SchrödingerDarstellung

SchrödingerDarstellung refers to the Schrödinger picture in quantum mechanics. In this formulation, the quantum state vector evolves in time according to the Schrödinger equation, while the operators representing observable quantities remain constant. This is in contrast to the Heisenberg picture, where the operators evolve in time and the state vector is fixed. The Schrödinger picture is often the most intuitive way to understand the time evolution of quantum systems, particularly when dealing with stationary states or when visualizing the evolution of wave functions.

The Schrödinger equation, central to this picture, is given by iħ ∂/∂t |ψ(t)⟩ = H |ψ(t)⟩, where |ψ(t)⟩ is the time-dependent state vector, ħ is the reduced Planck constant, and H is the Hamiltonian operator of the system. The Hamiltonian represents the total energy of the system. Solutions to this equation describe how the probability distribution of measurement outcomes changes over time. For systems with a time-independent Hamiltonian, the solution can be expressed as |ψ(t)⟩ = e^(-iHt/ħ) |ψ(0)⟩, where |ψ(0)⟩ is the initial state. This exponential operator is known as the time evolution operator.

The Schrödinger picture is particularly useful for calculating expectation values of observables. The expectation value of an observable A at time t is given by ⟨A⟩(t) = ⟨ψ(t)|Â|ψ(t)⟩, where  is the corresponding operator. Since the state vector evolves and the operators are constant, one directly substitutes the time-evolved state into this expression. The mathematical equivalence between the Schrödinger and Heisenberg pictures ensures that physical predictions are the same regardless of the chosen representation.