Quantizing Schrodinger Field continued
Classical Action: Started with a classical system described by the action: � = ∫ � � ∫ � 3 � � S = ∫ d t ∫ d 3 x L where � = � ∗ ( � ℏ ∂ � − � ^ ) � L = ψ ∗ ( i ℏ ∂ t − H ^ ) ψ . Schrödinger equation arises from the classical action. Expansion and Eigenfunctions: Expanded � ψ in terms of eigenfunctions � � u n of the Hamiltonian � ^ H ^ : � = ∑ � � � � � ψ = ∑ n a n u n . � � u n satisfies � ^ � � = � � � � H ^ u n = e n u n . Quantization: � � a n and � � ∗ a n ∗ were taken as generalized coordinates and momenta. Replaced functions with operators, imposing commutation relations: [ � � , � � ∗ ] = � � � [ a n , a m ∗ ] = δ mn . Quantum Hamiltonian: Constructed the quantum Hamiltonian: � ^ = ∑ � � � � � ∗ � � H ^ = ∑ n e n a n ∗ a n . Infinite Harmonic Oscillators: Described the resulting quantum system as an infinite collection of harmonic oscillators with different frequencies � � = � � ℏ ω n = ℏ e n . No interaction between oscillator...