Sudden Approximation in Quantum Mechanics: This technique is called a sudden approximation in order to examine quantum systems with fast variations. It becomes most useful in a situation when the Hamiltonian of the system is varying that quickly that not enough time has elapsed for the system to modify its state between two points.
Instantaneous Change: According to the sudden approximation, the Hamiltonian of the system changes abruptly at some point in time.
Wavefunction Continuity: Right at the time of change, the wavefunction of the system is continuous. However, it does not instantaneously become an eigenstate of the new Hamiltonian after the change.
Applications:
Quantum Jumps: Quantum jumps are described by using the sudden approximation where a particle changes its state by jumping from one state to another because of the rapid change of the potential.
Molecular and Atomic Physics: This approach is useful when the external fields or interactions are changing rapidly, for example, in collisions or sudden pulses of electric or magnetic fields.
Transition Amplitudes: It helps in the computation of transition amplitudes and probabilities in different quantum systems which experience sudden changes.
Example:
Let us consider a particle in a square well which changes its shape. If the particle was initially at the ground state of the initial potential, then the sudden approximation allows us to determine the probability of finding the particle in a particular eigenstate of the final potential immediately after the change has been made.
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