4.1 Time-dependent and time-independent Schrödinger equations
2 min read•august 9, 2024
The Schrödinger equation is the cornerstone of quantum mechanics, describing how particles behave at the atomic level. It comes in two flavors: time-dependent and time-independent. Each version helps us understand different aspects of quantum systems.
The time-dependent equation shows how quantum states change over time, useful for non-stationary situations. The time-independent version, simpler to solve, deals with and energy levels, key for understanding atomic structure and bonding.
Time-Dependent Schrödinger Equation
Fundamental Concepts and Components
Top images from around the web for Fundamental Concepts and Components
quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original
Is this image relevant?
Schrödinger Equation [The Physics Travel Guide] View original
Is this image relevant?
Equation de Schrödinger - Wikipedia, le encyclopedia libere View original
Is this image relevant?
quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original
Is this image relevant?
Schrödinger Equation [The Physics Travel Guide] View original
Is this image relevant?
1 of 3
Top images from around the web for Fundamental Concepts and Components
quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original
Is this image relevant?
Schrödinger Equation [The Physics Travel Guide] View original
Is this image relevant?
Equation de Schrödinger - Wikipedia, le encyclopedia libere View original
Is this image relevant?
quantum mechanics - How to "read" Schrodinger's equation? - Physics Stack Exchange View original
Is this image relevant?
Schrödinger Equation [The Physics Travel Guide] View original
Is this image relevant?
1 of 3
- describes quantum system evolution over time expressed as
- represents quantum state of system contains all measurable information about particle or system
- Wave function complex-valued function of position and time determines probability density of finding particle at specific location
- represents total energy of system includes kinetic and terms
- Hamiltonian operator for particle in potential field given by
Applications and Interpretations
- Time-dependent Schrödinger equation predicts behavior of quantum systems in non-stationary states (tunneling effect)
- Equation describes phenomena (double-slit experiment)
- Solutions to time-dependent Schrödinger equation reveal states (Schrödinger's cat thought experiment)
- Time evolution of wave function determines probability amplitudes for measurement outcomes
- Equation forms basis for understanding quantum dynamics in various fields (quantum chemistry, condensed matter physics)
Time-Independent Schrödinger Equation
Derivation and Properties
- derived from time-dependent equation when potential energy independent of time
- Equation takes form
- Stationary states represent quantum states with constant probability density over time
- corresponds to total energy of system in stationary state
- technique used to solve time-independent Schrödinger equation splits wave function into spatial and temporal parts
Solutions and Applications
- Time-independent Schrödinger equation solved for various potential energy functions (infinite square well, harmonic oscillator, hydrogen atom)
- Solutions yield energy levels and corresponding wave functions for bound states
- Equation predicts quantization of energy levels in atoms and molecules
- Time-independent solutions form basis for understanding atomic orbitals and chemical bonding
- Equation used to analyze through potential barriers (alpha decay, scanning tunneling microscopy)
Key Terms to Review (20)
Boundary Conditions: Boundary conditions are constraints applied to a physical system at its boundaries, which play a crucial role in determining the behavior of quantum systems. They specify how the wave function or other physical properties behave at the edges of a region where a particle may exist, and they help ensure that solutions to equations like the Schrödinger equation are physically meaningful and mathematically consistent. Properly set boundary conditions can lead to quantization of energy levels and influence phenomena such as tunneling.
Classical Limit: The classical limit refers to the conditions under which quantum mechanical systems behave like classical systems, typically as Planck's constant approaches zero or when quantum numbers become very large. This transition indicates that quantum effects become negligible, allowing the predictions of classical physics to emerge from quantum mechanics. Understanding the classical limit helps in bridging the gap between the two domains, showing how classical mechanics can be derived from quantum principles in certain situations.
Determinism vs. Indeterminism: Determinism is the philosophical concept that every event, including human actions, is determined by preceding events in accordance with the natural laws, suggesting a predictable universe. In contrast, indeterminism allows for the possibility that not all events are determined, introducing an element of randomness or unpredictability, particularly relevant in quantum mechanics where certain outcomes cannot be precisely predicted. This distinction is crucial when considering the implications of time-dependent and time-independent equations in quantum mechanics, as it raises questions about the nature of reality and the behavior of particles at a fundamental level.
Energy Eigenvalue: An energy eigenvalue is a specific value of energy associated with a quantum system that corresponds to a stable state described by a wave function. In the context of quantum mechanics, these eigenvalues are derived from the time-independent Schrödinger equation, which helps to determine the allowed energy levels of a quantum system, and they play a crucial role in understanding how systems evolve over time when considered alongside the time-dependent Schrödinger equation.
Energy Quantization: Energy quantization is the concept in quantum mechanics that energy can only take on discrete values rather than a continuous range. This principle is foundational in explaining the behavior of particles at the quantum level, where systems like electrons in atoms or particles in potential wells exhibit specific, quantized energy states. As a result, transitions between these energy states result in the absorption or emission of energy in fixed amounts, often represented by the equation $$E_n = rac{n^2 h^2}{8mL^2}$$ for particles in confined spaces.
