In the realm of quantum mechanics, the evolution and behavior of systems after a measurement can seem counterintuitive. Specifically, when a quantum system is measured, its wavefunction collapses to an eigenstate of the observable being measured, in this case, energy. After this collapse, the evolution of the system's wavefunction is governed by the Schr?dinger equation. This article delves into the step-by-step process, providing a comprehensive understanding of the dynamics involved.

Measurement and Collapse of the Wavefunction

When a quantum system's energy is measured, the wavefunction collapses to one of the eigenstates of the Hamiltonian operator, which corresponds to the total energy of the system. This specific eigenstate is associated with a particular energy eigenvalue.

Post-Measurement State

After the measurement, the wavefunction can be expressed as:

ψ(t0) φn

Here, φn represents the eigenstate corresponding to the measured energy En.

Time Evolution of the Wavefunction

After the measurement, the wavefunction evolves according to the time-dependent Schr?dinger equation:

?i? ?ψ/

partial t ?ψ

For a stationary state, an energy eigenstate, the solution can be expressed as:

ψ(t) φn e?iEnt/?

This indicates that the wavefunction acquires a phase factor over time but retains its spatial form if it remains in the same energy eigenstate.

Superposition and Future Measurements

If the system is subsequently left to evolve freely without further measurement, it may remain in the energy eigenstate. However, if it was initially in a superposition of energy states before measurement, it could evolve into a new superposition of these states over time. This leads to different probabilities for future measurements of energy.

Further Measurements

If a subsequent measurement of energy is performed, the wavefunction will collapse again to another energy eigenstate based on the probabilities determined by the evolved wavefunction. This process highlights the probabilistic nature of quantum mechanics and the role of the Schr?dinger equation in describing the dynamics of a quantum system.

In summary, after a measurement of energy, the wavefunction collapses to an energy eigenstate and then evolves in time according to the Schr?dinger equation. Under the condition that it remains in the same eigenstate, the wavefunction acquires a phase factor while preserving its spatial form.

Understanding these principles is crucial for gaining a deeper insight into the behavior of quantum systems and the fundamental concepts of quantum mechanics.