12.2.3 Analyzing Schrödinger's wave equation

• Schrödinger's equation is the cornerstone of quantum mechanics.
• The time-independent Schrödinger equation is used to determine stationary states and their energy levels.
• The time-dependent Schrödinger equation describes the evolution of quantum states over time.
• The wave function ψ describes the probability amplitude of a particle's location.


  True
• What does the time-dependent Schrödinger equation describe that the time-independent equation does not?

  Evolution of quantum states
• The time-dependent Schrödinger equation is used for systems that change with time.
• What does the time-independent Schrödinger equation help determine?

  Stationary states and energy levels
• The time-independent Schrödinger equation describes systems in equilibrium.


  True
• The Hamiltonian operator in the time-independent Schrödinger equation represents the total energy of the system.
• The full Hamiltonian operator is expressed as H = -ħ²/2m ∇² + V(r).
• The time-independent Schrödinger equation can be used to find discrete energy levels in a one-dimensional potential well.


  True
• What is the purpose of the time-dependent Schrödinger equation?

  Describes quantum state evolution
• The time-independent Schrödinger equation helps find stationary states and their corresponding energy levels.
• In the time-independent Schrödinger equation, the symbol 'E' represents the energy
• What does the wave function $\psi$ describe in the Schrödinger equation?


  Probability amplitude
• The time-dependent Schrödinger equation is used to find stationary states.

  False
• The kinetic energy component in the Hamiltonian operator is expressed as -ħ²/2m ∇²
• What is the full expression for the Hamiltonian operator in the time-independent Schrödinger equation?

  $H =$$- \frac{\hbar^{2}}{2m} \nabla^{2} +$$V(r)$
• The Hamiltonian operator includes both kinetic and potential energy terms.

  True
• What are the energy levels for a particle in a box of length $L$?


  $E_{n} =$$\frac{n^{2} h^{2}}{8mL^{2}}$
• Stationary states are solutions to the Schrödinger equation where the probability density changes over time.

  False
• The wave function $\psi$ represents the probability amplitude
• What is the time dependence of stationary states?

  None
• What type of energy levels do stationary states have?

  Fixed and quantized
• The ground state of a hydrogen atom is a stationary state.


  True
• The probability density is proportional to the square of the wave function, $|\psi|^{2}$, indicating the likelihood of finding the particle in a particular region
• Match the type of Schrödinger equation with its form:

  Time-Dependent ↔️ $iħ \frac{\partial \Psi}{\partial t} =$$H\Psi$

  Time-Independent ↔️ $E\psi =$$H\psi$
• What does the time-dependent Schrödinger equation describe?

  Evolution of quantum states
• Match the type of Schrödinger equation with its purpose:

  Time-Dependent ↔️ Describes the evolution of quantum states over time

  Time-Independent ↔️ Determines stationary states and energy levels
• ψ in the time-independent Schrödinger equation describes the probability amplitude of the particle's location.


  True
• What are the two components of the Hamiltonian operator?

  Kinetic and potential energy
• The potential energy component of the Hamiltonian is denoted as V(r) and depends on the position of the particle.


  True
• What equation is used to analyze specific potentials in quantum mechanics?

  Time-independent Schrödinger equation
• The Hamiltonian operator represents the total energy
• What is the shape of the potential for a particle in a box?

  Confined to a region
• The wave functions for the hydrogen atom involve both radial and angular components.

  True
• Stationary states have fixed and quantized energy levels.

  True
• The probability density of a stationary state is constant
• What is proportional to the probability density of finding a particle at a given location?

  $|\psi|^{2}$
• To calculate the probability of finding a particle between two points, we integrate the probability density over that interval.

  True