Glossary

4.1 Time-dependent and time-independent Schrödinger equations

3 min read•august 9, 2024

The Schrödinger equation is the foundation of quantum mechanics. It comes in two flavors: time-dependent and time-independent. These equations describe how particles behave at the quantum level, helping us understand atomic structure and energy levels.

Time-dependent Schrödinger equations show how quantum states change over time. Time-independent equations deal with stationary states and energy levels. Both are crucial for grasping quantum behavior and solving real-world problems in chemistry and physics.

Schrödinger Equation Fundamentals

Core Components of the Schrödinger Equation

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  • Schrödinger equation describes quantum mechanical behavior of physical systems
  • Wave function Ψ\Psi represents the quantum state of a system
  • H^\hat{H} corresponds to the total energy of the system
  • General form of the Schrödinger equation: H^Ψ=EΨ\hat{H}\Psi = E\Psi
  • ∣Ψ∣2|\Psi|^2 gives the likelihood of finding a particle in a specific region

Wave Function Properties and Interpretation

  • Wave function Ψ\Psi contains all information about a quantum system
  • Complex-valued function of position and time
  • Must be continuous, single-valued, and square-integrable
  • Normalization condition: ∫−∞∞∣Ψ∣2dx=1\int_{-\infty}^{\infty} |\Psi|^2 dx = 1
  • Describes the amplitude of the quantum state at each point in space and time

Hamiltonian Operator and Energy Calculations

  • Hamiltonian operator H^\hat{H} represents the total energy of the system
  • Consists of kinetic energy T^\hat{T} and potential energy V^\hat{V} operators
  • General form: H^=T^+V^\hat{H} = \hat{T} + \hat{V}
  • For a particle in one dimension: H^=−ℏ22md2dx2+V(x)\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)
  • Eigenvalue equation H^Ψ=EΨ\hat{H}\Psi = E\Psi determines the energy levels of the system

Time-Dependent and Time-Independent Equations

Time-Dependent Schrödinger Equation

  • Describes the evolution of quantum states over time
  • General form: iℏ∂Ψ∂t=H^Ψi\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi
  • Accounts for systems with time-dependent potentials
  • Solutions represent wave functions that change with time
  • Used to study dynamic processes (electron transitions in atoms)

Time-Independent Schrödinger Equation

  • Applies to systems with time-independent Hamiltonians
  • Derived from the time-dependent equation when potential energy is constant
  • General form: H^Ψ=EΨ\hat{H}\Psi = E\Psi
  • Solutions represent stationary states with definite energy
  • Used to find energy levels and wave functions for bound states (hydrogen atom)

Separation of Variables Technique

  • Method to solve partial differential equations
  • Assumes the wave function can be separated into spatial and temporal parts
  • Ψ(x,t)=ψ(x)ϕ(t)\Psi(x,t) = \psi(x)\phi(t)
  • Spatial part ψ(x)\psi(x) satisfies the
  • Temporal part ϕ(t)\phi(t) has the form e−iEt/ℏe^{-iEt/\hbar}
  • Allows conversion of time-dependent to time-independent equation
  • Simplifies solving complex quantum mechanical problems (particle in a box)

Stationary States and Eigensystems

Characteristics of Stationary States

  • Quantum states with time-independent probability densities
  • Eigenstates of the Hamiltonian operator
  • Energy remains constant over time
  • Wave function evolves only by a phase factor
  • Ψ(x,t)=ψ(x)e−iEt/ℏ\Psi(x,t) = \psi(x)e^{-iEt/\hbar}
  • Fundamental to understanding quantum systems (energy levels in atoms)

Eigenvalues and Their Physical Significance

  • Eigenvalues represent observable quantities in quantum mechanics
  • Energy eigenvalues correspond to allowed energy levels of the system
  • Discrete spectrum for bound states (hydrogen atom energy levels)
  • Continuous spectrum for unbound states (free particle)
  • Determined by solving the eigenvalue equation H^ψ=Eψ\hat{H}\psi = E\psi
  • Selection rules govern transitions between energy levels (atomic spectra)

Eigenfunctions and Quantum State Representation

  • Eigenfunctions are solutions to the time-independent Schrödinger equation
  • Form a complete orthonormal basis for the Hilbert space
  • Any wave function can be expressed as a linear combination of eigenfunctions
  • Ψ(x,t)=∑ncnψn(x)e−iEnt/ℏ\Psi(x,t) = \sum_n c_n\psi_n(x)e^{-iE_nt/\hbar}
  • Coefficients cnc_n determined by initial conditions
  • allows for complex quantum states (coherent states)

Key Terms to Review (18)

