In the time interval (t+δt, t) the Hamiltonian H of some system varies in such a way that |H| ψ⟩

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In the time interval (t+δt, t) the Hamiltonian H of some...

$\mathrm{In}$ the time interval $(t+\delta t, t)$ the Hamiltonian $H$ of some system varies in such a way that $|H| \psi\rangle \mid$ remains finite. Show that under these circumstances $|\psi\rangle$ is a continuous function of time. A harmonic oscillator with frequency $\omega$ is in its ground state when the stiffness of the spring is instantaneously reduced by a factor $f^{4}<$ 1, so its natural frequency becomes $f^{2} \omega .$ What is the probability that the oscillator is subsequently found to have energy $\frac{3}{2} \hbar f^{2} \omega ?$ Discuss the classical analogue of this problem.

$\mathrm{In}$ the time interval $(t+\delta t, t)$ the Hamiltonian $H$ of some system varies in such a way that $|H| \psi\rangle \mid$ remains finite. Show that under these circumstances $|\psi\rangle$ is a continuous function of time.
A harmonic oscillator with frequency $\omega$ is in its ground state when the stiffness of the spring is instantaneously reduced by a factor $f^{4}<$ 1, so its natural frequency becomes $f^{2} \omega .$ What is the probability that the oscillator is subsequently found to have energy $\frac{3}{2} \hbar f^{2} \omega ?$ Discuss the classical analogue of this problem.

Key Concepts

Continuity of the Quantum State

In quantum mechanics, the time evolution of the state vector is governed by the Schrödinger equation. The condition that the norm of the product H|?? remains finite over a time interval ensures that the time derivative of |?? is also finite. This implies that |?? changes in a controlled manner, making it a continuous function of time. This concept is fundamental to ensuring that physical observables, which are derived from the wavefunction, do not exhibit unphysical jumps or discontinuities.

Sudden Approximation

The sudden approximation is used when a system’s Hamiltonian is changed instantaneously. Under this approximation, the wavefunction does not have time to adjust during the change and remains the same immediately after the perturbation. Subsequent probabilities for finding the system in new energy eigenstates are determined by projecting the unchanged wavefunction onto the new eigenbasis. This concept is critical in connecting the instantaneous change of system parameters to the probabilities of various outcomes.

Transition Probability via Overlap Integrals

When the Hamiltonian changes suddenly, the probability for the system to be found in a particular new eigenstate is given by the square of the absolute value of the overlap (inner product) between the initial state and that new eigenstate. This method of computing transition probabilities, through overlap integrals, is a staple in quantum mechanics and is directly related to the sudden approximation.

Classical Analogue of Sudden Parameter Change

The classical analogue of a sudden change in the Hamiltonian, such as an instantaneous reduction in the stiffness of the spring in a harmonic oscillator, involves an abrupt change in the system’s parameters while retaining the same initial conditions (position and momentum). This results in the oscillator beginning to oscillate with a new frequency, and energy is redistributed between potential and kinetic energy. The classical perspective helps in visualizing the impact of an abrupt parameter change on the dynamics of a system, complementing the quantum mechanical description.

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