12.2.3 Analyzing Schrödinger's wave equation

Cards (67)

What does Schrödinger's wave equation describe?
Quantum mechanical systems
What does the energy eigenvalue in Schrödinger's equation represent?
Total energy of the particle
What does the time-independent form of Schrödinger's equation describe?
Energy levels of the system
The time-independent form of Schrödinger's equation is used to describe the energy levels of a quantum system. 
True
The time-independent form of Schrödinger's equation is Hψ = Eψ. 
True
Match the type of boundary condition with its description:
Continuity ↔️ ψ must be continuous at the boundaries
Smoothness ↔️ First derivative of ψ must be continuous
Finite Value ↔️ ψ must be finite
What is the continuity boundary condition for the wave function ψ?
ψ must be continuous
What does the Hamiltonian operator in Schrödinger's equation represent?
Total energy of the system
The equation for Schrödinger's wave equation is Hψ = Eψ
The time-independent Schrödinger equation is written as Hψ = Eψ
What does the Hamiltonian operator represent in Schrödinger's equation?
Total energy of the system
Match the boundary condition with its description:
Continuity ↔️ ψ must be continuous
Smoothness ↔️ dψ/dx must be continuous
Finite Value ↔️ ψ must be finite
What does the wave function (ψ) describe in quantum mechanics?
Probability distribution
The wave function (ψ) describes the physical location of a particle.
False
The wave function and eigenvalues provide a complete description of the quantum mechanical system.
True
The probability density is given by |ψ|².
True
Schrödinger's equation relates the total energy of the system to the Hamiltonian operator, wave function, and energy. 
True
The time-independent form of Schrödinger's equation describes how the wave function evolves over time.
False
Boundary conditions ensure wave function solutions are physically meaningful 
True
Allowed energy levels are described by the Hamiltonian operator 
True
Eigenvalues represent the allowed energy levels of a particle
True
The probability density is given by |ψ|²
In quantum mechanics, properties like position and momentum are described by probabilities
The Hamiltonian operator in Schrödinger's equation represents the total energy
The time-dependent form of Schrödinger's equation includes the partial derivative with respect to time
The time-dependent form of Schrödinger's equation is written as iħ∂ψ/∂t = Hψ, which describes how the wave function evolves over time
The time-dependent form of Schrödinger's equation describes how the wave function changes over time
The time-dependent form of Schrödinger's equation is written as iħ∂ψ/∂t = Hψ, which describes how the wave function evolves over time
The continuity boundary condition for a particle in a box requires ψ(0) = 0 and ψ(L) = 0.
True
The smoothness boundary condition requires the first derivative of ψ to be continuous. 
True
Schrödinger's wave equation can be used to describe the behavior of quantum mechanical systems. 
True
The Hamiltonian operator (H) represents the total energy of the quantum system
True
The time-independent Schrödinger equation describes how the wave function evolves over time
False
The time-dependent Schrödinger equation is written as iℏ∂ψ/∂t = Hψ
Boundary conditions restrict the wave functions to specific shapes that correspond to the allowed energy levels
Eigenvalues (E) represent the allowed energy levels
The wave function (ψ) describes the probability distribution of finding the particle in a particular state
For a particle in a box, the eigenvalues are given by $E =$ $(n^{2} \pi^{2} \hbar^{2}) / (2mL^{2})$ , where L is the box's length
How do you calculate the probability of finding a particle in a specific region of space?
Integrate the probability density
Match the form of Schrödinger's equation with its purpose:
Time-independent ↔️ Describes energy levels
Time-dependent ↔️ Describes wave function evolution