Srednicki — Quantum Field Theory · Problem 1.2
Attempts at relativistic quantum mechanics — Mark Srednicki
Physics — QFT · quantum-field-theory-schrodinger-equation
Problem
With the hamiltonian of eq. (1.32), show that the state defined in eq. (1.33) obeys the abstract Schr\"odinger equation, eq. (1.1), if and only if the wave function obeys eq. (1.30). Your demonstration should apply both to the case of bosons, where the particle creation and annihilation operators obey the commutation relations of eq. (1.31), and to fermions, where the particle creation and annihilation operators obey the anticommutation relations of eq. (1.38).Intuitive explanation
The key is recognizing that the abstract Schrödinger equation must hold for any state built from creation operators acting on the vacuum. By expanding the state as a superposition of -particle configurations and applying the Hamiltonian, you'll see that consistency requires the wave function coefficients to satisfy a specific differential equation—this is the bridge between the abstract operator formalism and the concrete wave-function dynamics. The beauty is that the same argument works for both bosons and fermions; only the commutation versus anticommutation relations change, but the logical structure of matching coefficients remains identical.
Intuitive explanation
The key is recognizing that the abstract Schrödinger equation must hold for any state built from creation operators acting on the vacuum. By expanding the state as a superposition of -particle configurations and applying the Hamiltonian, you'll see that consistency requires the wave function coefficients to satisfy a specific differential equation—this is the bridge between the abstract operator formalism and the concrete wave-function dynamics. The beauty is that the same argument works for both bosons and fermions; only the commutation versus anticommutation relations change, but the logical structure of matching coefficients remains identical.