Closed System: A closed system is a physical system that does not exchange matter with its surroundings but can exchange energy in the form of heat or work. In the context of quantum mechanics, a closed system is essential as it allows for the application of the Schrödinger equations, which describe how the quantum state of a physical system changes over time. This concept helps in simplifying complex interactions by focusing solely on the energy exchanges and maintaining the system's internal properties without external interference.
Eigenstate: An eigenstate is a specific quantum state of a system that corresponds to a definite value, or eigenvalue, of an observable, such as energy or momentum. In quantum mechanics, the concept of eigenstates is essential because it describes how systems behave under measurement, with each eigenstate representing a possible outcome when the system is observed. The relationship between eigenstates and observables is fundamental for both time-dependent and time-independent scenarios in quantum mechanics.
Energy Eigenvalue: An energy eigenvalue is a specific value of energy that corresponds to a quantum state of a system as described by the Schrödinger equation. In quantum mechanics, energy eigenvalues arise when solving the time-independent Schrödinger equation, yielding discrete energy levels for bound states in a potential. These values play a crucial role in determining the allowed energy states of a particle and can be thought of as the quantized energies that the system can occupy.
Hamiltonian Operator: The Hamiltonian operator is a mathematical operator used in quantum mechanics to represent the total energy of a system, including both kinetic and potential energy. It plays a crucial role in both time-dependent and time-independent formulations of the Schrödinger equations, allowing us to derive the behavior of quantum systems over time and analyze their stationary states.
Molecular Orbital Theory: Molecular orbital theory is a method for determining the electronic structure of molecules by combining atomic orbitals to form molecular orbitals, which can be occupied by electrons. This theory allows for the prediction of molecular properties and behaviors, including bond order and magnetic characteristics, by describing the distribution of electrons across the molecule. It connects to fundamental principles of quantum mechanics and linear algebra to analyze how atomic orbitals interact to form these molecular orbitals.
Momentum Operator: The momentum operator is a key mathematical tool in quantum mechanics, represented in one dimension as \\(-i \hbar \frac{d}{dx}\\), where \(\hbar\) is the reduced Planck's constant. It plays a crucial role in both time-dependent and time-independent formulations of quantum mechanics, acting on wave functions to yield information about a particle's momentum. This operator connects the wave-like behavior of particles with their measurable properties, linking them to observable quantities such as momentum through the formalism of operators.
Open System: An open system is a physical system that can exchange both matter and energy with its surroundings. This concept is important in understanding how systems interact with their environment and allows for the study of dynamic processes, such as those described by time-dependent and time-independent Schrödinger equations, where changes in energy levels and probabilities are influenced by external factors.
Perturbation Theory: Perturbation theory is a mathematical approach used in quantum mechanics to find an approximate solution to a problem that cannot be solved exactly. It involves starting with a known solution of a simpler system and adding small changes, or perturbations, to account for more complex interactions. This method is particularly useful for analyzing systems where the effects of interactions or external fields can be treated as small corrections to an idealized scenario.
Planck's Constant: Planck's constant is a fundamental physical constant that relates the energy of a photon to the frequency of its electromagnetic wave, represented as $$h = 6.62607015 \times 10^{-34} \text{ Js}$$. This constant is crucial in quantum mechanics and helps to bridge classical and quantum physics, influencing concepts such as quantization and the behavior of particles at the atomic level. Its significance extends into both time-dependent and time-independent formulations of wave functions, making it a foundational element in the study of quantum systems.
Probability Density: Probability density refers to the likelihood of finding a particle in a specific position or state within a quantum system. This concept is crucial in quantum mechanics, as it allows us to describe the behavior of particles using wave functions, which represent the quantum states. The probability density is obtained by squaring the absolute value of the wave function, providing insight into where particles are likely to be located in both time-dependent and time-independent scenarios.
Quantum Tunneling: Quantum tunneling is a quantum mechanical phenomenon where a particle has a probability of passing through a potential energy barrier, even when it does not have enough energy to overcome that barrier classically. This phenomenon highlights the unique behaviors of particles at the quantum level, connecting deeply to foundational principles of quantum mechanics, including the Schrödinger equations that describe how quantum states evolve and the Born-Oppenheimer approximation, which simplifies calculations in molecular systems by separating nuclear and electronic motion.
Reduced Mass: Reduced mass is a concept in physics that simplifies the analysis of two-body systems by allowing them to be treated as a single particle with an effective mass. This effective mass is given by the formula $$\mu = \frac{m_1 m_2}{m_1 + m_2}$$, where $m_1$ and $m_2$ are the masses of the two particles. By using reduced mass, complex problems in quantum mechanics, particularly those involving time-dependent and time-independent Schrödinger equations, become more manageable.
Separation of Variables: Separation of variables is a mathematical technique used to solve differential equations by breaking them down into simpler, manageable parts, allowing for each variable to be treated independently. This method is particularly useful in quantum mechanics, where it helps in deriving solutions for both time-dependent and time-independent equations, facilitating the understanding of eigenvalues and eigenfunctions, and playing a key role in approximations used in molecular systems.
Superposition Principle: The superposition principle states that a quantum system can exist in multiple states simultaneously, and the overall state of the system is represented as a linear combination of these individual states. This principle is fundamental in quantum mechanics and plays a crucial role in both the time-dependent and time-independent Schrödinger equations, allowing for complex behaviors and phenomena such as interference and entanglement.
Time-dependent Schrödinger equation: The time-dependent Schrödinger equation describes how the quantum state of a physical system changes over time. It is a fundamental equation in quantum mechanics, allowing for the prediction of a system's behavior by incorporating both spatial and temporal variables, which distinguishes it from the time-independent version that focuses solely on energy states in a stationary system.
Time-independent Schrödinger equation: The time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system at a specific energy level, independent of time. It relates the wave function of a particle to its energy and the potential energy in which it is moving, providing critical insights into the system's allowed energy states and spatial distribution. This equation is essential for understanding eigenvalues and eigenfunctions, which are key concepts in determining measurable quantities in quantum systems.
Uncertainty Principle: The uncertainty principle, formulated by Werner Heisenberg, states that it is impossible to simultaneously know both the exact position and exact momentum of a particle. This principle fundamentally challenges classical notions of determinism and introduces a limit to the precision with which certain pairs of physical properties, such as position and momentum, can be known.
Wavefunction: A wavefunction is a mathematical description of the quantum state of a system, representing the probability amplitude of finding a particle in a particular state or position. It is a fundamental concept in quantum mechanics, encapsulating all the information about a system's behavior and is crucial for understanding phenomena such as energy levels, molecular interactions, and electronic structures.
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