativized the equation and predicted antimatter.

• Quantum chemistry: the Schrodinger equation for molecules is solved by DFT and Hartree-Fock methods
• Quantum computers: a qubit is a two-level quantum system governed by H
• Solid state physics: band structure of semiconductors from the Bloch equation

Предварительные знания

• Previous lesson

Schrodinger Equation and Hamiltonian

Schrodinger in 1926 derived his equation from the analogy with Hamiltonian mechanics: i*hbar*d_t psi = H psi, where H = -hbar^2/(2m)*Delta + V is the energy operator. For one particle psi in L^2(R^3). Normalization: integral |psi|^2 = 1. Stationary states H psi_n = E_n psi_n are energy levels. Harmonic oscillator: E_n = hbar*omega*(n+1/2).

Spectral Theory and Quantization

Self-adjointness of H guarantees real spectrum and unitary evolution U(t) = e^{-iHt/hbar} (Stone's semigroup). Discrete spectrum = bound states; continuous spectrum = scattering. Rellich-Kato theorem: if V in L^2 + L^inf, then H is self-adjoint. Quantization: x_hat = x*, p_hat = -i*hbar*d_x, [x_hat, p_hat] = i*hbar.

Key Ideas

• i*hbar*d_t psi = H psi, H = -hbar^2/(2m)*Delta + V(x)
• Stationary states: H phi_n = E_n phi_n, psi_n = e^{-iE_n t/hbar} phi_n
• Oscillator: E_n = hbar*omega*(n+1/2), zero-point E_0 = hbar*omega/2
• Uncertainty principle: Delta x * Delta p >= hbar/2
• Self-adjointness of H <-> unitary group U(t) = e^{-iHt/hbar}

Further Directions

These ideas open paths to deeper mathematics.

• de-28-wave-equation — extends

Вопросы для размышления

• Give a concrete example.
• How does this connect to other areas of mathematics?