Erwin Schrödinger: Erwin Schrödinger was an Austrian physicist who made foundational contributions to quantum mechanics, most notably through the development of the Schrödinger equation. His work established a wave-based perspective of particles, linking the concepts of eigenstates and eigenvalues to the behavior of quantum systems and helping to describe the nature of atomic and subatomic particles.
Hamiltonian Operator: The Hamiltonian operator is a fundamental operator in quantum mechanics that represents the total energy of a quantum system, encompassing both kinetic and potential energy. It plays a crucial role in the formulation of the Schrödinger equations, which describe how quantum states evolve over time and their associated wave functions. Understanding the Hamiltonian operator is essential for analyzing the behavior of quantum systems, such as particles in a potential field or systems like the quantum harmonic oscillator.
Max Planck: Max Planck was a German physicist who is best known for his role in the development of quantum theory, which revolutionized our understanding of atomic and subatomic processes. His introduction of the concept of quantized energy levels laid the groundwork for many fundamental principles in quantum mechanics, influencing various topics such as energy states in potentials and the behavior of oscillators.
Normalization: Normalization is the process of adjusting a wave function so that its total probability across all space equals one. This ensures that the probability of finding a particle within a defined region can be properly interpreted, linking the mathematical description of quantum states to physical reality. It is crucial for understanding concepts like eigenstates, probability distributions, and the behavior of systems such as the infinite square well potential.
Particle in a Box: A particle in a box is a fundamental quantum mechanics model that describes a particle constrained within a perfectly rigid and impenetrable potential well, often represented as an infinite square well. This model helps illustrate the behavior of particles at quantum levels, where the particle can only occupy certain discrete energy levels. The concept is crucial for understanding wave functions, probability distributions, and the quantization of energy states in quantum systems.
Potential Energy: Potential energy is the energy stored in an object due to its position or configuration. It plays a crucial role in various physical systems, determining how particles interact and behave under different forces. In quantum mechanics, potential energy influences wavefunctions and their probabilities, shaping the solutions to fundamental equations governing the dynamics of quantum systems.
Quantum Harmonic Oscillator: The quantum harmonic oscillator is a fundamental model in quantum mechanics that describes a particle subject to a restoring force proportional to its displacement from an equilibrium position. This concept plays a crucial role in understanding various physical systems, including molecular vibrations, and connects to the time-dependent and time-independent Schrödinger equations, energy levels derived from the Hamiltonian, WKB approximation, and the use of creation and annihilation operators.
Quantum Interference: Quantum interference is a phenomenon that occurs when two or more quantum states overlap, resulting in a redistribution of probabilities and the reinforcement or cancellation of certain outcomes. This fundamental aspect of quantum mechanics reveals how particles can behave like waves, leading to observable effects in experiments. It plays a crucial role in various quantum systems, illustrating the non-classical behavior of particles as they traverse multiple paths simultaneously.
Quantum superposition: Quantum superposition is a fundamental principle of quantum mechanics that states a quantum system can exist in multiple states or configurations simultaneously until it is measured. This idea is crucial for understanding phenomena like interference patterns and the behavior of particles at the quantum level.
Quantum tunneling: Quantum tunneling is a phenomenon where a particle can pass through a potential energy barrier that it classically should not be able to overcome. This occurs due to the wave-like nature of particles, allowing them to have a probability of being found on the other side of the barrier, despite not having sufficient energy to overcome it classically. The implications of quantum tunneling are vast, affecting everything from nuclear processes to advanced imaging technologies.
Separation of Variables: Separation of variables is a mathematical method used to solve partial differential equations by expressing a function as a product of functions, each depending on a single variable. This technique simplifies complex problems by allowing one to isolate the influence of each variable, making it particularly useful in solving the time-dependent and time-independent Schrödinger equations and in analyzing systems like the hydrogen atom, where spatial and temporal components can be treated independently.
Stationary States: Stationary states refer to quantum states of a system that have a definite energy and do not change in time, except for an overall phase factor. In these states, the probability distribution of a particle's position remains constant over time, which makes them particularly important when solving the time-independent Schrödinger equation. Understanding stationary states helps in analyzing the behavior of quantum systems, as they represent solutions that can describe bound systems and their energy levels.
Time-Dependent Schrödinger Equation: The time-dependent Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It provides a mathematical framework for understanding the behavior of particles at a quantum level, incorporating both their wave-like properties and their time evolution. This equation is crucial for analyzing dynamic systems and is typically expressed as $$i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r}, t) = \hat{H}\Psi(\mathbf{r}, t)$$, where $$\hat{H}$$ is the Hamiltonian operator representing the total energy of the system, $$\Psi(\mathbf{r}, t)$$ is the wave function, and $$\hbar$$ is the reduced Planck's constant.
Time-independent Schrödinger equation: The time-independent Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes in space, independent of time. This equation is crucial for analyzing stationary states, where the system's properties do not vary over time, making it particularly relevant for problems like potential wells and atomic structures, including hydrogen. It allows for the derivation of energy eigenvalues and corresponding wavefunctions, forming the basis for much of quantum theory.
Wave function: The wave function is a mathematical description of the quantum state of a system, encapsulating all the information about the system's properties and behaviors. It is essential in predicting the likelihood of finding a particle in a given position and time, connecting directly to various quantum phenomena such as energy levels, probabilities, and duality.