Write down the time-independent (TISE) and the...

Name: Problem 6, Chapter 2: The Physics of Quantum Mechanics
Uploaded: 2021-06-07T15:42:13-08:00
Duration: 3 min 46 s
Channel: Sheh Lit Chang
Description: Write down the time-independent (TISE) and the time-dependent (TDSE) Schrödinger equations. Is it necessary for the wavefunction of a system to satisfy the TDSE? Under what circumstances does the wavefunction of a system satisfy the TISE?

Write down the time-independent (TISE) and the time-dependent (TDSE) Schrödinger equations. Is it necessary for the wavefunction of a system to satisfy the TDSE? Under what circumstances does the wavefunction of a system satisfy the TISE?

Key Concepts

Time-Dependent Schrödinger Equation (TDSE)

The TDSE is a fundamental equation in quantum mechanics that governs the full time evolution of a quantum state. It is a partial differential equation involving both spatial and temporal derivatives, and it describes how a wavefunction evolves over time under the influence of a given Hamiltonian. This equation is essential to dynamic problems in quantum mechanics where time dependence cannot be ignored.

Time-Independent Schrödinger Equation (TISE)

The TISE is derived by assuming solutions to the TDSE that separate time and space, typically leading to an eigenvalue problem for the Hamiltonian operator. It is applicable to systems in a stationary state, where the probability distribution does not change over time. The TISE is widely used in finding energy eigenstates and their corresponding energies in quantum systems with time-independent potentials.

Stationary States

Stationary states are specific solutions to the TISE where the wavefunction, aside from a time-dependent phase factor, remains constant in its spatial distribution. These states represent the energy eigenstates of a quantum system, and when a system is in a stationary state, its probability density is time-independent, making the TISE applicable.

Wavefunction

The wavefunction is the central mathematical object in quantum mechanics that encodes the state of a quantum system. It provides information about probabilities of measuring various physical properties, and it must satisfy the TDSE for dynamic evolution. When the system is in a stationary state or when time dependence is factored out, the spatial part of the wavefunction satisfies the TISE.

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